Tag: Physics

📚 IB Physics: Properties and Calculations of Gravitational Fields | IB物理：引力场的性质与计算

Gravitational fields are one of the fundamental concepts in IB Physics. They describe how mass interacts with mass across space, governing everything from an apple falling to Earth to the motion of galaxies. In this article, we will explore the key properties of gravitational fields, derive essential formulas, and discuss how to solve problems involving gravitational forces and potentials.

引力场是IB物理中的核心概念之一。它描述了质量与质量之间如何隔着空间相互作用，支配着从苹果落地到星系运动的种种现象。本文将探讨引力场的关键性质，推导重要公式，并讨论如何解决涉及引力与引力势的问题。

1. The Concept of a Gravitational Field | 引力场的概念

A gravitational field is a region of space in which a mass experiences a force due to the presence of another mass. The field is a vector field: at every point, it has a direction, normally toward the source mass, and a magnitude that depends on distance from that mass.

引力场是空间中一个质量因另一个质量的存在而受到力的区域。引力场是矢量场：在每一点上，它都有方向，通常指向源质量，并且大小取决于与该质量的距离。

We often represent gravitational fields with field lines. Around a spherical mass these lines point radially inward, becoming more spread out as distance increases. A wider spacing between field lines indicates a weaker field, while a denser spacing indicates a stronger field.

我们常用引力场线来描绘引力场。在球形质量周围，场线指向球心，且随距离增加而愈发分散。场线间距较大表示场较弱，间距较密则表示场较强。

2. Newton’s Law of Universal Gravitation | 牛顿万有引力定律

The magnitude of the gravitational force between two point masses m₁ and m₂ separated by a distance r is given by Newton’s law of universal gravitation. This law is one of the cornerstones of classical physics and applies to all masses, from subatomic particles to planets and stars.

两个质点 m₁ 和 m₂ 相距 r 时的万有引力大小由牛顿万有引力定律给出。该定律是经典物理学的基石之一，适用于从亚原子粒子到行星、恒星的一切质量。

F = G m₁ m₂ / r²

Here G is the gravitational constant, approximately equal to 6.674 × 10⁻¹¹ N m² kg⁻². The force is always attractive, acts along the line joining the two masses, and is independent of the surrounding medium. For spherical bodies with uniform density, the same equation applies when r is the distance between their centres.

其中 G 是万有引力常量，约等于 6.674 × 10⁻¹¹ N m² kg⁻²。该力总是吸引力，方向沿两质点的连线，且与周围介质无关。对于密度均匀的球体，当 r 为两球心之间的距离时，上述方程同样适用。

3. Gravitational Field Strength | 引力场强度

Gravitational field strength g at a point is defined as the gravitational force per unit mass experienced by a small test mass placed at that point. Mathematically this is written as g = F/m. Since force is a vector, gravitational field strength is also a vector.

引力场强度 g 定义为在该点放置的试探质量所受到的引力与试探质量之比，数学上写作 g = F/m。由于力是矢量，引力场强度也是矢量。

g = F/m = GM/r²

For a point mass M, substituting Newton’s law gives g = GM/r². The units are N kg⁻¹, which are equivalent to m s⁻². Near the surface of Earth, g ≈ 9.81 m s⁻² and is directed toward the centre of Earth. Notice that g depends on the source mass M and the distance r, but not on the mass of the test object.

对于质点 M，代入牛顿定律可得 g = GM/r²。其单位为 N kg⁻¹，等价于 m s⁻²。在地球表面附近，g ≈ 9.81 m s⁻²，方向指向地心。注意，g 取决于源质量 M 和距离 r，而与试探质量无关。

4.

Published by TutorHao | IB Physics Revision Series | aleveler.com

Find IB Physics Textbooks on eBay UK

New, used and second-hand copies of textbooks and revision guides are often much cheaper than retail — check current listings and prices before you buy.

Browse on eBay UK →

更多咨询请联系16621398022（同微信）

Share this:

• Email
• More

• LinkedIn
• X
• Facebook
• Tumblr
• Reddit
• Pinterest
• Telegram
• WhatsApp

Like this:

Like Loading…

📚 IB Physics: Nuclear Fusion & Stellar Evolution | IB物理：核聚变与恒星演化

Nuclear fusion is the process that powers the Sun and all other stars, converting mass into energy according to Einstein’s famous relation E = mc². Understanding fusion is essential for explaining how stars are born, how they shine for billions of years, and how they ultimately die—producing the heavy elements that make life possible.

核聚变是太阳和所有其他恒星的能源来源，它根据爱因斯坦的著名方程 E = mc² 将质量转化为能量。理解聚变对于解释恒星的诞生、它们在数十亿年间的发光过程，以及它们最终如何消亡——产生使生命成为可能的重元素——至关重要。

1. Binding Energy and Mass Defect | 结合能与质量亏损

The nucleus of an atom is held together by the strong nuclear force. When protons and neutrons combine to form a nucleus, the total mass of the nucleus is less than the sum of the masses of its individual nucleons. This difference is called the mass defect, Δm.

原子核由强核力将质子和中子束缚在一起。当质子和中子结合形成原子核时，原子核的总质量小于其各核子质量之和。这个差值称为质量亏损，记作 Δm。

The binding energy of a nucleus is the energy required to completely separate it into individual nucleons. It is calculated using the mass-energy equivalence:

原子核的结合能是将原子核完全拆分为独立核子所需的能量。它通过质能等价关系计算：

E_b = Δm × c²

where Δm is the mass defect and c is the speed of light (3.0 × 10⁸ m s⁻¹).

其中 Δm 是质量亏损，c 是光速（3.0 × 10⁸ m s⁻¹）。

For example, in a helium-4 nucleus, two protons and two neutrons have a combined mass of 4.0330 u, but the actual mass of the helium-4 nucleus is 4.0026 u. The mass defect of 0.0304 u corresponds to a binding energy of about 28.3 MeV.

例如，在氦-4 原子核中，两个质子和两个中子的总质量为 4.0330 u，但氦-4 原子核的实际质量为 4.0026 u。0.0304 u 的质量亏损对应约 28.3 MeV 的结合能。

2. Binding Energy per Nucleon Curve | 核子平均结合能曲线

When we plot the binding energy per nucleon against the mass number A, we obtain a curve that reveals which nuclei are most stable. The peak of this curve occurs around iron-56 (A ≈ 56), which has the highest binding energy per nucleon (about 8.8 MeV).

我们将核子平均结合能对质量数 A 作图，得到一条揭示哪些原子核最稳定的曲线。该曲线的峰值出现在铁-56 附近（A ≈ 56），其核子平均结合能最高（约 8.8 MeV）。

Key observations from the curve include:

从该曲线得出的关键观察包括：

• Nuclei with very low mass numbers (like hydrogen and helium) have relatively low binding energy per nucleon.
• 核素质量数很低（如氢和氦）的原子核，其核子平均结合能相对较低。
• Nuclei around iron-56 have the maximum binding energy per nucleon, making them the most stable.
• 铁-56 附近的原子核拥有最大的核子平均结合能，因此它们最稳定。
• Very heavy nuclei (like uranium-235) have lower binding energy per nucleon, making them prone to fission.
• 很重的原子核（如铀-235）的核子平均结合能较低，使它们易于发生裂变。

Two energy-releasing processes are evident from this curve: fusion of light nuclei and fission of heavy nuclei. In both cases, the products move towards the peak of the curve, releasing energy in the process.

从该曲线可以清楚地看出两种释放能量的过程：轻核的聚变和重核的裂变。在这两种过程中，产物都朝向曲线的峰值移动，同时释放能量。

3. Conditions Required for Nuclear Fusion | 核聚变所需的条件

For two nuclei to fuse, they must come within the range of the strong nuclear force (about 10⁻¹⁵ m). However, both nuclei are positively charged, so they experience a large electrostatic repulsion (Coulomb barrier) that must be overcome.

要让两个原子核融合，它们必须进入强核力的作用范围（约 10⁻¹⁵ m）。然而，两个原子核都带正电，因此它们会受到巨大的静电斥力（库仑势垒）的阻碍，必须被克服。

The conditions required for fusion are:

实现聚变所需的条件是：

• Extremely high temperature: In the core of the Sun, temperatures reach about 15 million Kelvin (1.5 × 10⁷ K), giving nuclei sufficient kinetic energy to overcome the Coulomb barrier.
• 极高的温度：太阳核心的温度达到约 1500 万开尔文（1.5 × 10⁷ K），赋予原子核足够的动能来克服库仑势垒。
• Extremely high pressure: The immense gravitational pressure in stellar cores compresses matter to very high densities, increasing the probability of collisions between nuclei.
• 极高的压强：恒星核心中巨大的引力压强将物质压缩到非常高的密度，增加了原子核之间碰撞的概率。
• Confinement time: The nuclei must be held together long enough for fusion reactions to occur at a sustained rate.
• 约束时间：原子核必须被约束足够长的时间，使聚变反应能够以持续速率发生。

At these temperatures, matter exists in the state of plasma, where electrons are stripped from their nuclei, forming a soup of charged particles.

在这些温度下，物质以等离子体状态存在，电子从原子核上剥离，形成带电粒子的”汤”。

4. Proton-Proton Chain | 质子-质子链反应

The Sun and other lower-mass stars (up to about 1.3 solar masses) primarily generate energy through the proton-proton chain. This is a series of fusion reactions that convert four hydrogen nuclei into one helium-4 nucleus.

太阳和其他较低质量的恒星（质量约为太阳质量的 1.3 倍以内）主要通过质子-质子链反应产生能量。这是一系列将四个氢核转化为一个氦-4 核的聚变反应。

The overall net reaction is:

总的净反应为：

4 ¹H → ⁴He + 2 e⁺ + 2 νₑ + energy (26.7 MeV)

The detailed steps of the proton-proton chain (PP I branch) are:

质子-质子链（PP I 分支）的详细步骤是：

Note that in Step 3, two ³He nuclei fuse, meaning that Steps 1 and 2 must occur twice before Step 3 can proceed. The positrons (e⁺) produced annihilate with electrons, releasing additional gamma-ray energy.

注意在第三步中，两个 ³He 核融合，意味着第一步和第二步必须各自发生两次，才能进行第三步。产生的正电子（e⁺）与电子湮灭，释放额外的伽马射线能量。

5. CNO Cycle | CNO 循环

For stars more massive than about 1.3 solar masses, the core temperature exceeds 17 million Kelvin, enabling a more efficient fusion pathway known as the CNO cycle. In this cycle, carbon, nitrogen, and oxygen nuclei act as catalysts to convert hydrogen into helium.

对于质量超过约 1.3 倍太阳质量的恒星，核心温度超过 1700 万开尔文，使得一种更高效的聚变途径成为可能，即CNO 循环。在这个循环中，碳、氮和氧原子核作为催化剂，将氢转化为氦。

The overall net result of the CNO cycle is identical to that of the proton-proton chain:

CNO 循环的总净结果与质子-质子链相同：

4 ¹H → ⁴He + 2 e⁺ + 2 νₑ + energy

However, the CNO cycle operates through a sequence of reactions involving ¹²C, ¹³N, ¹³C, ¹⁴N, ¹⁵N, and ¹⁵O isotopes, with ¹²C being regenerated at the end of the cycle.

然而，CNO 循环通过涉及 ¹²C、¹³N、¹³C、¹⁴N、¹⁵N 和 ¹⁵O 同位素的一系列反应运作，其中 ¹²C 在循环结束时被再生。

An important difference between the two processes is that the CNO cycle is much more temperature-sensitive than the proton-proton chain. The energy generation rate of the CNO cycle scales approximately as T¹⁸, whereas the proton-proton chain scales as T⁴.

这两个过程的一个重要区别是，CNO 循环对温度的敏感程度远高于质子-质子链。CNO 循环的能量产生率大约与 T¹⁸ 成正比，而质子-质子链的能量产生率与 T⁴ 成正比。

6. Hydrostatic Equilibrium | 流体静力学平衡

A star in its main-sequence phase is in a state of hydrostatic equilibrium. This means that two opposing forces are exactly balanced:

处于主序阶段的恒星处于流体静力学平衡状态。这意味着两个相反的力恰好平衡：

• Gravitational force: The inward pull of gravity, which compresses the stellar material toward the center.
• 引力：向内的引力作用，将恒星物质向中心压缩。
• Radiation pressure and gas pressure: The outward force caused by the enormous thermal energy released from fusion reactions and the pressure of the hot gas.
• 辐射压和气体压：由聚变反应释放的巨大热能和炽热气体的压强产生的向外作用力。

This balance can be expressed as:

这种平衡可以表示为：

Pressure gradient = -ρg

When the star is in perfect balance, it maintains a constant size and luminosity. If fusion reactions speed up (e.g., due to a slight temperature increase), the outward pressure increases, causing the star to expand. The expansion cools the core, slowing fusion, and the star settles back into equilibrium. This is the remarkable stellar thermostat that keeps stars stable for billions of years.

当恒星处于完美平衡时，它保持恒定的尺寸和光度。如果聚变反应加速（例如由于温度略微升高），向外的压强增大，导致恒星膨胀。膨胀使核心冷却，减慢聚变，恒星重新回到平衡状态。这就是了不起的恒星恒温器机制，使恒星稳定数十亿年。

7. Main Sequence Stars | 主序恒星

Stars spend about 90% of their lifetime on the main sequence, where they fuse hydrogen into helium in their cores. The exact position of a star on the main sequence depends on its mass, as shown by the Hertzsprung-Russell (H-R) diagram.

恒星大约将 90% 的生命时间花在主序阶段，在此期间它们在核心将氢聚变为氦。恒星在主序上的确切位置取决于其质量，如赫罗图（H-R 图）所示。

Key characteristics of main sequence stars include:

主序恒星的关键特征包括：

• Mass-luminosity relation: More massive stars are dramatically more luminous. The relationship is approximately L ∝ M³.⁵.
• 质光关系：质量更大的恒星光度也显著更高。近似关系为 L ∝ M³.⁵。
• Hydrogen burning: Fusion of hydrogen into helium occurs in the core, either via the proton-proton chain or the CNO cycle.
• 氢燃烧：氢在核心通过质子-质子链或 CNO 循环聚变为氦。
• Stable temperature: Surface temperatures range from about 3,000 K (red dwarfs) to over 40,000 K (blue giants).
• 稳定的温度：表面温度从约 3,000 K（红矮星）到超过 40,000 K（蓝巨星）不等。

A star’s main-sequence lifetime depends on the amount of fuel (mass) and the rate of consumption (luminosity). Since L ∝ M³.⁵, the lifetime t ∝ M/L ∝ M/M³.⁵ = 1/M².⁵. Therefore, massive stars live much shorter lives than low-mass stars.

恒星的主序寿命取决于燃料量（质量）和消耗速率（光度）。由于 L ∝ M³.⁵，寿命 t ∝ M/L ∝ M/M³.⁵ = 1/M².⁵。因此，大质量恒星的寿命比低质量恒星短得多。

8. Evolution of a Low-Mass Star | 低质量恒星的演化

When a low-mass star (up to about 2 solar masses) exhausts the hydrogen in its core, hydrogen fusion stops in the core but continues in a shell surrounding the core. The star expands and cools, becoming a red giant.

当低质量恒星（约 2 倍太阳质量以内）核心中的氢耗尽时，核心的氢聚变停止，但在核心周围的壳层中继续进行。恒星膨胀并冷却，成为红巨星。

The subsequent stages are:

随后的阶段是：

• Helium flash: When the core temperature reaches about 100 million Kelvin, helium begins fusing into carbon and oxygen via the triple-alpha process: 3 ⁴He → ¹²C.
• 氦闪：当核心温度达到约 1 亿开尔文时，氦开始通过三阿尔法过程聚变为碳和氧：3 ⁴He → ¹²C。
• Horizontal branch: The star’s core stabilizes as helium fuses, and the outer layers contract slightly.
• 水平分支：随着氦聚变的进行，恒星核心稳定，外层略微收缩。
• Asymptotic giant branch (AGB): After core helium is exhausted, the star expands again, becoming a second red giant. Helium and hydrogen both burn in shells around an inert carbon-oxygen core.
• 渐近巨星分支（AGB）：核心氦耗尽后，恒星再次膨胀，成为第二代红巨星。氦和氢都在惰性的碳-氧核心周围的壳层中燃烧。

During the AGB phase, the star loses a significant fraction of its mass through strong stellar winds, expelling its outer layers to form a planetary nebula. The exposed core collapses into a white dwarf—an extremely dense object supported by electron degeneracy pressure.

在渐近巨星分支阶段，恒星通过强烈的恒星风失去其大部分质量，将外层物质抛射形成行星状星云。裸露的核心坍缩形成白矮星——一种由电子简并压支持的极端致密天体。

9. Evolution of a Massive Star | 大质量恒星的演化

Stars with masses greater than about 8 solar masses follow a very different evolutionary path. These stars possess sufficient gravitational pressure to fuse successively heavier elements in their cores, creating an “onion-skin” layered structure.

质量超过约 8 倍太阳质量的恒星遵循一条截然不同的演化路径。这些恒星拥有足够的引力压强，可以在核心中依次聚变更重的元素，形成”洋葱皮”式分层结构。

The stages of nuclear fusion in a massive star are:

大质量恒星中核聚变的阶段是：

Iron is the end point of fusion because fusing iron into heavier elements requires energy input rather than releasing energy. This is because iron-56 has the maximum binding energy per nucleon, as discussed earlier.

铁是聚变的终点，因为将铁聚变为更重的元素需要输入能量而不是释放能量。这是因为如前所述，铁-56 具有最大的核子平均结合能。

10. Supernovae and Neutron Stars / Black Holes | 超新星与中子星/黑洞

When a massive star reaches the iron-burning stage, the core can no longer produce energy through fusion. Without outward pressure to balance gravity, the core collapses catastrophically in less than a second.

当大质量恒星达到铁燃烧阶段时，核心无法再通过聚变产生能量。失去了平衡引力的向外压强，核心在不到一秒的时间内灾难性地坍缩。

The collapse produces a supernova explosion, one of the most energetic events in the universe. Key aspects include:

这种坍缩产生超新星爆炸——宇宙中能量最巨大的事件之一。关键方面包括：

• The imploding core reaches densities comparable to that of an atomic nucleus, and the sudden halt of the collapse creates a shockwave that blows off the outer layers of the star.
• 坍缩的核心达到与原子核相当的密度，坍缩的突然停止产生冲击波，吹散恒星的外层。
• The temperature during a supernova can briefly reach billions of Kelvin, enabling the r-process (rapid neutron capture), which produces elements heavier than iron, including gold, silver, and uranium.
• 超新星期间的温度可以短暂达到数十亿开尔文，使得r-过程（快中子俘获）得以发生，产生比铁更重的元素，包括金、银和铀。
• The supernova outshines an entire galaxy for a short period and can briefly emit more energy than the star did in its entire lifetime.
• 超新星的亮度在短时间内超过整个星系，并且可以在短时间内释放比这颗恒星一生中发射的总能量还要多的能量。

The remnant left behind depends on the initial mass of the star:

留下的残余天体取决于恒星的初始质量：

A neutron star is an incredibly dense object with a radius of about 10–15 km but a mass comparable to that of the Sun. A teaspoon of neutron star material would weigh about 10 million tons on Earth.

中子星是一种极其致密的天体，半径约 10–15 km，但质量可与太阳相当。一茶匙中子星物质在地球上重约 1000 万吨。

If the core mass exceeds the Tolman–Oppenheimer–Volkoff limit (about 2–3 solar masses), not even neutron degeneracy pressure can support it, and the collapse continues to form a black hole.

如果核心质量超过 Tolman–Oppenheimer–Volkoff 极限（约 2–3 倍太阳质量），即使是中子简并压也无法支撑它，坍缩将继续进行，形成黑洞。

11. Nucleosynthesis: Origin of the Elements | 核合成：元素的起源

The elements we see around us—the oxygen we breathe, the carbon in our bodies, the iron in our blood—were all forged in the interiors of stars or during supernova explosions. This process is called stellar nucleosynthesis.

我们周围看到的元素——我们呼吸的氧、我们身体中的碳、我们血液中的铁——都是在恒星内部或超新星爆炸期间锻造而成。这个过程被称为恒星核合成。

The origins of elements by atomic number are summarized below:

按原子序数排列的元素来源总结如下：

• Hydrogen and helium: Produced in the Big Bang ~13.8 billion years ago.
• 氢和氦：约 138 亿年前在大爆炸中产生。
• Elements up to iron: Produced via nuclear fusion in stellar cores and shells.
• 直到铁的元素：通过恒星核心和壳层中的核聚变产生。
• Elements heavier than iron: Produced via the s-process (slow neutron capture) in AGB stars and the r-process in supernovae and neutron star mergers.
• 比铁更重的元素：通过渐近巨星分支恒星中的 s-过程（慢中子俘获）以及超新星和中子星合并中的 r-过程产生。

The famous phrase “we are made of star stuff” is literally true: the calcium in our bones and the iron in our hemoglobin were synthesized in ancient stars that exploded billions of years before the Solar System formed.

那句名言”我们是由星尘组成的”在字面上是真的：我们骨骼中的钙和血红蛋白中的铁，是在太阳系形成前数十亿年爆炸的古老恒星中合成的。

12. Fusion vs. Fission as Energy Sources | 聚变与裂变作为能源的比较

For IB Physics, it is important to compare nuclear fusion and fission as potential energy sources for human society:

对于 IB 物理，比较核聚变和核裂变作为人类社会潜在能源是非常重要的：

• Energy density: Fusion releases about 3–4 times more energy per kilogram of fuel than fission. Fusion of 1 kg of hydrogen releases roughly 6.4 × 10¹⁴ J, while fission of 1 kg of uranium-235 releases about 8.2 × 10¹³ J.
• 能量密度：聚变每千克燃料释放的能量约为裂变的 3–4 倍。1 kg 氢聚变释放约 6.4 × 10¹⁴ J，而 1 kg 铀

  Published by TutorHao | IB Physics Revision Series | aleveler.com

  Find IB Physics Textbooks on eBay UK

  New, used and second-hand copies of textbooks and revision guides are often much cheaper than retail — check current listings and prices before you buy.

  Browse on eBay UK →

  更多咨询请联系16621398022（同微信）

  Share this:

  • Email
  • More

  • LinkedIn
  • X
  • Facebook
  • Tumblr
  • Reddit
  • Pinterest
  • Telegram
  • WhatsApp

  Like this:

  Like Loading…

📚 IB Physics: A Guide to the Internal Assessment (IA) Investigation Process | IB物理：内部评估（IA）探究流程指南

The Internal Assessment (IA) is a crucial component of the IB Physics course, contributing 20% to your final grade. It is not just a lab report; it is a complete scientific investigation that demonstrates your ability to plan, execute, analyze, and evaluate a research question of your own choosing. This guide will walk you through every stage of the IA process, from selecting a topic to submitting your final report.

内部评估（IA）是IB物理课程中至关重要的一部分，占最终成绩的20%。它不仅仅是一份实验报告，而是一项完整的科学探究，旨在展示你独立规划、实施、分析和评估所选研究问题的能力。本指南将带你逐步了解IA的完整流程，从选题到提交最终报告。

1. Understanding the IA Criteria | 理解IA的评分标准

Before you begin, you must understand exactly how your work will be assessed. The IB Physics IA is marked against five criteria, each worth a specific number of marks. Knowing these criteria will shape every decision you make during the investigation.

在开始之前，你必须确切了解你的工作将如何被评分。IB物理IA依据五项标准进行评分，每项标准对应特定分值。了解这些标准将指导你在探究过程中做出的每一个决策。

• Personal Engagement (2 marks): Evidence that you have taken initiative, shown intellectual curiosity, and made the investigation your own. This is often shown through your choice of topic, your design decisions, and your independent thinking.
• 个人参与（2分）： 证明你展现了主动性、求知欲，并使这项探究成为你自己的作品。这通常通过你的选题、设计决策和独立思考来体现。
• Exploration (6 marks): The background research, the clarity of your research question, and the appropriateness of your method. You must justify your choices and demonstrate an understanding of the physics involved.
• 探索（6分）： 背景研究、研究问题的清晰度以及方法的适当性。你必须为自己的选择提供理由，并展示对相关物理原理的理解。
• Analysis (6 marks): How well you process your data, including uncertainty analysis, graphical representation, and the interpretation of your results in the context of your research question.
• 分析（6分）： 你处理数据的能力，包括不确定度分析、图形表示，以及结合研究问题对结果进行解释。
• Evaluation (6 marks): The quality of your conclusion, your critical evaluation of the method, and the extent to which you identify limitations and suggest realistic improvements.
• 评价（6分）： 结论的质量、对方法的批判性评价，以及你识别局限并提出切实可行改进建议的程度。
• Communication (4 marks): The clarity and logical organization of your report. This includes the use of appropriate terminology, labelled diagrams, and a consistent layout. It is not about word count; it is about effective presentation.
• 交流（4分）： 报告的清晰度和逻辑组织。这包括使用适当的术语、标注图表以及一致的排版布局。这不是关于字数，而是关于有效的呈现。

2. Choosing a Research Question | 选择研究问题

The research question is the heart of your IA. It must be focused, answerable through an experiment you can perform, and directly related to physics. A vague question like ‘how does gravity affect falling objects?’ is far too broad. A better question is ‘How does the mass of a spherical pendulum bob affect the period of a simple pendulum?’

研究问题是IA的核心。它必须是聚焦的、可通过你能进行的实验来回答的，并且与物理直接相关。像“重力如何影响下落物体？”这样模糊的问题太过宽泛。更好的问题是“球形摆锤的质量如何影响单摆的周期？”

Start by exploring a context that interests you. This could be from a physics topic you enjoyed, a real-world application, or a personal hobby. Once you have a general area, narrow it down to a specific, measurable variable. You will need one independent variable (which you change) and one dependent variable (which you measure), while controlling all other factors. The focus should be on the relationship between these two variables, which you can investigate quantitatively.

首先探索你感兴趣的情境。这可以是你喜欢的物理主题、现实世界应用或个人爱好。一旦确定了大方向，就将其缩小到一个具体的、可测量的变量。你需要一个自变量（你改变的变量）和一个因变量（你测量的变量），同时控制所有其他因素。重点应是研究这两个变量之间的定量关系。

3. Background Research and Theoretical Framework | 背景研究与理论框架

Your exploration must show that you understand the physics underpinning your research question. This is not a repetition of textbook theory; it is a targeted review that directly supports your choice of variables and your predictions. For example, for a pendulum investigation, you would derive or state the equation for the period T = 2π√(L/g) and explain how this leads to your prediction that the period is independent of mass.

你的探索部分必须展示你理解研究问题背后的物理原理。这不是对教科书理论的重复；而是直接支持你选择变量和预测的针对性综述。例如，对于单摆研究，你会推导或写出周期公式 T = 2π√(L/g)，并解释这如何引出你的预测：周期与质量无关。

Cite sources appropriately, but do not just list references. Use theory to justify your method, your choice of equipment, and the range of your measurements. A well-developed theoretical framework will help you design a better experiment and will also impress the examiner by showing that you know what you are doing and why.

适当引用来源，但不要只列出参考文献。要用理论来证明你的方法、设备选择以及测量范围是合理的。一个完善的理论框架将帮助你设计更好的实验，并通过向考官展示你清楚自己在做什么以及为什么这样做，从而给他们留下深刻印象。

4. Experimental Design and Methodology | 实验设计与方法

The exploration criterion also assesses the appropriateness and safety of your method. You must describe your experimental setup clearly, often with a labelled diagram. Your procedure should be detailed enough that another student could replicate it exactly. It must include the range and number of measurements, and most importantly, a thorough treatment of uncertainties.

探索标准还评估方法的适当性和安全性。你必须清晰地描述实验装置，通常附有标注图表。你的步骤应足够详细，让其他学生可以精确复现。步骤必须包括测量的范围和次数，最重要的是，对不确定度的全面处理。

Consider potential systematic errors. For example, reaction time when starting and stopping a timer, parallax error when reading a scale, and heat loss in a calorimeter experiment. Also, think about the range of your independent variable. Too narrow a range will make it difficult to identify a trend, while too wide a range may lead to unrealistic conditions. You should aim to take at least ten different values of the independent variable and repeat the measurement at least five times at each value to obtain a reliable mean and standard deviation.

考虑潜在的系统误差。例如，启动和停止计时器时的反应时间、读数时产生的视差误差，以及量热计实验中的热量损失。还要考虑自变量的范围。范围太窄将难以识别趋势，而范围太宽可能导致条件不现实。你应至少在十个不同的自变量值处进行测量，每个值处重复测量至少五次，以获得可靠的均值和标准差。

5. Data Collection and Uncertainty Analysis | 数据收集与不确定度分析

This is where your careful planning pays off. Record all raw data in a clear table, with units and absolute uncertainties. Every measurement has an uncertainty, whether from the instrument precision or from readout noise. For repeated measurements, calculate the mean and the standard deviation, which gives you a statistical measure of the random uncertainty.

这是你周密计划的回报时刻。将所有原始数据清晰记录在表格中，包含单位和绝对不确定度。每个测量都有不确定度，无论来自仪器精度还是读数波动。对于重复测量，计算均值和标准差，这为你提供了随机不确定度的统计量度。

When combining uncertainties, use appropriate rules: for addition and subtraction, add absolute uncertainties; for multiplication and division, add percentage uncertainties. When a quantity is raised to a power, multiply the percentage uncertainty by the power. For example, if V = r³, then the percentage uncertainty in V is three times the percentage uncertainty in r.

在合并不确定度时，使用适当的规则：对于加减法，绝对不确定度相加；对于乘除法，百分比不确定度相加。当某个量被取幂时，百分比不确定度乘以幂次。例如，如果 V = r³，那么 V 的百分比不确定度是 r 的百分比不确定度的三倍。

Use the following rules for uncertainty propagation:

对于不确定度传播，使用以下规则：

Addition/Subtraction: ΔZ = ΔA + ΔB | 加减法：ΔZ = ΔA + ΔB

Multiplication/Division: (ΔZ/Z) = (ΔA/A) + (ΔB/B) | 乘除法：(ΔZ/Z) = (ΔA/A) + (ΔB/B)

6. Data Processing and Graphical Presentation | 数据处理与图形呈现

With your raw data in hand, you must process it to find the relationship between your variables. This often involves linearizing the data. For example, if you are investigating how the period of a pendulum depends on its length, you would plot T² against L, which should give a straight line through the origin if the relationship is T = 2π√(L/g). Use the gradient of the line to extract a physical quantity, such as g, and compare it with the accepted value.

手中有原始数据后，你必须对其进行处理，以找到变量之间的关系。这通常涉及数据的线性化。例如，如果你在探究单摆周期如何依赖摆长，你应将 T² 对 L 作图，如果关系为 T = 2π√(L/g)，则应得到一条过原点的直线。用直线的斜率提取物理量，如重力加速度 g，并与公认值进行比较。

Graphs must be computer-drawn, with correct labels and units. Include error bars on your data points. When drawing a best-fit line, use a maximum-slope line and a minimum-slope line to find the uncertainty in the gradient. This is a powerful way to assess the reliability of your results. Remember to calculate the uncertainties in all processed data, not just the raw data. If you calculate T², you must also calculate the uncertainty in T².

图表必须用计算机绘制，标注正确并注明单位。在数据点上添加误差棒。在绘制最佳拟合线时，使用最大斜率线和最小斜率线来找到梯度的不确定度。这是评估结果可靠性的有力方法。记得计算所有处理数据的不确定度，而不仅仅是原始数据。如果你计算了 T²，你也必须计算 T² 的不确定度。

7. Analyzing Results and Drawing Conclusions | 分析结果并得出结论

The analysis section must go beyond simply stating your results. You must interpret them in the context of your research question. Does your data support your theoretical prediction? If you have plotted a linearized graph, discuss the significance of the gradient and the intercept. Calculate the percentage error between your experimental value and the accepted value, and consider whether your result is within the uncertainty range.

分析部分必须超越简单陈述结果。你必须结合研究问题对其进行解释。你的数据是否支持理论预测？如果你绘制了线性化图表，讨论斜率和截距的意义。计算实验值与公认值之间的百分比误差，并考虑你的结果是否在不确定度范围内。

Your conclusion should directly answer your research question. State whether the expected relationship was verified, and to what degree. A simple statement like ‘the results match theory’ is not enough. You must refer to specific quantitative findings, such as the gradient value with its uncertainty, and discuss the strength of the evidence. Do not overstate your conclusion; use qualifying phrases like ‘the data suggests’ or ‘within the limits of experimental uncertainty’.

你的结论应直接回答研究问题。说明预期关系是否得到验证，以及验证到什么程度。像“结果与理论相符”这样简单的陈述是不够的。你必须引用具体的定量结果，如带有不确定度的梯度值，并讨论证据的力度。不要夸大结论；使用“数据表明”或“在实验不确定度范围内”等限定性措辞。

8. Evaluation and Improvements | 评价与改进

The evaluation criterion demands a critical analysis of your method and a well-reasoned set of improvements. Identify the main sources of random and systematic error in your experiment. For each limitation, state its effect on your results. For example, air resistance in a pendulum experiment would cause the period to be slightly longer than theoretical, and this effect would be larger for higher speeds.

评价标准要求对方法进行批判性分析，并提出合理改进建议。识别实验中随机和系统误差的主要来源。对于每个局限，说明其对结果的影响。例如，单摆实验中的空气阻力会导致周期略长于理论值，且速度越高，这种影响越大。

For every limitation, suggest a specific, realistic improvement. Instead of saying ‘use better equipment’, be precise: ‘use a light gate to start and stop the timer automatically, eliminating reaction time’ or ‘perform the experiment in a vacuum chamber to reduce air resistance’. Improvements can also include extending the range of data, increasing the number of repetitions, or using a different data analysis technique.

对于每个局限，提出具体、可行的改进。不要说“使用更好的设备”，而要具体：“使用光电门自动启动和停止计时器，消除反应时间”或“在真空室进行实验以减少空气阻力”。改进还可以包括扩大数据范围、增加重复次数或使用不同的数据分析技术。

9. Presentation and Communication | 呈现与交流

Your final report must be clear, concise, and logically structured. Use a standard format: title page, introduction/research question, background theory, methodology, results (data tables and graphs), analysis, conclusion, and evaluation. Use subheadings to guide the reader. Label all figures and tables with numbers and titles. Ensure that every term is defined and every symbol is explained.

你的最终报告必须清晰、简洁、结构合理。使用标准格式：标题页、引言/研究问题、背景理论、方法、结果（数据表格和图表）、分析、结论和评价。使用小标题引导读者。为所有图表和表格加上编号和标题。确保每个术语都被定义，每个符号都被解释。

Do not include raw data that is not relevant to your analysis. You can put large tables of data in an appendix, but the main body must focus on processed data and analysis. Use proper significant figures, and be consistent with units. The communication criterion rewards clarity, so avoid unnecessary jargon and long-winded explanations. Every sentence should serve a purpose.

不要包含与分析无关的原始数据。你可以将大型数据表放在附录中，但正文必须侧重于处理后的数据和分析。使用适当的有效数字，并保持单位一致。交流标准奖励清晰性，因此避免不必要的术语和冗长解释。每个句子都应有其作用。

10. Final Checklist and Submission | 最终检查清单与提交

Before you submit, go through this checklist to ensure you have not missed anything critical. This final review can significantly improve your mark.

在提交之前，请逐项检查此清单，以确保你没有遗漏任何关键内容。这最后的审查可以显著提高你的分数。

• Research question: Is it focused and answerable?
• 研究问题： 是否聚焦且可回答？
• Personal engagement: Have you shown independent thinking and curiosity?
• 个人参与： 你是否展示了独立思考和求知欲？
• Exploration: Is your background research relevant and properly cited?
• 探索： 你的背景研究是否相关且引用得当？
• Method: Could another student replicate your experiment?
• 方法： 其他学生能否复现你的实验？
• Uncertainties: Have you included uncertainties at every stage?
• 不确定度： 你是否在每一阶段都包含了不确定度？
• Graphs: Are they complete with labels, units, and error bars?
• 图表： 是否包含标签、单位和误差棒？
• Conclusion: Does it answer the research question?
• 结论： 是否回答了研究问题？
• Evaluation: Are the limitations and improvements specific?
• 评价： 局限和改进是否具体？

This process may seem daunting, but it is an opportunity to demonstrate your skills as a scientist. A well-planned, well-executed IA is within your reach if you follow this structured approach. Good luck with your investigation!

这个过程可能看起来令人望而生畏，但这是一个展示你科学素养的机会。如果你遵循这种结构化的方法，你完全可以完成一份规划周密、执行良好的IA。祝你的探究顺利！

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 IB Physics: The Laws and Applications of Radioactive Decay | IB物理：放射性衰变的规律与应用

Radioactive decay is one of the most important topics in IB Physics because it connects quantum mechanics, nuclear structure, and practical technology. The behaviour of unstable nuclei follows a beautifully simple exponential law, yet the process itself is fundamentally random. This article introduces the key equations, decay modes, half-life calculations, and the real-world uses that you need for revision.

放射性衰变是IB物理中最重要的主题之一，因为它将量子力学、核结构和实际应用连接在一起。不稳定原子核的行为遵循一条简洁的指数规律，但这一过程本身在本质上是随机的。本文将介绍关键方程、衰变模式、半衰期计算以及备考所需的实际应用。

1. The Random and Spontaneous Nature of Decay | 衰变的随机性与自发性

Radioactive decay is a spontaneous process: an unstable nucleus emits radiation without any external trigger, because it is trying to reach a more stable energy state. It is also a random process: it is impossible to predict exactly which nucleus will decay next, or when it will decay.

放射性衰变是一个自发过程：不稳定的原子核无需外部触发就会发射辐射，因为它试图达到更稳定的能量状态。它同时也是一个随机过程：无法准确预测下一个会衰变的是哪一个原子核，也无法预测它何时衰变。

Although individual decays are unpredictable, the behaviour of a very large number of nuclei is statistically regular. This allows us to define a decay constant λ, which is the probability that a single nucleus will decay per unit time.

虽然单个衰变不可预测，但大量原子核的行为在统计上是规则的。因此我们可以定义衰变常数 λ，即单个原子核在单位时间内发生衰变的概率。

dN/dt = −λN

dN/dt = −λN

Here N is the number of undecayed nuclei remaining, and the negative sign shows that N decreases with time. The minus sign must not be forgotten when you write the differential form.

其中 N 是尚未衰变的原子核数，负号表示 N 随时间减少。在写微分形式时，一定不要遗漏负号。

2. Alpha, Beta and Gamma Emissions | α、β、γ 三种辐射

Alpha (α) radiation is a stream of helium-4 nuclei, written as ⁴₂He. When a nucleus emits an alpha particle, its mass number decreases by 4 and its atomic number decreases by 2.

α 辐射是氦-4原子核流，写作 ⁴₂He。当原子核发射一个α粒子时，其质量数减少4，原子序数减少2。

²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He

²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He

Beta-minus (β⁻) decay involves a neutron changing into a proton, an electron and an antineutrino. The mass number stays the same, but the atomic number increases by 1. For example, carbon-14 decays to nitrogen-14.

β⁻ 衰变涉及一个中子转变为质子，同时释放一个电子和一个反中微子。质量数不变，原子序数增加1。例如，碳-14衰变为氮-14。

¹⁴₆C → ¹⁴₇N + ⁰₋₁e + ν̄

¹⁴₆C → ¹⁴₇N + ⁰₋₁e + ν̄

Gamma (γ) radiation is high-energy electromagnetic radiation. It usually accompanies alpha or beta decay when the daughter nucleus is left in an excited state. Gamma emission changes neither the mass number nor the atomic number.

γ 辐射是高能电磁波。当子核处于激发态时，γ辐射通常伴随α或β衰变出现。γ发射既不改变质量数，也不改变原子序数。

• Alpha particles are highly ionising but weakly penetrating. They can be stopped by a sheet of paper or a few centimetres of air.

  α粒子电离能力很强，但穿透能力很弱，一张纸或几厘米空气就能阻挡。

• Beta particles are less ionising than alpha particles but more penetrating. A few millimetres of aluminium are usually enough to stop them.

  β粒子的电离能力比α粒子弱，但穿透能力更强，通常几毫米厚的铝就能阻挡。

• Gamma rays are weakly ionising but highly penetrating. Dense materials such as lead or several centimetres of concrete are needed to reduce their intensity.

  γ射线电离能力弱，但穿透能力很强，需要铅等致密材料或数厘米厚的混凝土来减弱其强度。

3. The Exponential Law of Decay | 指数衰变定律

Solving the differential equation dN/dt = −λN gives the exponential decay law for the number of undecayed nuclei:

求解微分方程 dN/dt = −λN 可得到未衰变原子核数的指数衰变定律：

N = N₀ e^(−λt)

N = N₀ e^(−λt)

Here N₀ is the initial number of nuclei, λ is the decay constant, and t is the elapsed time. The activity A is the number of decays per second, measured in becquerel (Bq), where 1 Bq = 1 s⁻¹. Since A = λN, the activity also decays exponentially:

其中 N₀ 是初始原子核数，λ 是衰变常数，t 是经过的时间。活度 A 是每秒发生的衰变次数，单位为贝克勒尔（Bq），1 Bq = 1 s⁻¹。由于 A = λN，活度也按指数规律衰变：

A = A₀ e^(−λt)

A = A₀ e^(−λt)

An exponential curve approaches zero but never actually reaches zero. In practice, after enough half-lives the remaining amount becomes so small that it is undetectable.

指数曲线会趋近于零，但永远不会真正等于零。实际上，经过足够多的半衰期后，剩余量会小到无法探测。

4. Half-Life and Decay Constant | 半衰期与衰变常数

The half-life T½ is the time required for half of the original nuclei to decay, or equivalently, the time for the activity to fall to half of its initial value. The half-life is related to the decay constant by:

半衰期 T½ 是初始原子核数的一半发生衰变所需的时间，也等于活度降至初始值一半所需的时间。半衰期与衰变常数的关系为：

T½ = ln 2 / λ ≈ 0.693 / λ

T½ = ln 2 / λ ≈ 0.693 / λ

For example, if a sample has a half-life of 5.0 years, its decay constant is λ = 0.693 / 5.0 = 0.139 yr⁻¹. After 15 years, which is three half-lives, only one-eighth of the original nuclei remain.

例如，如果某样品的半衰期为5.0年，则其衰变常数为 λ = 0.693 / 5.0 = 0.139 yr⁻¹。15年后，即3个半衰期后，只剩下原来原子核数的八分之一。

A useful equivalent form is N = N₀(1/2)ⁿ, where n = t/T½ is the number of half-lives that have elapsed. This is often quicker than using the exponential form in simple problems.

一个有用的等价形式是 N = N₀(1/2)ⁿ，其中 n = t/T½ 是经过的半衰期个数。在简单问题中，这通常比使用指数形式更快。

5. Measuring Half-Life | 测量半衰期

For a short-half-life source, place a sample near a Geiger-Müller tube and record the count rate at regular time intervals. Before taking measurements, always measure and subtract the background radiation count.

对于短半衰期放射源，可将样品放在盖革-米勒管附近，每隔一定时间记录计数率。在测量之前，务必先测量本底辐射并扣除。

Because a detector does not catch every decay, the observed count rate is usually less than the true activity. However, as long as the detection efficiency is constant, the count rate is proportional to activity and follows the same exponential decay.

由于探测器无法捕捉到每一次衰变，观察到的计数率通常小于真实活度。但只要探测效率恒定，计数率就与活度成正比，并遵循相同的指数衰减规律。

For a long half-life, the decay over a short observation period is too small to measure directly. Instead, you can measure the current activity A and estimate the number N of nuclei from the mass of the sample, then use λ = A/N.

对于长半衰期样品，在较短观测时间内衰变量太小，无法直接测量。此时可以测量当前活度 A，并根据样品质量估算原子核数 N，然后利用 λ = A/N 计算。

Another graphical method is to plot ln A against t. The result is a straight line with slope −λ, so the decay constant can be found from the gradient.

另一种作图方法是将 ln A 对 t 作图，得到一条斜率为 −λ 的直线，从而求出衰变常数。

6. Decay Chains and Radioactive Equilibrium | 衰变链与放射性平衡

Many radioactive nuclei do not decay directly to a stable isotope. They form a decay chain, passing through a series of daughter nuclei until a stable end product is reached. The well-known uranium-238 series ends with stable lead-206.

许多放射性原子核并非直接衰变为稳定同位素，而是形成衰变链，经过一系列子核直到达到稳定的最终产物。著名的铀-238衰变链最终是稳定的铅-206。

In a decay chain, if the parent has a very long half-life, the activity of some shorter-lived daughters can become almost equal to the parent activity over time. This condition is called secular equilibrium, and it is important in mineral dating and environmental monitoring.

在衰变链中，如果母核半衰期很长，经过足够时间，某些短寿命子核的活度几乎等于母核活度。这种状态称为长期平衡，它在矿物定年和环境监测中非常重要。

For IB problems, you usually only consider a single decay step. But understanding chains explains why old uranium ores contain many different radioactive isotopes, not just uranium.

在IB题目中，通常只考虑单步衰变。但理解衰变链可以解释为什么古老的铀矿石中含有多种不同的放射性同位素，而不只是铀。

7. Radiocarbon Dating | 放射性碳定年法

Carbon-14 is continuously produced in the upper atmosphere by cosmic-ray neutrons reacting with nitrogen-14. Living organisms absorb carbon-14 as carbon dioxide through photosynthesis and the food chain, so the ratio of carbon-14 to carbon-12 in a living organism stays roughly constant.

碳-14由宇宙射线中的中子与大气上层氮-14反应不断产生。生物体通过光合作用和食物链以二氧化碳形式吸收碳-14，因此活生物体内碳-14与碳-12的比值大致保持不变。

When an organism dies, it stops exchanging carbon with the environment. The carbon-14 it contains decays with a half-life of about 5730 years, so its activity decreases. By comparing the remaining carbon-14 activity with that in a living sample, the time since death can be found:

当生物死亡后，它不再与外界交换碳。其体内的碳-14以约5730年的半衰期衰变，因此活度降低。通过比较剩余碳-14活度与活体样品中的活度，即可算出死亡至今的时间：

t = (1/λ) ln(A₀/A)

t = (1/λ) ln(A₀/A)

For older objects, carbon-14 dating is not practical because the remaining activity becomes too small. Instead, methods such as potassium-argon dating are used for geological samples, because potassium-40 has a half-life of about 1.25 billion years.

对于更古老的物体，碳-14定年法不再适用，因为剩余活度太小。此时可采用钾-氩定年法等地学方法，因为钾-40的半衰期约为12.5亿年。

8. Medical Applications | 医学应用

Radioactive tracers are used to diagnose medical conditions. A small amount of a radioactive isotope is injected into the body, and its progress is followed using a gamma camera. The isotope must have a short half-life so that the patient is not exposed to radiation for a long time.

放射性示踪剂用于医学诊断。将少量放射性同位素注入体内，然后用γ相机追踪其运动过程。同位素必须具有短半衰期，以免患者长时间受到辐射。

• Technetium-99m has a half-life of about 6 hours and emits gamma rays ideal for imaging. It is widely used for bone, heart and brain scans.

  锝-99m的半衰期约为6小时，发射适用于成像的γ射线，广泛用于骨骼、心脏和脑部扫描。

• Iodine-131 is absorbed by the thyroid gland. It can be used to diagnose and treat thyroid conditions such as hyperthyroidism.

  碘-131会被甲状腺吸收，可用于诊断和治疗甲状腺功能亢进等疾病。

• In radiotherapy, focused beams of gamma radiation from sources such as cobalt-60 are used to destroy cancerous tissue. Rapidly dividing cells are especially sensitive to radiation.

  在放射治疗中，来自钴-60等放射源的γ射线束被用来摧毁癌变组织。快速分裂的细胞对辐射尤其敏感。

9. Industrial and Archaeological Applications | 工业与考古应用

Smoke detectors contain a small americium-241 source that emits alpha particles. The alpha particles ionise the air in a detection chamber, creating a small electric current. If smoke enters the chamber, it absorbs the alpha particles and reduces the current, triggering the alarm.

烟雾报警器中含有少量镅-241α放射源。α粒子使探测室中的空气电离，产生微弱电流。当烟雾进入探测室时，会吸收α粒子并减弱电流，从而触发警报。

Thickness gauges use the absorption of beta or gamma radiation to measure the thickness of paper, plastic or metal sheets during production. As the material becomes thicker, more radiation is absorbed, allowing a detector to send a feedback signal to adjust the rollers.

厚度计利用β或γ辐射的吸收情况来测量纸张、塑料或金属板材在生产过程中的厚度。材料越厚，吸收的辐射越多，探测器可向轧辊发送反馈信号进行调节。

In archaeology and geology, the same exponential decay law is used to date samples. Radiocarbon dating is ideal for once-living materials up to about 50,000 years old, while uranium-lead and potassium-argon methods are used for much older rocks.

在考古学和地质学中，相同的指数衰变规律被用于样品定年。碳-14定年适用于约5万年以内的生物材料，而铀-铅法和钾-氩法适用于更古老的岩石。

10. Radiation Safety and Biological Effects | 辐射安全与生物效应

Ionising radiation can remove electrons from atoms and damage DNA. High doses can kill cells, while lower doses increase the long-term risk of cancer. Because alpha particles deposit their energy over a very short distance, they are especially damaging if ingested or inhaled.

电离辐射能使原子失去电子并损伤DNA。高剂量可杀死细胞，低剂量则增加长期患癌风险。由于α粒子在很短距离内释放能量，如果被摄入或吸入，其危害尤其大。

The absorbed dose D is defined as the energy absorbed per unit mass:

吸收剂量 D 的定义是单位质量吸收的能量：

D = E / m

D = E / m

The unit of absorbed dose is the gray (Gy), where 1 Gy = 1 J kg⁻¹. However, different types of radiation have different biological effects, so physicists also define the equivalent dose H = wR × D, measured in sieverts (Sv). For gamma and beta radiation, wR = 1; for alpha radiation, wR = 20.

吸收剂量的单位是戈瑞（Gy），1 Gy = 1 J kg⁻¹。但不同辐射的生物效应不同，因此物理学家还定义等效剂量 H = wR × D，单位为希沃特（Sv）。对于γ和β辐射，wR = 1；对于α辐射，wR = 20。

Practical safety measures include minimising time near a source, maximising distance, using appropriate shielding, and handling sources with tongs behind lead screens. Monitoring devices such as film badges are used by workers to track accumulated exposure.

实际安全措施包括缩短在放射源附近的时间、尽量加大距离、使用适当屏蔽，并使用长柄工具和铅屏操作。工作人员还会佩戴胶片剂量计等监测设备来记录累积照射量。

11. Common IB Exam Pitfalls and Tips | 常见IB考试误区与提示

• Do not say that a particular nucleus ‘will decay after one half-life’. Half-life is a statistical concept; it applies to a large population of nuclei, not to a single nucleus.

  不要说某个特定原子核“会在一个半衰期后衰变”。半衰期是统计概念，适用于大量原子核，而不是单个原子核。

• Remember that λ is the probability of decay per nucleus per unit time. It is not the same as the activity, unless N = 1.

  记住 λ 是每个原子核在单位时间内的衰变概率。除非 N = 1，否则它不等同于活度。

Published by TutorHao | IB Physics Revision Series | aleveler.com

Find IB Physics Textbooks on eBay UK

New, used and second-hand copies of textbooks and revision guides are often much cheaper than retail — check current listings and prices before you buy.

Browse on eBay UK →

更多咨询请联系16621398022（同微信）

Share this:

• Email
• More

• LinkedIn
• X
• Facebook
• Tumblr
• Reddit
• Pinterest
• Telegram
• WhatsApp

Like this:

Like Loading…

📚 IB Physics: Core Concepts of Quantum Physics | IB物理：量子物理核心概念

Quantum physics is arguably the most revolutionary and counterintuitive branch of physics. For IB Physics students, mastering the core concepts of quantum theory is not only essential for examination success but also for understanding the modern technological world, from semiconductors to medical imaging. This article systematically presents the fundamental ideas that form the backbone of the IB Quantum and Nuclear Physics topic.

量子物理可以说是物理学中最具革命性、最反直觉的分支。对于IB物理学生而言，掌握量子理论的核心概念不仅是考试成功的关键，更是理解现代科技世界——从半导体到医学成像——的基础。本文系统性地阐述构成IB量子与核物理主题主干的那些基本思想。

1. The Photon Model of Light | 光的光子模型

Quantum physics begins with a radical departure from classical wave theory. In 1905, Albert Einstein proposed that light is not a continuous wave but consists of discrete packets of energy called photons. Each photon carries an energy directly proportional to its frequency, expressed as E = hf, where h is Planck’s constant (6.63 × 10⁻³⁴ J·s) and f is the frequency of the electromagnetic radiation.

量子物理始于对经典波动理论的彻底背离。1905年，阿尔伯特·爱因斯坦提出光并非连续波，而是由称为光子的离散能量包组成。每个光子携带的能量与其频率成正比，表示为 E = hf，其中 h 是普朗克常数（6.63 × 10⁻³⁴ J·s），f 是电磁辐射的频率。

This model explains phenomena that classical wave optics could not, such as the photoelectric effect. It establishes that energy exchange between light and matter occurs in discrete quantum jumps, not continuously. A beam of monochromatic light is thus a stream of identical photons, each with the same energy.

该模型解释了经典波动光学无法解释的现象，如光电效应。它确立了光与物质之间的能量交换是以离散的量子跃迁方式发生的，而非连续的。因此，一束单色光就是一系列相同的光子流，每个光子具有相同的能量。

2. The Photoelectric Effect | 光电效应

The photoelectric effect is the emission of electrons from a metal surface when illuminated by electromagnetic radiation above a certain threshold frequency. This phenomenon could not be explained by classical wave theory, which predicted that any frequency of light would eventually eject electrons if the intensity were sufficient. In reality, no electrons are emitted below the threshold frequency, regardless of intensity.

光电效应是当金属表面受到高于某一阈值频率的电磁辐射照射时，电子从金属表面发射的现象。这一现象无法用经典波动理论解释，后者预言只要有足够的强度，任何频率的光最终都能打出电子。而事实上，低于阈值频率的光无论强度多大都不会发射电子。

Einstein’s explanation treated light as photons: one photon interacts with one electron. The work function W₀ is the minimum energy needed to liberate an electron from the surface. The maximum kinetic energy of emitted electrons follows the equation:

爱因斯坦的解释将光视为光子：一个光子与一个电子相互作用。逸出功 W₀ 是将电子从表面释放所需的最小能量。发射电子的最大动能满足方程：

KE_max = hf − W₀

This equation is central to IB examinations. Students must be able to use it to calculate threshold frequency, investigate the gradient of a KE_max versus frequency graph, and explain why intensity affects only the number of electrons, not their maximum kinetic energy.

该方程是IB考试的核心。学生必须会用其计算阈值频率，分析 KE_max 对频率关系图的斜率，并解释为什么光的强度只影响发射电子的数量，而不影响其最大动能。

3. de Broglie Wavelength and Matter Waves | 德布罗意波长与物质波

If light, previously considered a wave, exhibits particle-like properties, then perhaps matter, previously considered particle-like, exhibits wave-like properties. In 1924, Louis de Broglie proposed this symmetrical hypothesis: any particle with momentum p has an associated wavelength given by:

如果说光——此前被视为波——表现出粒子性质，那么或许物质——此前被视为粒子——会表现出波动性质。1924年，路易·德布罗意提出了这一对称性假说：任何具有动量 p 的粒子都伴生一个波长，由下式给出：

λ = h ⁄ p = h ⁄ (mv)

Electron diffraction experiments confirmed this prediction: electrons accelerated through a potential difference produce interference patterns analogous to X-rays or light waves. The de Broglie wavelength of macroscopic objects is so small that their wave nature is unobservable; for example, a cricket ball of mass 0.15 kg moving at 30 m/s would have a wavelength of approximately 1.5 × 10⁻³⁴ m.

电子衍射实验证实了这一预言：加速电子穿过电势差后会产生与X射线或光波类似的干涉图样。宏观物体的德布罗意波长极小，其波动性无法被观察到；例如，一个质量为0.15 kg、以30 m/s运动的板球，其波长约为1.5 × 10⁻³⁴ m。

In the IB syllabus, students must calculate the de Broglie wavelength of electrons accelerated through a known potential difference, using the relationship E = eV = ½mv² to find the speed before computing wavelength.

在IB教学大纲中，学生必须会计算加速穿过已知电势差的电子的德布罗意波长，先利用关系式 E = eV = ½mv² 求出速度，再进行波长计算。

4. Wave-Particle Duality | 波粒二象性

Wave-particle duality is the principle that all quantum entities exhibit both wave-like and particle-like properties, depending on the experimental context. Light shows interference and diffraction (wave behaviour) in double-slit experiments, yet demonstrates discrete energy transfer (particle behaviour) in the photoelectric effect. Electrons, conversely, behave as particles in deflection experiments but exhibit interference patterns when passed through crystalline lattices.

波粒二象性是所有量子实体根据实验情境同时表现出波动性和粒子性的原则。光在双缝实验中显示干涉和衍射（波动行为），而在光电效应中表现出离散的能量传递（粒子行为）。相反，电子在偏转实验中表现为粒子，但在穿过晶格时却呈现干涉图样。

Niels Bohr formulated the complementarity principle: wave and particle descriptions are complementary aspects of the same reality, never observed simultaneously in a single experiment. This duality is not a limitation of experimental technique but a fundamental feature of nature at the quantum scale.

尼尔斯·玻尔提出了互补性原理：波动描述和粒子描述是同一实在的两个互补方面，在单次实验中永远无法同时观察到。这种二象性不是实验技术的局限，而是量子尺度下自然界的基本特征。

Students should be able to discuss how the double-slit experiment demonstrates this duality, and how “which-path” measurements destroy interference patterns—a profound insight that observation affects the system.

学生应当能够讨论双缝实验如何展示这种二象性，以及”哪条路径”测量如何破坏干涉图样——这是一个深刻的洞察，表明观察会影响系统本身。

5. Atomic Energy Levels and Line Spectra | 原子能级与线状光谱

Bohr’s model of the hydrogen atom introduced quantized energy levels: electrons occupy discrete orbits with specific energies, rather than any arbitrary energy. When an electron transitions from a higher energy level E₂ to a lower level E₁, it emits a photon of energy hf = E₂ − E₁. Conversely, absorption of a photon of precisely matching energy can excite an electron to a higher level.

玻尔的氢原子模型引入了量子化能级：电子占据具有特定能量的离散轨道，而非任意能量。当电子从高能级 E₂ 跃迁到低能级 E₁ 时，会发射能量为 hf = E₂ − E₁ 的光子。反之，吸收能量精确匹配的光子可将电子激发到更高的能级。

The energy levels of hydrogen are given by:

氢原子的能级由下式给出：

Eₙ = −13.6 eV ⁄ n²

where n = 1, 2, 3, … is the principal quantum number. This explains why atomic spectra consist of discrete lines rather than continuous bands. The Lyman series (transitions to n = 1) lies in the ultraviolet; the Balmer series (transitions to n = 2) is in the visible spectrum; the Paschen series (transitions to n = 3) is in the infrared.

其中 n = 1, 2, 3, … 为主量子数。这解释了为什么原子光谱由离散谱线而非连续带构成。莱曼系（跃迁到 n = 1）位于紫外区；巴尔末系（跃迁到 n = 2）位于可见光谱区；帕邢系（跃迁到 n = 3）位于红外区。

6. The Wave Function and Probability Interpretation | 波函数与概率诠释

Erwin Schrödinger’s wave mechanics describes quantum systems using a wave function Ψ (psi). Unlike classical waves, Ψ itself carries no direct physical meaning; rather, the square of its magnitude |Ψ|² represents the probability density of finding the particle at a particular position.

埃尔温·薛定谔的波动力学使用波函数 Ψ（psi）描述量子系统。与经典波不同，Ψ 本身没有直接的物理意义；其模的平方 |Ψ|² 表示在特定位置找到粒子的概率密度。

This probabilistic interpretation, championed by Max Born, revolutionised our understanding of determinism. In the quantum world, we cannot predict exactly where an electron will be found; we can only specify the probability distribution. The same quantum state, prepared identically, can yield different measurement outcomes on different trials—a stark departure from classical determinism.

这一由马克斯·玻恩倡导的概率诠释彻底改变了我们对决定论的理解。在量子世界中，我们无法精确预知电子会出现在哪里；只能给出概率分布。相同的量子态，以相同方式制备，在不同次的测量中可能产生不同的结果——这与经典决定论形成了鲜明对比。

For IB students, the key point is understanding that |Ψ|² is a probability density: the probability of finding the particle in a small region of length dx is |Ψ|² dx. The wave function must be continuous, finite, single-valued, and normalised—meaning the total probability of finding the particle somewhere in space equals 1.

对于IB学生，关键在于理解 |Ψ|² 是概率密度：在长度为 dx 的小区域内找到粒子的概率为 |Ψ|² dx。波函数必须是连续的、有限的、单值的，并且归一化——即在整个空间找到粒子的总概率等于1。

7. Heisenberg’s Uncertainty Principle | 海森堡测不准原理

Werner Heisenberg’s uncertainty principle is one of the most profound results of quantum mechanics. It states that certain pairs of physical properties cannot both be known with arbitrary precision—the more precisely one is known, the less precisely the other can be determined. For position and momentum, the principle is expressed as:

维尔纳·海森堡的测不准原理是量子力学最深刻的成果之一。它指出某些物理性质的成对组合不能同时以任意精度被确定——其中一个知道得越精确，另一个能确定的精度就越低。对于位置与动量，该原理表达为：

Δx · Δpₓ ≥ h ⁄ 4π

where Δx is the uncertainty in position and Δpₓ is the uncertainty in momentum. Analogously, for energy and time: ΔE · Δt ≥ h ⁄ 4π, meaning that a state of very short lifetime has a correspondingly uncertain energy.

其中 Δx 是位置的不确定度，Δpₓ 是动量的不确定度。类似地，对于能量与时间：ΔE · Δt ≥ h ⁄ 4π，这意味着寿命极短的状态其能量相应地具有不确定性。

This principle is not a statement about measurement limitation imposed by technology; it is a fundamental property of nature. Attempting to measure an electron’s position with a photon inevitably transfers momentum to the electron, altering its motion. The uncertainty is intrinsic, not merely practical.

该原理并非关于技术限制的测量论断；它是自然界的基本属性。试图用光子测量电子的位置不可避免地将动量转移给电子，从而改变其运动。这种不确定性是内禀的，不仅仅是实际层面的。

In IB examinations, students are expected to apply these inequalities numerically to estimate uncertainties and to appreciate that the product of uncertainties is of the order of Planck’s constant, which is why quantum effects are invisible at macroscopic scales.

在IB考试中，学生需要数值上应用这些不等式来估算不确定度，并认识到不确定度的乘积量级为普朗克常数，这正是量子效应在宏观尺度上不可见的原因。

8. Quantum Tunnelling | 量子隧穿

Quantum tunnelling is a phenomenon in which a particle passes through a potential energy barrier that would be classically impenetrable because the particle’s energy is less than the barrier height. Classically, this is impossible—a ball cannot roll over a wall taller than its kinetic energy permits.

量子隧穿是一种现象：粒子穿过在其能量低于势垒高度时经典情况下不可穿透的势能壁垒。经典上这是不可能的——一个球无法滚过高于其动能所允许高度的墙壁。

In quantum mechanics, the wave function does not abruptly vanish at the barrier; instead, it decays exponentially within the barrier region. If the barrier is sufficiently thin, a non-zero portion of the wave function emerges on the other side, meaning there is a finite probability the particle tunnels through. The tunnelling probability increases with lower barrier height, thinner barrier width, and lower particle mass.

在量子力学中，波函数不会在势垒处戛然而止；而是在势垒区域内指数衰减。如果势垒足够薄，波函数有一非零部分从另一侧溢出，意味着粒子有有限的概率隧穿过去。隧穿概率随势垒高度降低、势垒宽度变薄和粒子质量减少而增大。

Applications are abundant: scanning tunnelling microscopes rely on tunnelling current to image surfaces at atomic resolution; nuclear fusion in stars occurs at temperatures lower than classical predictions would require because protons tunnel through the Coulomb barrier; and modern flash memory devices exploit quantum tunnelling for data storage.

量子隧穿的应用非常广泛：扫描隧道显微镜依靠隧穿电流以原子分辨率对表面成像；恒星中的核聚变在低于经典预言所需温度下发生，这是因为质子隧穿过库仑势垒；现代闪存设备也利用量子隧穿来进行数据存储。

9. The Complementary Nature of Wave and Particle Models | 波模型与粒子模型的互补性

Bohr’s complementarity principle deserves special attention in IB Study. It asserts that wave and particle presentations of quantum entities are not contradictory but complementary—each provides a valid description of a different experimental scenario, and both are needed for a complete understanding.

玻尔的互补性原理在IB学习中值得特别关注。它主张量子实体的波动呈现与粒子呈现并不矛盾，而是互补的——每一种呈现都有效描述了不同实验场景，全面的理解需要两者。

Consider how we might choose models for different contexts: for interference and diffraction calculations, the wave model is appropriate; for photoelectric effect and photon energy calculations, the particle model is appropriate. Neither model is “correct” in an absolute sense; each is a tool for a specific aspect of quantum behaviour.

想想我们在不同情境中如何选择模型：对于干涉和衍射计算，波动模型是合适的；对于光电效应和光子能量计算，粒子模型是合适的。这两种模型都不是绝对意义上的”正确”；每一种都是处理量子行为某一特定方面的工具。

This philosophical framing helps students understand why quantum mechanics resists intuitive classical pictures. The quantum world operates according to its own logic, and the mathematics of wave functions provides the most complete description currently available.

这种哲学框架帮助学生理解为什么量子力学抗拒直觉的经典图景。量子世界按照自身的逻辑运作，而波函数的数学提供了当前最完整的描述。

10. Applications of Quantum Physics in Technology | 量子物理在技术中的应用

Quantum physics is not merely theoretical; it underpins a vast array of modern technologies. The light-emitting diode (LED) operates on the principle of electron transitions between energy bands in semiconductors, emitting photons of precisely engineered energies determined by the semiconductor band gap.

量子物理不仅仅是理论性的；它支撑着大量现代技术。发光二极管（LED）的工作原理是电子在半导体能带之间跃迁，发射出由半导体带隙精确设计的光子能量。

Lasers rely on stimulated emission—a quantum process in which incoming photons trigger identical photons to be emitted, producing coherent, monochromatic light. Without quantum physics, laser interferometry, optical communication, barcode scanners, and laser surgery would all be impossible.

激光依赖受激辐射——一种量子过程：入射光子触发发射相同的光子，从而产生相干、单色的光。没有量子物理，激光干涉测量、光通信、条码扫描仪和激光手术都将是不可想象的。

Additionally, magnetic resonance imaging (MRI) relies on nuclear spin quantum states of hydrogen nuclei, and positron emission tomography (PET) exploits the annihilation of a positron-electron pair into two photons. The IB curriculum emphasises connecting theoretical concepts to these practical applications, encouraging students to view quantum mechanics as an active, living discipline.

此外，磁共振成像（MRI）依赖于氢核的核自旋量子态，正电子发射断层扫描（PET）则利用正负电子对湮灭为两个光子的过程。IB课程强调将理论概念与实际应用联系起来，鼓励学生将量子力学视为一门活跃的、充满生命力的学科。

Published by TutorHao | IB Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 IB Physics: The Doppler Effect — Principles and Applications | IB物理：多普勒效应的原理与应用

The Doppler effect is one of the most intuitive yet deeply physical phenomena in wave mechanics. It describes how the observed frequency of a wave changes when there is relative motion between a source and an observer. From the changing pitch of an ambulance siren to the redshift of distant galaxies, this effect bridges everyday experience and cutting-edge astrophysics.

多普勒效应是波动学中既直观又极具物理深度的重要现象。它描述了当波源与观察者之间存在相对运动时，观察到的波动频率如何发生改变。从救护车警笛音调的变化到遥远星系的红移，这一效应将日常生活体验与前沿天体物理学紧密相连。

1. Historical Background | 历史背景

Christian Doppler, an Austrian physicist, first proposed the effect in 1842. He hypothesized that the colour of binary stars might be influenced by their motion relative to Earth. His idea was initially controversial, but later verified experimentally by Dutch scientist Buys Ballot in 1845, who used a locomotive and a group of musicians with perfect pitch to confirm the pitch change of sound waves.

奥地利物理学家克里斯蒂安·多普勒于1842年首次提出这一效应。他假设双星的颜色可能受到其相对于地球运动的影响。这一想法最初备受争议，但随后在1845年由荷兰科学家拜斯·巴洛特通过实验证实——他利用火车机车和一组具有绝对音感的音乐家，验证了声波音调的变化。

2. The Basic Principle | 基本原理

When a wave source moves towards a stationary observer, the wavefronts are compressed ahead of the source. This results in a shorter wavelength and therefore a higher observed frequency. Conversely, when the source moves away, the wavefronts are stretched, producing a longer wavelength and a lower observed frequency. The wave speed itself remains unchanged — only the wavelength and frequency perceived by the observer change.

当波源朝向静止观察者运动时，波前在波源前方被压缩，导致波长变短，因而观察频率升高。相反，当波源远离时，波前被拉长，产生更长的波长和更低的观察频率。波速本身保持不变——改变的仅仅是观察者感知到的波长和频率。

f′ = f × v / (v ∓ vₛ)

Here, f′ is the observed frequency, f is the source frequency, v is the wave speed in the medium, and vₛ is the speed of the source. The minus sign applies when the source moves towards the observer, and the plus sign when it moves away.

其中 f′ 是观察频率，f 是波源频率，v 是波在介质中的传播速度，vₛ 是波源的运动速度。当波源朝向观察者运动时取减号，远离时取加号。

3. Doppler Effect with Moving Observer | 观察者运动的情形

When the observer moves relative to a stationary source, the effect arises from the observer encountering wavefronts at a different rate. If the observer moves towards the source, they intercept wavefronts more frequently, resulting in a higher frequency. The formula becomes f′ = f × (v ± vₒ) / v, where vₒ is the observer’s speed. The plus sign is used when moving towards the source.

当观察者相对于静止波源运动时，效应源于观察者以不同的速率遭遇波前。如果观察者朝向波源运动，他们会更频繁地截获波前，导致频率升高。此时公式变为 f′ = f × (v ± vₒ) / v，其中 vₒ 是观察者的速度。朝向波源运动时取加号。

A crucial point in IB Physics is distinguishing between these two scenarios. When the source moves, wavelengths physically change in the medium. When the observer moves, the wavelength in the medium remains unchanged, but the relative speed of the waves with respect to the observer changes, altering the frequency of arrival.

IB物理中的一个关键点在于区分这两种情形。当波源运动时，介质中的波长发生实际改变。当观察者运动时，介质中的波长不变，但波相对于观察者的速度发生了变化，从而改变了到达频率。

4. General Formula for Sound Waves | 声波的通用公式

For sound waves, both source and observer may be moving simultaneously. The general formula combines both effects:

对于声波，波源和观察者可能同时运动。通用公式将两种效应结合起来：

f′ = f × (v ± vₒ) / (v ∓ vₛ)

The convention for signs is systematic: the upper signs (plus for vₒ and minus for vₛ) are used when source and observer approach each other; the lower signs when they move apart. It is essential to recognise which sign corresponds to the physical situation described in the question.

符号约定具有系统性：当波源与观察者相互靠近时，使用上方符号（vₒ 取加号，vₛ 取减号）；相互远离时使用下方符号。必须准确判断题目所描述的物理情境对应哪一个符号。

For electromagnetic waves such as light, the situation is more subtle. Light does not require a medium, and relativistic effects must be considered. The observed frequency is given by the relativistic Doppler formula, involving the Lorentz factor γ and the relative speed between source and observer.

对于电磁波（如光），情况更为微妙。光不需要介质传播，必须考虑相对论效应。光的观察频率由相对论性多普勒公式给出，涉及洛伦兹因子 γ 和波源与观察者之间的相对速度。

5. Applications in Astronomy: Redshift and Blueshift | 天文学应用：红移与蓝移

In astrophysics, the Doppler effect manifests as redshift and blueshift of spectral lines. When a galaxy moves away from Earth, its spectral lines shift towards longer wavelengths — this is redshift. When it moves towards Earth, the lines shift towards shorter wavelengths — blueshift. This phenomenon provides direct evidence for the expansion of the Universe.

在天体物理学中，多普勒效应表现为谱线的红移与蓝移。当星系远离地球时，其光谱线向更长波长方向移动——即红移。当星系朝向地球运动时，谱线向更短波长方向移动——即蓝移。这一现象为宇宙膨胀提供了直接证据。

Edwin Hubble’s observation of the relationship between galactic redshift and distance led to the formulation of Hubble’s Law, which states that the recessional velocity of a galaxy is proportional to its distance from us. This is one of the cornerstones of modern cosmology and is directly derived from Doppler shift measurements.

埃德温·哈勃对星系红移与距离关系的观测促成了哈勃定律的提出。该定律指出，星系的退行速度与其到我们的距离成正比。这是现代宇宙学的基石之一，直接源于多普勒位移的测量。

z = Δλ / λ₀ = v / c (for v ≪ c)

Here z is the redshift parameter, Δλ is the wavelength shift, λ₀ is the rest wavelength, v is the recessional velocity, and c is the speed of light.

其中 z 是红移参数，Δλ 是波长变化量，λ₀ 是静止波长，v 是退行速度，c 是光速。

6. Radar Guns and Speed Measurement | 雷达测速仪

Police radar guns exploit the Doppler effect with microwave radiation. A beam of microwaves is directed at a moving vehicle; the reflected wave returns with a frequency shift proportional to the vehicle’s speed. By measuring this shift, the instrument can calculate the vehicle’s velocity accurately.

警用雷达测速仪利用微波的多普勒效应。微波束射向行驶中的车辆；反射波返回时带有与车辆速度成正比的频移。通过测量这一频移，仪器便可精确计算出车辆的速度。

The formula used in radar applications is Δf = 2v f₀ / c, where the factor of 2 accounts for the double Doppler shift — once when the wave reaches the moving vehicle, and again when the reflected wave returns to the stationary detector. The angle between the radar beam and the direction of motion must also be considered for accurate results.

雷达应用中使用的公式为 Δf = 2v f₀ / c，其中因子 2 代表双重多普勒频移——第一次是波到达运动车辆时，第二次是反射波返回静止探测器时。精确测量还需考虑雷达波束与运动方向之间的夹角。

7. Medical Imaging: Doppler Ultrasound | 医学成像：多普勒超声

In medicine, Doppler ultrasound is widely used to assess blood flow. High-frequency sound waves are directed at blood vessels, and the reflected echoes from red blood cells exhibit a frequency shift proportional to the blood flow velocity. This non-invasive technique helps diagnose conditions such as deep vein thrombosis, carotid artery stenosis, and fetal circulation problems.

在医学中，多普勒超声广泛用于评估血流状况。高频声波射向血管，红细胞反射的回声带有与血流速度成正比的多普勒频移。这种无创技术有助于诊断深静脉血栓、颈动脉狭窄和胎儿循环异常等疾病。

By analysing both the magnitude and direction of the frequency shift — and by colour-coding the velocity information — medical practitioners can create detailed maps of blood circulation. Modern imaging combines traditional ultrasound with Doppler analysis to provide both anatomical structure and functional flow information in real time.

通过分析频移的大小和方向——并以颜色编码速度信息——医生可以绘制出详细的血液循环图。现代成像将传统超声与多普勒分析相结合，实时提供解剖结构和功能血流信息。

8. Ionospheric and Atmospheric Remote Sensing | 电离层与大气遥感

Doppler radars in meteorology measure wind speeds by tracking the frequency shift of microwaves reflected by precipitation particles. These systems provide critical data for weather forecasting, especially for storm detection and tornado warnings.

气象学中的多普勒雷达通过跟踪降水粒子反射的微波频移来测量风速。这些系统为天气预报提供了关键数据，尤其在风暴探测和龙卷风预警方面具有重要意义。

Acoustic radar and lidar systems also use the Doppler effect to measure atmospheric temperature profiles and wind patterns at various altitudes. These measurements support climate research and aviation safety by providing real-time wind shear detection near airports.

声雷达和激光雷达系统也利用多普勒效应测量不同高度的大气温度廓线和风场。这些测量支持气候研究，并通过提供机场附近的风切变实时探测保障航空安全。

9. Sonar and Oceanography | 声呐与海洋学

In the ocean, the Doppler effect is employed in sonar (Sound Navigation and Ranging) systems. Submarines and research vessels use acoustic signals to detect the speed of underwater objects or ocean currents. ADCP (Acoustic Doppler Current Profiler) instruments measure water current velocities at different depths by analysing the Doppler shift of sound pulses scattered by particles in the water.

在海洋中，声呐（声音导航与测距）系统利用多普勒效应。潜艇和研究船使用声学信号来探测水下目标的速度或洋流的速度。声学多普勒流速剖面仪通过分析水中颗粒散射的声脉冲的多普勒频移，测量不同深度的海流速度。

This technology also contributes to tsunami warning systems. When seismic waves pass through the ocean, changes in sea level are tracked via Doppler-based radar systems on satellites, providing early detection of potentially dangerous waves.

该技术也有助于海啸预警系统。当地震波穿越海洋时，卫星上的多普勒雷达系统可追踪海平面的变化，为潜在危险海浪提供早期探测。

10. IB Physics Examination Points | IB物理考点分析

In IB Physics examinations, the Doppler effect is typically assessed in the topic of waves (Topic 4) and astrophysics (Topic 12). Students are expected to solve numerical problems using the formula for sound waves and qualitative questions on light. The most common mistakes include incorrect sign selection, confusing source velocity with observer velocity, and forgetting that the wave speed is determined by the medium, not by the source or observer motion.

在IB物理考试中，多普勒效应通常出现在波动（Topic 4）和天体物理（Topic 12）中。学生需要运用声波公式解决数值问题，回答关于光的定性问题。最常见的错误包括符号选择错误、混淆波源速度与观察者速度，以及忘记波速由介质决定而非由波源或观察者运动决定。

Exam questions often present real-life scenarios such as an ambulance passing a pedestrian, or a satellite emitting radio signals while orbiting Earth. Students must identify whether the source, the observer, or both are moving, then apply the appropriate formula with the correct sign convention. A systematic approach — drawing a diagram, labelling velocities, and stating the direction of motion — significantly reduces errors.

考试题目通常呈现现实场景，例如救护车经过行人，或卫星绕地运行并发射无线电信号。学生必须判断是波源、观察者还是两者都在运动，然后应用合适的公式并选择正确的符号。系统性的解题方法——绘制示意图、标注速度、说明运动方向——能显著减少错误。

11. Common Misconceptions Clarified | 常见误解辨析

A frequent misconception is that the Doppler effect only applies to sound or light. In fact, it applies to all types of waves, including water waves and seismic waves. Another misconception is that the perceived frequency change means the source’s emitted frequency actually changes — it does not. The source frequency remains constant; only the observed frequency varies with relative motion.

一个常见的误解是多普勒效应仅适用于声波或光波。事实上，它适用于所有类型的波，包括水波和地震波。另一个误解是，感知到的频率变化意味着波源发射的频率真的改变了——并非如此。波源的频率保持不变，变化的仅仅是与相对运动相关的观察频率。

Students sometimes believe that the Doppler effect can be detected instantly regardless of distance. In reality, the effect depends on the continuous emission of waves and the relative motion over time. The change in frequency directly reflects the component of relative velocity along the line joining source and observer, not the total velocity.

学生有时认为多普勒效应与距离无关，可以即时检测。实际上，该效应依赖于波的持续发射以及随时间变化的相对运动。频移直接反映的是沿波源与观察者连线方向的相对速度分量，而非总速度。

12. Summary and Study Strategies | 总结与学习策略

To master the Doppler effect for IB Physics, first ensure a solid understanding of basic wave properties — wavelength, frequency, and wave speed. Then practise deriving the Doppler formula from first principles by considering wavefront diagrams. This will help you intuitively know when the frequency increases or decreases, and which sign to use.

要在IB物理中掌握多普勒效应，首先需要扎实理解波的基本性质——波长、频率和波速。然后通过绘制波前图，练习从基本原理出发推导多普勒公式。这将帮助您直观判断频率何时升高或降低，以及应使用哪个符号。

Finally, engage with worked examples from past papers and real-world applications. Understand the meaning of redshift in cosmology and the use of Doppler ultrasound in medicine — these not only prepare you for exam questions but also deepen your appreciation of how a simple wave phenomenon shapes our understanding of the universe.

最后，多练习历年真题中的例题，并关注现实世界中的应用。理解宇宙学中红移的意义以及医学中多普勒超声的用途——这不仅帮助您应对考试问题，更能加深认识：一个简单的波动现象如何深刻塑造了我们对宇宙的理解。

Published by TutorHao | IB Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 IB Physics: Nuclear Physics & Quantum Physics | IB物理：核物理与量子物理专题

The study of nuclear and quantum physics marks the frontier where classical intuition breaks down and a new, probabilistic description of nature emerges. For IB Physics students, mastering these topics is essential not only for examinations but also for understanding the modern technological world — from medical imaging to nuclear power generation.

核物理与量子物理的研究标志着经典直觉失效、自然界新的概率性描述出现的前沿领域。对于IB物理学生而言，掌握这些专题不仅是考试的关键，更是理解现代科技世界——从医学影像到核能发电——的必要基础。

1. Atomic Structure & The Nucleus | 原子结构与原子核

The atom consists of a tiny, dense nucleus surrounded by a cloud of electrons. The nucleus itself contains protons and neutrons, collectively called nucleons. The number of protons defines the atomic number Z, while the total number of nucleons defines the mass number A. Isotopes are atoms of the same element with the same Z but different A.

原子由一个微小而致密的原子核及周围环绕的电子云组成。原子核本身包含质子和中子，统称为核子。质子数定义原子序数Z，而核子总数定义质量数A。同位素是具有相同Z但不同A的同种元素的原子。

Nuclear size can be estimated using the relationship:

R = R₀A¹ᐟ³

where R₀ ≈ 1.2 × 10⁻¹⁵ m (1.2 fm). This implies that nuclear volume is proportional to the number of nucleons — the nucleus behaves like an incompressible fluid.

其中R₀ ≈ 1.2 × 10⁻¹⁵ m（1.2飞米）。这意味着核体积与核子数成正比——原子核的行为类似于不可压缩流体。

2. Radioactive Decay | 放射性衰变

Unstable nuclei emit radiation to reach a more stable configuration. Three primary types of decay exist: alpha (α), beta (β), and gamma (γ). Alpha decay involves the emission of a helium-4 nucleus (²₄He), beta decay involves the conversion of a neutron to a proton with electron emission, and gamma decay releases excess energy as high-frequency photons.

不稳定的原子核通过发射辐射以达到更稳定的状态。存在三种主要衰变类型：阿尔法衰变（α）、贝塔衰变（β）和伽马衰变（γ）。阿尔法衰变涉及氦-4原子核（²₄He）的发射，贝塔衰变涉及中子转化为质子并发射电子，伽马衰变则以高频光子的形式释放多余能量。

In beta-minus decay, the equation is:

ⁿₐX → ᵧₐ₊₁Y + ₋₁⁰e + ν̄ₑ

where ν̄ₑ represents the antineutrino — a particle introduced to conserve energy, momentum, and angular momentum.

其中ν̄ₑ代表反中微子——引入该粒子是为了守恒能量、动量和角动量。

3. Half-Life & Activity | 半衰期与活度

The half-life (T₁/₂) is the time required for half of the radioactive nuclei in a sample to decay. It is a statistical measure that remains constant regardless of the sample size or external conditions — a fundamental characteristic of each radioisotope.

半衰期（T₁/₂）是样品中一半放射性原子核发生衰变所需的时间。它是一种统计量，无论样品大小或外部条件如何都保持不变——每种放射性同位素的基本特征。

The decay law is expressed as:

N = N₀e^(−λt)

where N is the remaining number of nuclei, N₀ is the initial number, λ is the decay constant, and t is the elapsed time. The decay constant relates to half-life by λ = ln2 / T₁/₂. Activity A = λN, measured in becquerels (Bq), where 1 Bq = 1 decay per second.

其中N是剩余原子核数，N₀是初始原子核数，λ是衰变常数，t是经过的时间。衰变常数与半衰期的关系为λ = ln2 / T₁/₂。活度A = λN，单位为贝克勒尔（Bq），1 Bq = 每秒一次衰变。

4. Nuclear Binding Energy & Mass-Energy Equivalence | 核结合能与质能方程

Einstein’s famous equation E = mc² reveals that mass and energy are interchangeable. The mass of a nucleus is always less than the sum of its constituent nucleons’ masses — this mass defect corresponds to the binding energy that holds the nucleus together.

爱因斯坦的著名方程E = mc²揭示了质量与能量可以相互转化。原子核的质量总是小于其组成核子质量之和——这个质量亏损对应于将原子核结合在一起的结合能。

The binding energy per nucleon is calculated as:

BE = Δm × c² = (Zmₚ + Nmₙ − m_nucleus) × c²

The binding energy per nucleon curve shows that iron-56 (⁵⁶Fe) has the highest value at approximately 8.8 MeV per nucleon. Elements lighter than iron can release energy through fusion, while elements heavier than iron can release energy through fission.

每个核子的结合能曲线显示，铁-56（⁵⁶Fe）具有最高值，约为每核子8.8 MeV。比铁轻的元素可以通过聚变释放能量，而比铁重的元素可以通过裂变释放能量。

5. Nuclear Fission & Fusion | 核裂变与核聚变

Nuclear fission occurs when a heavy nucleus, such as uranium-235, absorbs a neutron and splits into two lighter nuclei, releasing energy and additional neutrons. A typical fission reaction is:

核裂变发生在重核（如铀-235）吸收中子后分裂为两个较轻的原子核，释放能量和额外中子的过程。典型的裂变反应为：

²³⁵₉₂U + ¹₀n → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3¹₀n + Energy

Nuclear fusion, by contrast, involves the combination of light nuclei to form a heavier nucleus. The Sun’s energy comes from the fusion of hydrogen nuclei into helium. Fusion releases approximately three to four times more energy per kilogram of fuel than fission and produces less radioactive waste. However, achieving controlled fusion on Earth requires temperatures exceeding 10⁷ K to overcome electrostatic repulsion.

核聚变则相反，涉及轻核结合形成较重原子核的过程。太阳的能量来自氢核聚变为氦。聚变每千克燃料释放的能量约为裂变的3到4倍，且产生的放射性废物更少。然而，在地球上实现受控聚变需要超过10⁷ K的温度以克服静电排斥力。

6. Wave-Particle Duality | 波粒二象性

One of the most profound discoveries in quantum physics is that entities traditionally classified as particles exhibit wave-like behavior, and vice versa. Light, long considered a wave, behaves as discrete packets of energy called photons in the photoelectric effect. Conversely, electrons — clearly particles — exhibit interference patterns when passed through a double slit.

量子物理学最深刻的发现之一是：传统上被归类为粒子的实体表现出波动行为，反之亦然。光——长期被视为波——在光电效应中表现为称为光子的离散能量包。相反，电子——显然是粒子——在通过双缝时表现出干涉图样。

Louis de Broglie proposed that every particle has an associated wavelength:

λ = h / p = h / (mv)

where h is Planck’s constant (6.63 × 10⁻³⁴ J·s). For macroscopic objects, the de Broglie wavelength is so small that wave behavior is unobservable — this is why classical physics works well in everyday life.

其中h是普朗克常数（6.63 × 10⁻³⁴ J·s）。对于宏观物体，德布罗意波长极小以至于波动行为不可观测——这就是为什么经典物理在日常生活中很适用。

7. The Photoelectric Effect | 光电效应

The photoelectric effect provided crucial evidence for the particle nature of light. When light shines on a metal surface, electrons may be emitted. Classical wave theory predicted that increasing light intensity would increase electron energy — but experiments showed the opposite.

光电效应为光的粒子性提供了关键证据。当光照射金属表面时，电子可能会被发射出来。经典波动理论预测增加光强度会增加电子能量——但实验却显示相反的结果。

Einstein’s explanation invoked the photon model:

E_max = hf − φ

where E_max is the maximum kinetic energy of emitted electrons, f is the photon frequency, and φ is the work function — the minimum energy required to liberate an electron from the metal surface. Key observations explained by this equation include the existence of a threshold frequency (below which no electrons are emitted regardless of intensity) and the instantaneous emission of electrons.

其中E_max是发射电子的最大动能，f是光子频率，φ是逸出功——从金属表面释放电子所需的最小能量。该方程解释的关键观察包括阈值频率的存在（低于该频率，无论强度多大都不会发射电子）以及电子的即时发射。

8. The Bohr Model & Energy Levels | 玻尔模型与能级

Niels Bohr proposed a model of the hydrogen atom in which electrons orbit the nucleus only in specific, quantized energy levels. An electron transitions between levels by absorbing or emitting a photon whose energy exactly equals the energy difference between the two levels:

尼尔斯·玻尔提出了氢原子模型，其中电子仅在特定的量子化能级上绕核运动。电子通过吸收或发射光子在不同能级间跃迁，光子的能量精确等于两个能级之间的能量差：

ΔE = E_high − E_low = hf

For hydrogen, the energy levels are given by:

Eₙ = −13.6 / n² eV

where n = 1, 2, 3, … is the principal quantum number. Transitions between levels produce spectral lines: the Lyman series (n ≥ 2 → n = 1) lies in the ultraviolet, the Balmer series (n ≥ 3 → n = 2) in the visible, and the Paschen series (n ≥ 4 → n = 3) in the infrared.

其中n = 1, 2, 3, …是主量子数。能级之间的跃迁产生谱线：莱曼系（n ≥ 2 → n = 1）位于紫外区，巴尔末系（n ≥ 3 → n = 2）位于可见光区，帕邢系（n ≥ 4 → n = 3）位于红外区。

9. Heisenberg’s Uncertainty Principle | 海森堡不确定性原理

The Heisenberg uncertainty principle states that certain pairs of physical properties cannot be simultaneously known with arbitrary precision. The most commonly cited pair is position and momentum:

海森堡不确定性原理指出，某些物理属性对无法同时以任意精度被知晓。最常被引用的配对是位置和动量：

Δx · Δp ≥ ħ/2

where ħ = h/(2π). A similar relationship exists between energy and time:

其中ħ = h/(2π)。能量和时间之间也存在类似的关系：

ΔE · Δt ≥ ħ/2

This principle is not a limitation of measurement technology but a fundamental property of nature. It explains why electrons cannot collapse into the nucleus: if an electron were confined to a very small space (small Δx), its momentum uncertainty (Δp) would become enormous, giving it enough kinetic energy to escape.

这一原理不是测量技术的限制，而是自然界的基本属性。它解释了为什么电子不能塌缩进原子核：如果电子被限制在非常小的空间（小的Δx），其动量不确定性（Δp）将变得巨大，使其获得足够的动能逃离。

10. Quantum Tunneling & Applications | 量子隧穿与前沿应用

Quantum tunneling is a phenomenon in which a particle passes through a potential energy barrier that, according to classical physics, it does not have enough energy to surmount. This arises from the wave nature of particles — the wavefunction does not drop to zero abruptly at the barrier boundary but decays exponentially within it, allowing a small probability of transmission.

量子隧穿是一种粒子穿过势垒的现象，根据经典物理，粒子没有足够的能量跨越该势垒。这源于粒子的波动性——波函数在势垒边界不会突然降为零，而是在其中指数衰减，从而存在一定的透射概率。

The probability of tunneling depends exponentially on the barrier width and height. This principle underpins several cutting-edge technologies:

隧穿概率指数依赖于势垒的宽度和高度。这一原理支撑了多项前沿技术：

• Scanning Tunneling Microscopy (STM): A sharp conducting tip is brought close to a surface; the tunneling current between tip and surface reveals atomic-scale topography.
• 扫描隧道显微镜（STM）： 将尖锐的导电探针靠近表面；探针与表面之间的隧穿电流揭示原子尺度的形貌。
• Flash memory: Electrons tunnel through an insulating layer to store data.
• 闪存： 电子通过绝缘层隧穿以存储数据。
• Nuclear fusion in stars: Protons tunnel through the Coulomb barrier, enabling fusion at temperatures lower than classically required.
• 恒星中的核聚变： 质子隧穿穿过库仑势垒，使得在比经典所需更低的温度下发生聚变。

11. Nuclear Physics in Medicine & Society | 核物理在医学与社会中的应用

The practical applications of nuclear and quantum physics have transformed medicine and energy production. In radiotherapy, targeted gamma radiation is used to destroy cancerous tumors. Positron Emission Tomography (PET) scanning uses the annihilation of positrons and electrons to produce detectable gamma photons — the same E = mc² principle in reverse (mass created from energy).

核物理和量子物理的实际应用已经改变了医学和能源生产。在放射治疗中，靶向伽马辐射用于摧毁癌性肿瘤。正电子发射断层扫描（PET）利用正电子和电子的湮灭产生可检测的伽马光子——这是E = mc²原理的逆向应用（从能量产生质量）。

In energy production, nuclear fission reactors provide approximately 10% of the world’s electricity. However, concerns about radioactive waste disposal, reactor safety, and nuclear proliferation remain significant societal challenges. The pursuit of practical fusion energy — the “holy grail” of clean power — continues through international projects like ITER, with the promise of abundant energy from seawater-derived fuel.

在能源生产方面，核裂变反应堆提供了全球约10%的电力。然而，对放射性废物处理、反应堆安全和核扩散的担忧仍然是重大的社会挑战。通过ITER等国际项目，实用聚变能源——清洁能源的”圣杯”——的追求仍在继续，有望从海水提取的燃料中获得丰富的能源。

12. Exam Tips & Common Pitfalls | 考试技巧与常见错误

IB Physics students frequently encounter specific challenges when tackling nuclear and quantum physics questions. Being aware of these can significantly improve your exam performance:

IB物理学生在处理核物理和量子物理问题时经常遇到特定的挑战。了解这些可以显著提高你的考试成绩：

• Units: Always work in SI units — convert MeV to joules (1 MeV = 1.6 × 10⁻¹³ J) and angstroms to meters.
• 单位转换： 始终使用SI单位——将MeV转换为焦耳（1 MeV = 1.6 × 10⁻¹³ J），将埃转换为米。
• Mass defect: Remember that mass defect compares the nucleus to its separate nucleons — not to the atom’s total mass including electrons (though electron masses often cancel).
• 质量亏损： 记住质量亏损是将原子核与分离的核子进行比较——而不是与包含电子的原子总质量比较（尽管电子质量通常会抵消）。
• Photoelectric effect: Intensity determines the number of photoelectrons (current), not their kinetic energy; frequency determines kinetic energy.
• 光电效应： 强度决定光电子的数量（电流），而非其动能；频率决定动能。
• Half-life graph: Do not confuse activity (A) with number of nuclei (N) — both decay exponentially but represent different quantities.
• 半衰期图像： 不要混淆活度（A）和原子核数（N）——两者都按指数衰减但代表不同的量。
• Energy-level diagrams: Show the direction of photon emission (downward arrow) or absorption (upward arrow) clearly on diagrams.
• 能级图： 在图上清晰标注光子发射（向下箭头）或吸收（向上箭头）的方向。

Mastering nuclear and quantum physics requires both conceptual understanding and computational fluency. Practice past paper questions, memorize key constants, and always verify the physical plausibility of your answers — if a calculated energy seems absurdly large or small, reconsider your approach.

掌握核物理和量子物理既需要概念理解，也需要计算熟练度。练习历年真题，记住关键常数，并始终验证答案的物理合理性——如果计算出的能量似乎大得离谱或小得离谱，请重新审视你的方法。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 Atomic Structure Models in IB Physics | IB物理：原子结构模型解析

Atomic structure is a central topic in IB Physics, bridging classical ideas with the quantum world. Understanding how models of the atom evolved helps students grasp key concepts such as energy levels, spectra, and nuclear notation.

原子结构是IB物理的核心主题，连接了经典物理与量子世界。理解原子模型的演变过程，有助于学生掌握能级、光谱和核素符号等关键概念。

1. The Plum Pudding Model | 葡萄干布丁模型

Before Rutherford’s experiment, the atom was thought to be a sphere of positive charge with negatively charged electrons embedded inside, like raisins in a pudding. This model, proposed by J.J. Thomson in 1904, explained electrical neutrality but gave no insight into nuclear structure.

在卢瑟福实验之前，人们认为原子是一个正电荷球体，其中镶嵌着带负电的电子，就像布丁里的葡萄干。这个模型由J.J.汤姆逊在1904年提出，它能解释电中性，但无法揭示核结构。

The model predicted that alpha particles passing through a thin gold foil would be deflected only slightly, since the positive charge was spread out uniformly.

该模型预言，α粒子穿过薄金箔时只会发生微小偏转，因为正电荷均匀分布。

2. Rutherford’s Nuclear Model | 卢瑟福核式模型

In 1911, Geiger and Marsden, under Rutherford’s supervision, observed that a small fraction of alpha particles were scattered through angles greater than 90°. This was impossible under the plum pudding model.

1911年，盖革和马斯登在卢瑟福指导下观察到，一小部分α粒子被散射到大于90°的角度。这在葡萄干布丁模型下是不可能发生的。

Rutherford concluded that the atom must contain a tiny, dense, positively charged nucleus, with electrons orbiting at relatively large distances. Most of the atom is empty space.

卢瑟福由此推断，原子内部必然存在一个极小且致密的正电原子核，电子在较远距离处绕核运动。原子的大部分是空的。

3. The Bohr Model of Hydrogen | 玻尔氢原子模型

Rutherford’s model failed to explain why electrons do not spiral into the nucleus due to electromagnetic radiation. In 1913, Niels Bohr proposed that electrons occupy specific circular orbits with fixed angular momentum.

卢瑟福模型无法解释电子为何不会因电磁辐射而螺旋坠入原子核。1913年，尼尔斯·玻尔提出电子占据具有固定角动量的特定圆形轨道。

The key condition is that the angular momentum is quantised:

关键条件是角动量量子化：

mₑ v r = n × (h / 2π), 其中 n = 1, 2, 3, …

Here mₑ is the electron mass, v is its speed, r is the orbit radius, h is Planck’s constant, and n is the principal quantum number.

其中mₑ是电子质量，v是电子速率，r是轨道半径，h是普朗克常数，n是主量子数。

4. Energy Levels and Photon Emission | 能级与光子发射

In the Bohr model, an electron in a higher energy level can transition to a lower level by emitting a photon. The photon energy equals the difference between the two energy levels:

在玻尔模型中，处于高能级的电子跃迁到低能级时会发射光子。光子能量等于两个能级之差：

E = Eᵢ − E_f = h f

where Eᵢ is the initial energy, E_f is the final energy, h is Planck’s constant, and f is the frequency of the emitted photon.

其中Eᵢ是初态能量，E_f是末态能量，h是普朗克常数，f是发射光子的频率。

For hydrogen, the energy of level n is given by:

对于氢原子，第n能级的能量为：

Eₙ = −13.6 eV / n²

5. Atomic Absorption and Emission Spectra | 原子吸收与发射光谱

When atoms absorb energy, electrons jump to higher energy levels. When they return to lower levels, they emit photons at discrete frequencies. These frequencies form the characteristic line spectrum of the element.

当原子吸收能量时，电子跃迁到较高能级。当它们返回较低能级时，会以离散频率发射光子。这些频率构成了该元素特有的线状光谱。

The Lyman, Balmer, and Paschen series correspond to transitions ending at n = 1, n = 2, and n = 3 respectively. The Balmer series lies in the visible region.

莱曼系、巴尔末系和帕申系分别对应跃迁终态为n = 1、n = 2和n = 3的谱线系。其中巴尔末系位于可见光区域。

An absorption spectrum is produced when light passes through a cool gas; the missing wavelengths match the emission lines of that gas.

当光穿过冷气体时会产生吸收光谱；缺失的波长恰好与该气体的发射谱线对应。

6. De Broglie’s Matter Waves | 德布罗意物质波

Louis de Broglie proposed that particles such as electrons also exhibit wave-like properties. The wavelength associated with a particle of momentum p is:

路易·德布罗意提出，电子等粒子也具有波动性。与动量为p的粒子相联系的波长为：

λ = h / p = h / (m v)

This wave nature explains why only certain orbits are allowed: a standing wave must fit exactly around the circumference of the orbit.

这种波动性解释了为何只有某些轨道被允许：驻波必须恰好环绕轨道圆周一周。

2π r = n λ

7. The Schrödinger Model and Orbitals | 薛定谔模型与轨道

Erwin Schrödinger refined the Bohr model by treating the electron as a wave described by a wave function. The square of the wave function gives the probability density of finding the electron at a given location.

埃尔温·薛定谔通过将电子视为由波函数描述的波，对玻尔模型进行了改进。波函数的平方给出在某一位置找到电子的概率密度。

Instead of fixed orbits, we use atomic orbitals: regions in space where the electron is most likely to be found. These orbitals are characterised by quantum numbers n, l, and mₗ.

我们不再使用固定轨道，而是使用原子轨道：电子最可能出现空间区域。这些轨道由量子数n、l和mₗ表征。

In IB Physics, you are expected to understand the hydrogen atom energy levels, but not to solve the Schrödinger equation in detail.

在IB物理中，需要理解氢原子能级，但不需要详细求解薛定谔方程。

8. The Standard Model of Particles | 粒子物理标准模型

Modern physics describes the atom’s constituents in terms of fundamental particles. Protons and neutrons are made of quarks: a proton consists of two up quarks and one down quark (uud), while a neutron consists of two down quarks and one up quark (udd).

现代物理学用基本粒子来描述原子的组成。质子和中子由夸克组成：质子由两个上夸克和一个下夸克组成（uud），中子由两个下夸克和一个上夸克组成（udd）。

The electron is a fundamental lepton with charge −1. It participates in electromagnetic and weak interactions, but not in the strong nuclear force.

电子是一种基本轻子，电荷为−1。它参与电磁相互作用和弱相互作用，但不参与强核力。

Here is a summary of the key particles:

下面是关键粒子汇总：

9. Nuclear Notation and Isotopes | 核素符号与同位素

An atom is represented using the nuclear notation:

原子用核素符号表示：

ᴬ_Z X

where Z is the atomic number (number of protons), A is the mass number (protons + neutrons), and X is the chemical symbol. The number of neutrons is N = A − Z.

其中Z是原子序数（质子数），A是质量数（质子数+中子数），X是元素符号。中子数为N = A − Z。

Isotopes are atoms of the same element with the same Z but different A. For example, carbon-12 and carbon-14 both have Z = 6, but contain 6 and 8 neutrons respectively.

同位素是同一元素中Z相同但A不同的原子。例如，碳-12和碳-14的Z都是6，但分别含有6个和8个中子。

10. Nuclear Radius and Density | 原子核半径与密度

The nuclear radius is approximately given by:

原子核半径近似为：

r = r₀ A^(1/3)

where r₀ is a constant of about 1.2 × 10⁻¹⁵ m, and A is the mass number. This relationship implies that the volume of a nucleus is proportional to A, so all nuclei have roughly the same density.

其中r₀约为1.2 × 10⁻¹⁵ m的常数，A为质量数。这个关系表明原子核的体积与A成正比，因此所有原子核的密度大致相同。

This very high density means that a nucleus is about 10⁴ times denser than ordinary bulk matter.

这种极高的密度意味着原子核的密度大约是普通宏观物质的10⁴倍。

11. Applications and Evidence | 应用与实验证据

Atomic models are supported by multiple experimental observations. The line spectra of hydrogen match Bohr’s predictions very closely. Rutherford’s scattering experiment established the existence of the nucleus, while Frank–Hertz and photoelectric experiments further confirmed quantised energy levels.

原子模型得到了多项实验观察的支持。氢的线状光谱与玻尔的预言高度吻合。卢瑟福散射实验确立了原子核的存在，而弗兰克-赫兹实验和光电效应实验进一步证实了能级的量子化。

Understanding atomic structure is essential for topics such as radioactivity, nuclear fission and fusion, and medical imaging using isotopes.

理解原子结构对于放射性、核裂变与核聚变以及利用同位素进行医学成像等课题至关重要。

12. Exam Tips for IB Physics | IB物理考试建议

When answering questions on atomic structure, always state the full model name and its key assumptions. For energy level calculations, use E = h f and convert eV to joules when necessary (1 eV = 1.6 × 10⁻¹⁹ J).

在回答原子结构问题时，务必写出完整的模型名称及其关键假设。对于能级计算，使用E = h f，并在必要时将电子伏特转换为焦耳（1 eV = 1.6 × 10⁻¹⁹ J）。

Be careful to identify the initial and final energy levels in transitions, and remember that a photon is absorbed when an electron moves to a higher level, and emitted when it moves to a lower level.

注意识别跃迁的初态和末态能级，记住电子吸收光子跃迁到较高能级时是吸收，跃迁到较低能级时是发射。

Also memorise the charge and composition of protons, neutrons and electrons, and be able to calculate the number of neutrons in an isotope.

此外，要牢记质子、中子和电子的电荷与组成，并能计算同位素中的中子数。

Finally, understand the difference between the Bohr orbit and the Schrödinger orbital: the former is a definite path, while the latter is a probability distribution.

最后，要理解玻尔轨道与薛定谔轨道的区别：前者是确定的路径，后者是概率分布。

Published by TutorHao | IB Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 IB Physics: Motion of Charged Particles in Electric and Magnetic Fields | IB物理：带电粒子在电磁场中的运动

The motion of charged particles in electric and magnetic fields is a cornerstone topic in IB Physics. It appears in both SL and HL examinations, often as multi-part questions combining kinematics, circular motion, and energy conservation. Mastering this topic requires a clear understanding of when each force acts and how it affects the trajectory.

带电粒子在电场和磁场中的运动是IB物理的核心内容，在SL和HL考试中均频繁出现，常以多部分综合题考查运动学、圆周运动和能量守恒。掌握这一主题的关键在于清楚每种力何时起作用，以及它们如何改变粒子的运动轨迹。

1. The Lorentz Force | 洛伦兹力

A charged particle in an electromagnetic field experiences the Lorentz force, which is the vector sum of the electric and magnetic forces:

处于电磁场中的带电粒子受到洛伦兹力，它是电场力和磁场力的矢量和：

F = qE + qv × B

The electric force F = qE is parallel to the field direction and acts regardless of the particle’s motion. The magnetic force F = qv × B is given by a cross product: its magnitude is F = qvB sinθ, where θ is the angle between v and B, and its direction is given by the right-hand rule for positive charges.

电场力 F = qE 平行于场方向，与粒子的运动状态无关。磁场力 F = qv × B 由叉积定义：其大小为 F = qvB sinθ，其中 θ 为 v 与 B 的夹角，方向由右手定则确定（针对正电荷）。

Three key facts about the magnetic force:

关于磁场力的三个关键事实：

• It is always perpendicular to both v and B; hence it never does work.
• 它始终垂直于 v 和 B，因此永不做功。
• It changes only the direction of the velocity, not its magnitude.
• 它只改变速度的方向，不改变速度的大小。
• If v is parallel to B (θ = 0° or 180°), the magnetic force is zero.
• 若 v 平行于 B（θ = 0° 或 180°），磁场力为零。

2. Motion Parallel to a Uniform Electric Field | 平行于匀强电场的运动

When a charged particle moves parallel or antiparallel to a uniform electric field, the electric force is constant and collinear with the velocity. This produces uniform acceleration.

当带电粒子平行或反平行于匀强电场运动时，电场力恒定且与速度共线，产生匀加速运动。

For a particle of charge q in a field E, the acceleration is:

对于电荷量为 q、处于电场 E 中的粒子，其加速度为：

a = qE/m

This is analogous to projectile motion under gravity. If the particle starts from rest, its final speed after moving through a potential difference V is found from the work-energy theorem:

这与重力作用下的抛体运动类似。若粒子从静止出发，经过电势差 V 后的末速度可由功能定理求出：

qV = ½mv²

Since the electrostatic force is conservative, this result is independent of the path taken. The speed depends only on the potential difference and the charge-to-mass ratio q/m.

由于静电力是保守力，该结果与路径无关。末速度仅取决于电势差和荷质比 q/m。

3. Motion Perpendicular to a Uniform Electric Field | 垂直于匀强电场的运动

When a particle enters a uniform electric field with its initial velocity perpendicular to the field, its motion resembles projectile motion under gravity.

当粒子以垂直于电场方向的初速度进入匀强电场时，其运动类似于重力场中的抛体运动。

Taking the x-axis along the initial velocity and the y-axis along the field:

以初速度方向为 x 轴、电场方向为 y 轴：

• x-direction: constant velocity, x = v₀t
• x 方向：匀速直线运动，x = v₀t
• y-direction: uniform acceleration, y = ½at² = ½(qE/m)t²
• y 方向：匀加速运动，y = ½at² = ½(qE/m)t²

Eliminating t gives a parabolic trajectory:

消去 t 可得抛物线轨迹：

y = (qE/(2mv₀²))x²

The particle emerges from a field region of length L having been deflected by an angle φ where:

粒子穿出长度为 L 的场区域时，偏转角 φ 满足：

tan φ = v_y/v₀ = qEL/(mv₀²)

This deflection principle is used in cathode ray tubes and in the deflection plates of oscilloscopes.

这一偏转原理被应用于阴极射线管和示波器的偏转板中。

4. Motion in a Uniform Magnetic Field | 匀强磁场中的运动

When a charged particle enters a uniform magnetic field with velocity perpendicular to the field, the magnetic force acts as a

Published by TutorHao | IB Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 IB Physics: Core Concepts of Electric and Magnetic Fields | IB物理：电场与磁场的核心概念

Electric and magnetic fields form one of the most fundamental pillars of the IB Physics syllabus. From Coulomb’s law to the motion of charged particles in magnetic fields, these concepts explain everything from why a balloon sticks to a wall to how particle accelerators work. This revision guide consolidates the key definitions, equations, and relationships you need for your exams.

电场与磁场是IB物理课程中最基础的支柱之一。从库仑定律到带电粒子在磁场中的运动，这些概念解释了从气球吸附在墙上到粒子加速器如何工作的各种现象。本复习指南整合了你考试所需的关键定义、方程和关系。

1. The Concept of an Electric Field | 电场的概念

An electric field is a region of space around a charged object where another charged object experiences a force. It is a vector field, meaning it has both magnitude and direction at every point in space. The electric field strength E at a point is defined as the force per unit positive charge placed at that point:

电场是电荷周围存在的一个空间区域，在该区域内另一带电物体会受到力的作用。电场是矢量场，意味着它在空间每一点都有大小和方向。某一点的电场强度E定义为放置在该点的单位正电荷所受的力：

E = F / q

Where F is the force in newtons (N) and q is the test charge in coulombs (C). The SI unit of electric field strength is N C⁻¹, which is equivalent to V m⁻¹.

其中F是力，单位为牛顿（N），q是试探电荷，单位为库仑（C）。电场强度的国际单位是N C⁻¹，与V m⁻¹等价。

2. Coulomb’s Law | 库仑定律

Coulomb’s law describes the electrostatic force between two point charges. The magnitude of the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them:

库仑定律描述了两个点电荷之间的静电力。力的大小与电荷量的乘积成正比，与它们之间距离的平方成反比：

F = k |q₁q₂| / r²

Here, k is Coulomb’s constant, equal to 8.99 × 10⁹ N m² C⁻². In IB Physics, we often write this in terms of the permittivity of free space ε₀, where k = 1/(4πε₀). The force is attractive for opposite charges and repulsive for like charges. This is an inverse square law, meaning that doubling the distance reduces the force to one quarter of its original value.

其中k是库仑常数，等于8.99 × 10⁹ N m² C⁻²。在IB物理中，我们通常用真空介电常数ε₀来表示，即k = 1/(4πε₀)。异种电荷之间力为引力，同种电荷之间力为斥力。这是一个平方反比定律，意味着距离加倍时，力减小到原来的四分之一。

3. Electric Field of a Point Charge | 点电荷的电场

For a point charge Q, the electric field strength at a distance r can be derived directly from Coulomb’s law. Combining E = F/q with F = kQq/r² gives:

对于点电荷Q，距离r处的电场强度可以直接从库仑定律推导出来。将E = F/q与F = kQq/r²结合，得到：

E = kQ / r²

The direction of the field is radially outward from a positive charge and radially inward toward a negative charge. Note that the field strength decreases with the square of the distance — this is a key concept for sketching field patterns and solving problems involving multiple charges.

电场方向从正电荷径向向外，指向负电荷径向向内。注意电场强度随距离的平方而减小——这是描绘电场线和解决多电荷问题时的关键概念。

4. Electric Field Lines | 电场线

Electric field lines provide a visual representation of the electric field. The rules for drawing them are:

电场线为电场提供了直观的视觉表示。绘制电场线的规则如下：

• Field lines start on positive charges and end on negative charges (or at infinity).
• 电场线从正电荷出发，终止于负电荷（或无穷远处）。
• The tangent to a field line at any point gives the direction of the force on a positive test charge.
• 电场线上任意一点的切线方向表示正试探电荷在该点所受力的方向。
• The density of field lines indicates the magnitude of the field — closer lines mean a stronger field.
• 电场线的疏密表示电场强度的大小——线越密，场越强。
• Field lines never cross each other.
• 电场线永不相交。

Between two parallel plates with opposite charges, the field lines are parallel and equally spaced, creating a uniform electric field. In this region, the force on a charge is constant in magnitude and direction.

在两块带异种电荷的平行板之间，电场线平行且等距，形成匀强电场。在该区域内，电荷所受的力大小和方向恒定。

5. Electric Potential and Potential Energy | 电势与电势能

Electric potential V at a point is the work done per unit charge in bringing a positive test charge from infinity to that point. It is a scalar quantity measured in volts (V), where 1 V = 1 J C⁻¹. For a point charge Q:

某一点的电势V是将正试探电荷从无穷远处移到该点所做的功除以电荷量。它是标量，单位为伏特（V），其中1 V = 1 J C⁻¹。对于点电荷Q：

V = kQ / r

The electric potential energy of a charge q at a point is then simply U = qV. The potential difference between two points, ΔV, is equal to the work done per unit charge in moving a charge between those points. In a uniform field, the relationship between potential difference and field strength is ΔV = Ed, where d is the distance parallel to the field direction.

电荷q在某一点的电势能即为U = qV。两点之间的电势差ΔV等于将单位电荷在两点之间移动所做的功。在匀强电场中，电势差与场强的关系为ΔV = Ed，其中d是沿场方向的距离。

6. The Concept of a Magnetic Field | 磁场的概念

A magnetic field is a region of space where a moving charge or a magnetic dipole experiences a force. Magnetic fields are produced by moving charges — that is, by electric currents. The magnetic field strength, also called magnetic flux density, is denoted by B and measured in tesla (T). One tesla equals one newton per ampere per meter (N A⁻¹ m⁻¹).

磁场是运动电荷或磁偶极子受到力的作用的空间区域。磁场由运动的电荷产生，也就是由电流产生。磁感应强度（也称磁通量密度）用B表示，单位为特斯拉（T）。1特斯拉等于1牛顿每安培每米（N A⁻¹ m⁻¹）。

The direction of a magnetic field is conventionally the direction in which the north pole of a compass needle points. Magnetic field lines run from a north pole to a south pole outside the magnet, and they form closed loops through the interior of the magnet.

磁场的方向习惯上定义为指南针北极指向的方向。在磁体外部，磁感线从北极指向南极，并通过磁体内部形成闭合回路。

7. Force on a Moving Charge in a Magnetic Field | 磁场中运动电荷所受的力

A charged particle moving with velocity v in a magnetic field B experiences a force described by the equation:

在磁场B中以速度v运动的带电粒子所受的力由以下方程描述：

F = qvB sin θ

Where q is the charge and θ is the angle between the velocity and the magnetic field direction. The direction of this force is always perpendicular to both the velocity and the magnetic field, given by Fleming’s left-hand rule. If θ = 90°, the particle moves in a circular path because the magnetic force acts as a centripetal force:

其中q是电荷量，θ是速度方向与磁场方向之间的夹角。该力的方向始终垂直于速度和磁场方向，由左手定则确定。当θ = 90°时，粒子做圆周运动，因为磁场力充当向心力：

qvB = mv² / r

This leads to the radius of the circular path r = mv/(qB). If the velocity has a component parallel to the field, the particle follows a helical path.

由此可得圆周运动的半径r = mv/(qB)。如果速度有平行于磁场的分量，粒子将沿螺旋路径运动。

8. Force on a Current-Carrying Wire | 电流导体所受的安培力

A current-carrying conductor placed in a magnetic field also experiences a force. For a wire of length L carrying current I perpendicular to the magnetic field B, the magnitude of the force is:

放置于磁场中的载流导体也会受到力的作用。对于长度为L、电流为I且垂直于磁场B的导线，力的大小为：

F = BIL

More generally, when the angle between the wire and the field is θ, F = BIL sin θ. This principle is the basis of the electric motor — a current-carrying coil in a magnetic field experiences a torque that causes it to rotate.

更一般地，当导线与磁场的夹角为θ时，F = BIL sin θ。这一原理是电动机的基础——磁场中的载流线圈受到力矩作用而转动。

9. Magnetic Field of a Long Straight Wire | 长直导线的磁场

A long straight wire carrying current I produces a magnetic field whose field lines form concentric circles around the wire. The magnitude of the field at a distance r from the wire is:

载流I的长直导线产生的磁场，其磁感线环绕导线形成同心圆。距离导线r处的磁感应强度大小为：

B = μ₀I / (2πr)

Here, μ₀ is the permeability of free space, equal to 4π × 10⁻⁷ T m A⁻¹. The direction of the field is given by the right-hand grip rule: if you grip the wire with your right hand with the thumb pointing in the direction of the current, your fingers curl in the direction of the field lines.

其中μ₀是真空磁导率，等于4π × 10⁻⁷ T m A⁻¹。磁场方向由右手螺旋定则确定：右手握住导线，拇指指向电流方向，四指弯曲的方向即为磁感线方向。

10. Comparison: Electric Fields vs Magnetic Fields | 电场与磁场的对比

Understanding the similarities and differences between electric and magnetic fields is essential for constructing clear mental models. The table below summarizes the key comparisons:

理解电场与磁场的异同对于构建清晰的物理模型至关重要。下表总结了关键的对比：

A crucial consequence of the magnetic force doing no work is that a magnetic field can never change the speed of a charged particle — it can only change its direction. In contrast, an electric field can accelerate or decelerate particles, changing their kinetic energy.

磁场力不做功这一关键推论意味着磁场永远不能改变带电粒子的速率——它只能改变其方向。相比之下，电场可以加速或减速粒子，改变其动能。

11. Electromagnetic Induction | 电磁感应

Electromagnetic induction is the phenomenon where a changing magnetic field induces an electromotive force (EMF) in a conductor. Faraday’s law states that the induced EMF is equal to the negative rate of change of magnetic flux linkage:

电磁感应是指变化的磁场在导体中感应出电动势（EMF）的现象。法拉第定律表明，感应电动势等于磁通量变化率的负值：

ε = −N (ΔΦ / Δt)

Where N is the number of turns in the coil, ΔΦ is the change in magnetic flux (Φ = BA cos θ), and Δt is the time interval. Lenz’s law, which explains the negative sign, states that the induced current flows in a direction such that its magnetic field opposes the change that produced it. This is a direct consequence of the conservation of energy.

其中N是线圈匝数，ΔΦ是磁通量的变化量（Φ = BA cos θ），Δt是时间间隔。楞次定律解释了负号的物理意义：感应电流的方向使得其产生的磁场阻碍引起它的磁通量变化。这是能量守恒的直接结果。

12. Exam Tips and Common Pitfalls | 考试技巧与常见误区

To succeed in IB Physics exam questions on this topic, keep these points in mind:

要在IB物理考试中答好这类题目，请牢记以下几点：

• Always draw field line diagrams carefully — examiners look for arrows indicating direction and correct line density.
• 始终仔细绘制场线图——考官会检查表示方向的箭头和线的疏密是否正确。
• Convert units before substituting into equations. Remember that cm must become m, and µC must become C.
• 在代入方程前统一单位。记住厘米必须换算为米，微库仑必须换算为库仑。
• For magnetic force problems, first identify whether the charge moves parallel, perpendicular, or at an angle to the field.
• 对于磁场力问题，首先判断电荷的运动方向与磁场是平行、垂直还是成一定角度。
• Do not confuse the formulas for electric potential (V = kQ/r) and electric field (E = kQ/r²) — one falls off as 1/r, the other as 1/r².
• 不要混淆电势公式（V = kQ/r）和电场强度公式（E = kQ/r²）——一个随1/r变化，另一个随1/r²变化。
• When using Fleming’s left-hand rule, remember the order: Force (thumb), magnetic Field (index finger), Current (middle finger).
• 使用左手定则时，记住顺序：力（拇指）、磁场（食指）、电流（中指）。

Mastering electric and magnetic fields requires practice with both qualitative diagrams and quantitative calculations. Build a solid foundation by drawing field patterns, deriving key relationships, and working through past paper questions systematically.

掌握电场和磁场需要同时练习定性作图和定量计算。通过绘制场图、推导关键关系以及系统性地练习历年真题来打好坚实基础。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 Standing Waves and Resonance: IB Physics Key Points Explained | 驻波与共振考点详解

Standing waves and resonance are core topics in the IB Physics syllabus, appearing in both Standard Level (SL) and Higher Level (HL) papers. These concepts are not only frequently tested in Paper 1 and Paper 2 but also form the foundation for understanding sound, musical instruments, and even quantum mechanics. In this article, we will break down every essential point you need to know, from wave superposition to harmonic series, with exam-style insights throughout.

驻波与共振是 IB 物理课程中的核心考点，在标准级（SL）和高级级（HL）试卷中都会出现。这些概念不仅在 Paper 1 和 Paper 2 中频繁考查，更是理解声学、乐器原理乃至量子力学的基础。本文将从波的叠加到谐波序列，逐一拆解所有必考点，并穿插考试风格的解读。

1. Wave Superposition and Interference | 波的叠加与干涉

When two or more waves meet at a point in space, their displacements add vectorially. This is called the principle of superposition. If the waves are coherent (constant phase difference), they produce interference patterns with constructive and destructive interference.

当两个或多个波在空间中某点相遇时，它们的位移按矢量叠加，这称为叠加原理。如果波是相干的（相位差恒定），它们会产生包含相长干涉和相消干涉的干涉图样。

For two waves of the same frequency and amplitude travelling in the same direction, the resultant displacement at any point is given by:

y = 2A cos(Δφ/2) × sin(kx − ωt + Δφ/2)

where Δφ is the phase difference. When Δφ = 0, 2π, 4π… we get constructive interference (amplitude 2A). When Δφ = π, 3π, 5π… we get destructive interference (amplitude zero).

对于同频率、同振幅、同方向传播的两列波，任意点的合位移为：

y = 2A cos(Δφ/2) × sin(kx − ωt + Δφ/2)

其中 Δφ 为相位差。当 Δφ = 0, 2π, 4π… 时发生相长干涉（振幅为 2A）；当 Δφ = π, 3π, 5π… 时发生相消干涉（振幅为零）。

IB Exam Tip: Questions often ask you to distinguish between interference and superposition. Superposition is the general principle; interference is the observable result of superposition between coherent waves.

IB 考试提示：考题常要求区分叠加与干涉。叠加是一般原理；干涉是相干波叠加后产生的可观测结果。

2. Formation of Standing Waves | 驻波的形成

A standing wave is formed when two waves of identical frequency, amplitude, and speed travel in opposite directions through the same medium. This typically happens when a travelling wave reflects back upon itself from a boundary.

当两列频率、振幅和波速完全相同但传播方向相反的波在同一介质中相遇时，就形成驻波。这通常发生在行进波从边界反射并与自身相遇时。

Consider a wave travelling to the right: y₁ = A sin(kx − ωt), and a wave travelling to the left: y₂ = A sin(kx + ωt). Using the superposition principle:

设向右传播的波为 y₁ = A sin(kx − ωt)，向左传播的波为 y₂ = A sin(kx + ωt)。根据叠加原理：

y = y₁ + y₂ = 2A sin(kx) cos(ωt)

Notice that the position-dependent term sin(kx) and the time-dependent term cos(ωt) are separated. This means every particle in the medium oscillates with simple harmonic motion at the same frequency, but with an amplitude that depends on its position: A(x) = 2A sin(kx).

注意位置项 sin(kx) 与时间项 cos(ωt) 已经分离。这意味着介质中每个质点都以相同频率做简谐运动，但振幅取决于其位置：A(x) = 2A sin(kx)。

At positions where sin(kx) = 0, the amplitude is always zero — these are called nodes. At positions where sin(kx) = ±1, the amplitude is maximum (2A) — these are called antinodes.

在 sin(kx) = 0 的位置，振幅恒为零——这些点称为波节。在 sin(kx) = ±1 的位置，振幅最大（2A）——这些点称为波腹。

3. Characteristics of Standing Waves | 驻波的特征

Standing waves differ fundamentally from travelling waves in several ways. Understanding these differences is essential for multiple-choice questions and short-answer questions in IB Paper 1 and Paper 2.

驻波与行波在本质上有多方面的不同。理解这些差异对于 Paper 1 选择题和 Paper 2 简答题至关重要。

• In a standing wave, there is no net energy transfer — energy is stored in the oscillations, trapped between nodes.

  驻波中没有净能量传递——能量被束缚在波节之间的振荡中。

• All particles between two adjacent nodes oscillate in phase (they reach maximum displacement simultaneously), but particles on opposite sides of a node are in antiphase.

  相邻两个波节之间的所有质点同相振荡（同时达到最大位移），但波节两侧的质点反相振荡。

• The amplitude varies from zero at nodes to maximum at antinodes.

  振幅从波节处的零到波腹处的最大呈周期性变化。

• In a travelling wave, every particle has the same amplitude and adjacent particles are out of phase.

  行波中每个质点振幅相同，相邻质点之间存在相位差。

IB Exam Tip: A common trick question asks: “What is the phase difference between two particles on the same side of a node?” The answer is zero (in phase). Never say they are in antiphase — that only applies to particles on opposite sides of a node.

IB 考试提示：常见陷阱题问：”波节同侧的两个质点之间的相位差是多少？”答案是零（同相）。切勿说它们反相——反相只适用于波节两侧的质点。

4. The Standing Wave Equation | 驻波方程

The standard form of the standing wave equation is:

驻波方程的标准形式为：

y(x, t) = 2A sin(kx) cos(ωt)

where A is the amplitude of each individual travelling wave, k = 2π/λ is the wave number, and ω = 2πf is the angular frequency. The term 2A sin(kx) represents the maximum amplitude at position x, and cos(ωt) describes the time evolution of the oscillation.

其中 A 为每一列行波的振幅，k = 2π/λ 为波数，ω = 2πf 为角频率。2A sin(kx) 表示位置 x 处的最大振幅，cos(ωt) 描述振荡随时间的变化。

From this equation, we can derive the positions of nodes and antinodes. Nodes occur where sin(kx) = 0, i.e., kx = nπ, giving:

由此方程可以推导出波节和波腹的位置。波节出现在 sin(kx) = 0 处，即 kx = nπ：

x_node = nλ/2, n = 0, 1, 2, 3…

Antinodes occur where sin(kx) = ±1, i.e., kx = (n + ½)π, giving:

波腹出现在 sin(kx) = ±1 处，即 kx = (n + ½)π：

x_antinode = (n + ½)λ/2, n = 0, 1, 2, 3…

Therefore, the distance between two successive nodes is λ/2, and the distance between a node and the next antinode is λ/4. These spatial relationships are frequently tested in IB data-based questions.

因此，相邻两个波节之间的距离为 λ/2，相邻波节与波腹之间的距离为 λ/4。这些空间关系在 IB 数据题中经常考查。

5. Standing Waves on a String | 弦上的驻波

When a string is fixed at both ends and plucked, waves reflect at both boundaries, creating standing waves. For the string to vibrate with a standing wave pattern, there must be a node at each fixed end. This boundary condition restricts the allowed wavelengths to specific values.

当一根两端固定的弦被拨动时，波在两个边界处反射，形成驻波。要使弦以驻波模式振动，两端固定点必须为波节。这一边界条件将允许的波长限制为特定值。

For a string of length L fixed at both ends, the fundamental mode (first harmonic) has antinodes at the centre and nodes at both ends. The length L corresponds to λ/2, so:

对于长度为 L 的两端固定弦，基频模式（第一谐波）在中心有一个波腹，两端为波节。弦长 L 对应 λ/2，因此：

λ₁ = 2L (fundamental wavelength)

λ₁ = 2L（基波波长）

The general condition for the n-th harmonic is:

第 n 次谐波的通用条件为：

λₙ = 2L/n, fₙ = n v / (2L), n = 1, 2, 3…

where v is the wave speed on the string, given by v = √(T/μ), with T being the tension and μ the linear mass density (mass per unit length) of the string.

其中 v 为弦上的波速，由 v = √(T/μ) 给出，T 为弦的张力，μ 为弦的线密度（单位长度的质量）。

IB Exam Tip: The formula v = √(T/μ) is provided in the IB data booklet, but you must know how to combine it with fₙ = n v / (2L) to solve problems involving string instruments. For example, if tension doubles, speed increases by a factor of √2, and so does the frequency.

IB 考试提示：公式 v = √(T/μ) 在 IB 数据手册中给出，但你必须知道如何将其与 fₙ = n v / (2L) 结合来解涉及弦乐器的问题。例如，若张力加倍，波速增大为 √2 倍，频率也增大为 √2 倍。

6. Standing Waves in Pipes | 管中的驻波

Standing waves can also form in air columns inside pipes. There are two types of boundary conditions: open ends (where air molecules are free to move, creating an antinode) and closed ends (where air molecules cannot move, creating a node).

驻波也能在管内的空气柱中形成。有两种边界条件：开端（空气分子可自由移动，形成波腹）和闭端（空气分子不能移动，形成波节）。

Pipe open at both ends | 两端开口的管

Both ends are antinodes. The fundamental mode has a node at the centre, with length L = λ/2. The harmonics are:

两端都是波腹。基频模式在中心有一个波节，管长 L = λ/2。各次谐波为：

fₙ = n v / (2L), n = 1, 2, 3… (all harmonics present)

fₙ = n v / (2L), n = 1, 2, 3…（所有谐波均存在）

Pipe closed at one end | 一端封闭的管

The closed end is a node and the open end is an antinode. The fundamental mode has length L = λ/4. This means the wavelength of the fundamental is λ₁ = 4L, and the general condition is:

封闭端为波节，开端为波腹。基频模式的管长 L = λ/4。这意味着基波波长为 λ₁ = 4L，一般条件为：

fₙ = n v / (4L), n = 1, 3, 5, 7… (only odd harmonics)

fₙ = n v / (4L), n = 1, 3, 5, 7…（仅存在奇次谐波）

Common Mistake: Students often forget that a one-end-closed pipe only supports odd harmonics (1st, 3rd, 5th…). The 2nd and 4th harmonics simply cannot exist in such a pipe. This is a classic IB multiple-choice trap.

常见错误：学生经常忘记一端封闭的管只支持奇次谐波（第 1、3、5 次…）。第 2 和第 4 次谐波在这种管中根本不存在。这是 IB 选择题的经典陷阱。

7. Resonance and Natural Frequency | 共振与固有频率

Every object or system has one or more natural frequencies at which it tends to oscillate when disturbed. When a system is driven by an external periodic force, the amplitude of oscillation depends on how close the driving frequency is to the natural frequency.

每个物体或系统都有一个或多个固有频率，即受到扰动时倾向于振荡的频率。当系统受到外部周期性驱动力作用时，振荡的振幅取决于驱动频率与固有频率的接近程度。

Resonance occurs when the driving frequency matches the natural frequency of the system. At resonance, the system absorbs maximum energy from the driving force, resulting in maximum amplitude of oscillation.

共振发生在驱动频率等于系统固有频率时。共振时，系统从驱动力中吸收最大能量，导致振荡幅度达到最大。

In the context of standing waves, resonance occurs because the reflected waves constructively interfere with the incident waves only at specific frequencies. At other frequencies, destructive interference prevents large amplitudes from building up.

在驻波的语境中，共振之所以发生，是因为反射波与入射波只在特定频率下才能相长干涉。在其他频率下，相消干涉阻止了大振幅的建立。

Resonance condition: f_drive = f_natural

共振条件：f_驱动 = f_固有

IB Exam Tip: In Paper 2, you may be asked to draw a resonance curve (amplitude vs. driving frequency). The curve should peak sharply at the natural frequency, and the peak becomes sharper (more narrow) when damping is smaller.

IB 考试提示：在 Paper 2 中，你可能被要求绘制共振曲线（振幅对驱动频率）。曲线应在固有频率处出现尖峰，阻尼越小时峰越尖锐（越窄）。

8. Damping and its Effect on Resonance | 阻尼及其对共振的影响

Real systems are never perfectly isolated — friction, air resistance, and other dissipative forces remove energy from the system. This is called damping. Damping affects resonance in two important ways.

真实系统从不完全隔离——摩擦、空气阻力和其他耗散力会从系统中带走能量。这称为阻尼。阻尼在两个方面影响共振。

• Reduced amplitude: The maximum amplitude at resonance decreases as damping increases.

  振幅减小：共振时的最大振幅随阻尼增大而减小。

• Broader peak: The resonance peak becomes wider and flatter with increased damping. The resonance frequency also shifts slightly to a lower value with heavy damping.

  峰变宽：随阻尼增大，共振峰变得更宽更平。在强阻尼下，共振频率还会略微向低频移动。

For underdamped systems, the amplitude decays exponentially over time: A(t) = A₀e^(−bt/2m), where b is the damping constant and m is the mass. In IB Physics (HL), you should be able to interpret amplitude-time graphs showing exponential decay.

对于欠阻尼系统，振幅随时间呈指数衰减：A(t) = A₀e^(−bt/2m)，其中 b 为阻尼系数，m 为质量。IB 物理（HL）要求你能解读显示指数衰减的振幅-时间图像。

Critical damping is the minimum damping required to return a displaced system to equilibrium without any oscillation. This is relevant in real-world applications like car suspension systems.

临界阻尼是使偏离平衡的系统不发生振荡而直接回到平衡所需的最小阻尼。这在汽车悬挂系统等实际应用中非常重要。

9. Applications and Dangers of Resonance | 共振的应用与危害

Resonance has both beneficial applications and destructive consequences. IB exam questions often connect these real-world examples to the physics principles you have learned.

共振既有有益的应用，也有破坏性的后果。IB 考题经常将这些现实案例与所学物理原理联系起来。

Beneficial applications | 有益的应用

• Musical instruments rely on resonance to amplify sound — the air column resonates with the vibrating string or reed.

  乐器依赖共振放大声音——空气柱与振动的弦或簧片共振。

• Microwave ovens use resonance to heat food — microwaves at frequency ~2.45 GHz resonate with water molecules, transferring energy efficiently.

  微波炉利用共振加热食物——约 2.45 GHz 的微波与水分子共振，高效传递能量。

• Radio tuners work by adjusting the circuit’s natural frequency to match the desired station’s broadcast frequency.

  收音机调台通过调节电路的固有频率来匹配目标电台的广播频率。

• Structural engineers design buildings to avoid resonance with wind or earthquake frequencies.

  结构工程师设计建筑时避免与风或地震的频率发生共振。

Dangers of resonance | 共振的危害

• The Tacoma Narrows Bridge collapse in 1940 is a famous example. Wind-induced oscillations matched the bridge’s natural frequency, causing the amplitude to grow until structural failure occurred.

  1940 年塔科马海峡大桥坍塌是一个著名案例。风致振荡与桥梁固有频率匹配，振幅不断增大，直至结构破坏。

• Soldiers marching in step across a bridge can cause resonant vibrations — which is why they are ordered to break step when crossing.

  士兵齐步走过桥梁可能引发共振——这就是为何过桥时要下令便步走的原因。

• Machinery operating at speeds that cause resonance can suffer excessive vibration and premature failure.

  机械在引发共振的速度下运转会产生过度振动并过早失效。

10. Common Exam Question Types and Strategies | 常见题型与解题策略

To maximise your score on standing wave and resonance questions, you need to recognise the patterns in how IB frames these problems. Here are the most common question types.

要在驻波和共振题目上拿高分，你需要识别 IB 出题的模式。以下是最常见的题型。

Type 1 — Diagram-based questions: You are given a diagram of a standing wave on a string or in a pipe. Identify the harmonic number, wavelength, or frequency.

类型 1 — 图解型题目：给你一张弦或管中驻波的示意图，要求识别谐波次数、波长或频率。

Strategy: Count the number of loops (each loop = half a wavelength). For a string fixed at both ends, n = number of loops. For a pipe closed at one end, n is always odd, and n = 2N − 1 where N is the number of quarter-wavelength segments.

策略：数波腹段数（每段 = 半个波长）。对于两端固定的弦，n = 波腹段数。对于一端封闭的管，n 始终为奇数，且 n = 2N − 1，其中 N 为四分之一波长的段数。

Type 2 — Calculation questions: Given string length, tension, and linear density, calculate the fundamental frequency or the speed of waves.

类型 2 — 计算型题目：给定弦长、张力和线密度，计算基频或波速。

Strategy: Use v = √(T/μ) to find the wave speed first, then apply fₙ = n v / (2L). Never forget to convert the linear density to kg m⁻³ if it is given in g m⁻¹.

策略：先用 v = √(T/μ) 求波速，再应用 fₙ = n v / (2L)。如果线密度以 g m⁻¹ 给出，切勿忘记换算为 kg m⁻¹。

Type 3 — Comparison questions: Compare the harmonics of a pipe open at both ends with a pipe closed at one end of the same length.

类型 3 — 比较型题目：比较相同长度的两端开口管与一端封闭管的谐波。

Strategy: Open pipe frequencies are fₙ = n v / (2L) for all n; closed pipe frequencies are fₙ = n v / (4L) for odd n only. The closed pipe fundamental is one octave lower than the open pipe fundamental.

策略：开口管频率对所有 n 为 fₙ = n v / (2L)；闭管频率仅对奇数 n 为 fₙ = n v / (4L)。闭管基频比开口管基频低一个八度。

Type 4 — Graphical analysis: Interpret resonance curves showing amplitude vs. frequency under different damping conditions.

类型 4 — 图像分析：解读不同阻尼条件下振幅与频率的关系曲线。

Strategy: Higher damping → lower peak, wider curve. The resonance frequency is the location of the peak. A sharper peak means less damping.

策略：阻尼越大 → 峰越低、曲线越宽。共振频率即峰的位置。峰越尖锐意味着阻尼越小。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 IB Physics: Wave Model and Its Applications | IB物理：波动模型及其应用解析

The wave model is one of the most powerful conceptual frameworks in physics. In the updated IB Physics syllabus (first assessment 2025), the wave model is central to Unit C: Wave Behaviour. It connects mechanical oscillations to electromagnetic radiation, forming the basis for understanding everything from music to modern communications. Mastering this model is not just about memorising formulas; it is about developing a deep, intuitive understanding of how energy travels through space and matter.

波动模型是物理学中最强大的概念框架之一。在更新的IB物理课程大纲（2025年首考）中，波动模型是C单元：波动行为的核心。它连接了机械振动与电磁辐射，构成了理解从音乐到现代通信等一切现象的基础。掌握这一模型不仅仅在于记住公式，更在于对能量如何穿过空间和物质传播形成深刻的直觉理解。

1. Types of Waves: Transverse and Longitudinal | 波的类型：横波与纵波

Waves are broadly classified by the direction of particle oscillation relative to the direction of energy propagation. In a transverse wave, particles oscillate perpendicular to the direction of wave travel. Examples include electromagnetic waves, waves on a string, and S-waves (secondary waves) in earthquakes. In a longitudinal wave, particles oscillate parallel to the direction of wave travel, creating compressions and rarefactions. Sound waves and P-waves (primary waves) are classic examples of longitudinal waves.

波根据质点振动方向与能量传播方向的关系，大致分为两类。在横波中，质点振动方向垂直于波的传播方向，例如电磁波、绳波和地震中的S波（剪切波）。在纵波中，质点振动方向平行于波的传播方向，形成疏密相间的区域，声波和地震P波（纵波）是纵波的典型例子。

Another fundamental distinction exists between mechanical waves and electromagnetic waves. Mechanical waves require a medium (solid, liquid, or gas) to propagate; they cannot travel through a vacuum. Electromagnetic waves, however, are self-sustaining oscillations of electric and magnetic fields and can propagate perfectly through a vacuum. This distinction is a frequent source of multiple-choice questions in IB Paper 1.

另一个根本性区别是机械波与电磁波的区别。机械波需要介质（固体、液体或气体）才能传播；它们无法在真空中传播。而电磁波是电场和磁场的自持振荡，可以完美地在真空中传播。这一区别是IB Paper 1选择题的常见考点。

2. Wave Characteristics and the Wave Equation | 波的特性与波速公式

To fully describe a wave, four key parameters are essential: amplitude (A), wavelength (λ), frequency (f), and wave speed (v). The amplitude is the maximum displacement from equilibrium, determining the energy carried by the wave. The wavelength is the distance between two consecutive points in phase. The frequency is the number of complete oscillations per second, measured in hertz (Hz), and is determined solely by the source.

要完整描述一列波，四个关键参数必不可少：振幅（A）、波长（λ）、频率（f）和波速（v）。振幅是相对于平衡位置的最大位移，决定了波所携带的能量。波长是两个相邻同相点之间的距离。频率是每秒完整振动的次数，单位为赫兹（Hz），仅由波源决定。

The relationship between wave speed, frequency, and wavelength is known as the wave equation. This equation is universally applicable to all types of waves, whether mechanical or electromagnetic. The wave speed is determined by the properties of the medium (e.g., temperature, density, and tension).

波速、频率和波长之间的关系称为波速公式。该公式普遍适用于所有类型的波，无论是机械波还是电磁波。波速由介质的性质（如温度、密度和张力）决定。

v = f × λ

For IB exams, it is crucial to understand that when a wave passes from one medium to another, its frequency remains constant, but its speed and wavelength change accordingly. In Paper 2, students often lose marks by forgetting that the period (T) is the reciprocal of frequency (T = 1/f), and that the phase difference is directly related to the path difference.

对于IB考试，理解当波从一种介质进入另一种介质时，其频率保持不变，而速度和波长相应改变，这一点至关重要。在Paper 2中，学生常因忘记周期（T）是频率的倒数（T = 1/f），以及相位差与光程差直接相关而失分。

3. Wavefronts and Huygens’ Principle | 波前与惠更斯原理

A wavefront is an imaginary surface connecting all points of the same phase, such as a crest. The direction in which the wave travels is always perpendicular to the wavefront. Plane waves have parallel, straight wavefronts, while spherical waves have concentric circular (or spherical) wavefronts. The distance between consecutive wavefronts is equal to the wavelength.

波前是连接所有同相点（如波峰）的虚拟曲面。波的传播方向始终垂直于波前。平面波具有平行、直线的波前，而球面波具有同心的圆形（或球形）波前。相邻波前之间的距离等于波长。

Christiaan Huygens proposed a principle that elegantly explains wave propagation: every point on a wavefront can be considered a source of secondary spherical wavelets. After a time ‘t’, the new wavefront is the envelope of these secondary wavelets. Huygens’ principle provides a geometric construction that explains why waves spread out, reflect, refract, and diffract.

惠更斯提出了一个优雅地解释波传播的原理：波前上的每一点都可以看作是新的球面子波的波源。经过时间t后，新的波前就是这些子波的包络面。惠更斯原理提供了一种几何构造方法，能够解释波为何会拓展、反射、折射和衍射。

Specifically, Huygens’ construction can be used to derive the laws of reflection and refraction. When a plane wave strikes a boundary, the secondary wavelets generated at different points along the boundary produce new plane waves in the reflected and transmitted directions. The geometry of this construction directly leads to the law of reflection (angle of incidence equals angle of reflection) and Snell’s law for refraction.

具体来说，惠更斯作图法可以用来推导反射定律和折射定律。当平面波到达边界时，边界上不同点产生的子波会在反射和透射方向形成新的平面波。这种构造的几何关系直接推导出反射定律（入射角等于反射角）和折射斯涅尔定律。

4. Reflection, Refraction, and Total Internal Reflection | 反射、折射与全反射

Reflection occurs when a wave encounters a boundary and bounces back into the original medium. The law of reflection states that the angle of incidence is equal to the angle of reflection, with both angles measured with respect to the normal (a line perpendicular to the surface). Refraction, on the other hand, is the change in direction of a wave when it passes from one medium to another due to a change in its speed.

反射是波遇到边界并返回原介质的现象。反射定律指出，入射角等于反射角，两个角均相对于法线（垂直于表面的线）测量。折射则是波从一种介质进入另一种介质时，因速度变化而发生方向改变的现象。

Snell’s law quantitatively relates the angles of incidence and refraction to the refractive indices of the two media:

斯涅尔定律定量地关联了入射角、折射角与两种介质的折射率：

n₁ × sin θ₁ = n₂ × sin θ₂

When a wave moves from a denser medium (higher refractive index) to a less dense medium (lower refractive index), the refracted ray bends away from the normal. As the angle of incidence increases, the angle of refraction increases more quickly. At a specific angle of incidence, called the critical angle (θ_c), the angle of refraction becomes 90°. For angles of incidence greater than the critical angle, total internal reflection occurs. This phenomenon is the fundamental principle behind optical fibers used in high-speed internet and endoscopes in medicine.

当波从光密介质（折射率较高）进入光疏介质（折射率较低）时，折射光线偏离法线。随着入射角增大，折射角增大得更快。当入射角达到特定角度，即临界角（θ_c）时，折射角变为90°。当入射角大于临界角时，发生全反射。这一现象是高速互联网中使用的光纤和医学内窥镜的基本原理。

5. Diffraction and Single-Slit Interference | 衍射与单缝干涉

Diffraction is the spreading or bending of waves when they pass through an aperture or around an obstacle. The amount of diffraction is most noticeable when the size of the aperture or obstacle is comparable to the wavelength of the wave. For example, sound waves (with wavelengths around 1 meter) easily diffract around doorways, which is why you can hear someone speaking from another room. Light waves, however, have very short wavelengths, so their diffraction is only noticeable through very narrow slits or around sharp edges.

衍射是波穿过狭缝或绕过障碍物时发生的展宽或弯曲现象。当狭缝或障碍物的尺寸与波长相近时，衍射现象最为显著。例如，声波（波长约为1米）很容易绕过门口发生衍射，这就是为什么你能听到另一个房间的人说话。然而，光波的波长很短，因此只有在通过很窄的狭缝或尖锐边缘时，衍射现象才明显。

In IB Physics, single-slit diffraction produces a central maximum with alternate bright and dark fringes. The condition for destructive interference (dark fringes) in single-slit diffraction is given by:

在IB物理中，单缝衍射产生中央亮纹和明暗相间的条纹。单缝衍射中相消干涉（暗纹）的条件为：

a × sin θ = n × λ (where n = 1, 2, 3, …)

It is important to note that the central maximum is twice as wide as the secondary maxima. The spread of the diffraction pattern is inversely proportional to the slit width ‘a’. This means that narrowing the slit broadens the diffraction pattern, a concept that has significant implications for the wave nature of particles and modern physics.

需要注意的是，中央亮纹的宽度是次级亮纹的两倍。衍射条纹的扩展程度与狭缝宽度a成反比。这意味着减小狭缝宽度会使衍射图样变宽，这一概念对粒子的波动性和现代物理学具有重要意义。

6. Interference and Young’s Double-Slit Experiment | 干涉与杨氏双缝实验

Interference is the superposition of two or more waves, resulting in a new wave pattern. Constructive interference occurs when the crests of one wave align with the crests of another, leading to a larger amplitude. Destructive interference occurs when the crest of one wave aligns with the trough of another, leading to a smaller amplitude or complete cancellation. For interference to produce a stable pattern, the sources must be coherent—they must have the same frequency and a constant phase difference.

干涉是两列或多列波叠加形成新波形的现象。当一列波的波峰与另一列波的波峰重合时，发生相长干涉，振幅增大。当一列波的波峰与另一列波的波谷重合时，发生相消干涉，振幅减小或完全抵消。要产生稳定的干涉图样，波源必须是相干的——它们必须具有相同的频率和恒定的相位差。

Thomas Young’s double-slit experiment provides a classic demonstration of wave interference. When monochromatic light passes through two closely spaced slits, the overlapping waves create a pattern of alternating bright and dark fringes on a screen. For constructive interference (bright fringes), the path difference between the two waves must be an integer multiple of the wavelength:

杨氏双缝实验是波干涉的经典演示。当单色光通过两个相距很近的狭缝时，重叠的波在屏幕上形成明暗相间的条纹。对于相长干涉（明纹），两列波的光程差必须是波长的整数倍：

dsin θ = n × λ (where n = 0, 1, 2, …)

The fringe separation (Δy) on the screen is given by the formula:

屏幕上的条纹间距（Δy）由以下公式给出：

Δy = λD / d

Where ‘λ’ is the wavelength, ‘D’ is the distance from the slits to the screen, and ‘d’ is the slit separation. In IB exams, students are expected to apply this formula to various contexts, including determining the wavelength of light and explaining how changing D or d affects the fringe pattern.

其中λ是波长，D是狭缝到屏幕的距离，d是狭缝间距。在IB考试中，学生需要能够将该公式应用于各种情境，包括计算光的波长，以及解释改变D或d如何影响条纹图样。

7. Standing Waves and Resonance | 驻波与共振

A standing wave is the result of the superposition of two waves with the same frequency and amplitude traveling in opposite directions. Unlike traveling waves, standing waves do not transfer energy from one end to the other. Instead, they have specific points called nodes, where the amplitude is always zero, and antinodes, where the amplitude is maximum. The distance between two consecutive nodes (or antinodes) is half a wavelength.

驻波是由两列振幅和频率相同但传播方向相反的波叠加而成的。与行波不同，驻波不传递能量。驻波具有特定的点，称为波节，振幅始终为零；以及波腹，振幅最大。相邻两个波节（或波腹）之间的距离为半个波长。

Standing waves are fundamental to the operation of musical instruments. When a string is plucked or a column of air is blown, only certain modes of vibration (harmonics) are allowed, depending on the boundary conditions. The lowest possible frequency is called the fundamental frequency (first harmonic). The frequencies of the higher harmonics are integer multiples of the fundamental frequency: fₙ = n × f₁.

驻波是乐器运作的基础。当拨动弦或吹响气柱时，根据边界条件的限制，只允许特定的振动模式（谐波）存在。可能的最低频率称为基频（一次谐波）。高次谐波的频率是基频的整数倍：fₙ = n × f₁。

Resonance occurs when the driving frequency matches the natural frequency of the system, leading to a dramatic increase in amplitude. This concept is prevalent in IB Physics questions, exploring examples such as the Tacoma Narrows Bridge collapse and the breaking of a wine glass by an opera singer. In Paper 2, you may be asked to draw standing wave patterns for strings (fixed at both ends) and pipes (open or closed ends).

当驱动频率与系统的固有频率匹配时，发生共振，导致振幅急剧增大。这一概念在IB物理问题中很常见，常探讨塔科马海峡大桥坍塌和歌剧演唱者震碎酒杯等案例。在Paper 2中，你可能需要画出弦（两端固定）和气柱（开口或闭口）的驻波图样。

8. The Doppler Effect | 多普勒效应

The Doppler effect describes the apparent change in frequency of a wave due to relative motion between the source and the observer. It is observed in both sound waves and electromagnetic waves. When the source and observer are moving closer together, the observed frequency is higher than the emitted frequency. When they are moving apart, the observed frequency is lower. This occurs because the relative motion changes the number of wavefronts reaching the observer per unit time.

多普勒效应描述了由于波源和观察者之间的相对运动，导致观察到的波频率发生变化的现象。它在声波和电磁波中都可以观察到。当波源和观察者彼此靠近时，观测频率高于发射频率。当它们彼此远离时，观测频率低于发射频率。这是因为相对运动改变了单位时间内到达观察者的波前数量。

The general formula for the Doppler effect for sound is:

声波多普勒效应的通用公式为：

f’ = f × (v ± v₀) / (v ∓ vₛ)

In the formula, ‘f’ is the emitted frequency, ‘f” is the observed frequency, ‘v’ is the speed of sound in the medium, ‘v₀’ is the speed of the observer, and ‘vₛ’ is the speed of the source. The choice of signs depends on whether the source and observer are moving towards or away from each other. A common exam technique is to remember that “moving towards” always increases the observed frequency, so you choose the signs that make f’ larger.

在该公式中，f是发射频率，f’是观测频率，v是声波在介质中的速度，v₀是观察者的速度，vₛ是波源的速度。正负号的选择取决于波源和观察者是相向运动还是相背运动。一个常见的考试技巧是记住“相向运动”总是使观测频率增加，因此选择能使f’变大的符号。

For electromagnetic waves, the observed frequency shift is related to the relative speed along the line of sight. This principle is applied in radar speed guns used by police, Doppler ultrasound in medicine to measure blood flow speed, and in astronomy to determine the radial velocity of stars and galaxies (redshift and blueshift).

对于电磁波，观测频率的移动与视线方向的相对速度有关。这一原理应用于警察使用的雷达测速枪、医学中测量血流速度的多普勒超声，以及天文学中确定恒星和星系径向速度（红移和蓝移）等领域。

9. Applications of the Wave Model in Technology | 波动模型在科技中的应用

The wave model is not merely an abstract concept confined to textbooks; it is the foundational principle behind a vast array of modern technologies. In telecommunications, optical fibers use total internal reflection to transmit data over long distances with minimal loss.

Published by TutorHao | IB Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 IB Physics: Simple Harmonic Motion – Key Concepts and Model Analysis | IB物理：简谐运动考点与模型精讲

Simple Harmonic Motion (SHM) is one of the most important topics in IB Physics, appearing in both Standard Level and Higher Level examinations. It describes any oscillation where the net restoring force is proportional to the negative displacement, leading to sinusoidal motion in time.

简谐运动是IB物理最重要的考点之一，出现在标准级别和高级别的考试中。它描述了净回复力与位移的负值成正比的振荡形式，从而在时间上呈现正弦规律。

1. Definition and Conditions | 定义与条件

The defining mathematical condition of SHM is a = -ω²x, where a is acceleration, x is displacement from equilibrium, and ω is the angular frequency. This equation shows that acceleration is always opposite to displacement and proportional to it.

简谐运动的定义式是 a = -ω²x，其中 a 是加速度，x 是相对平衡位置的位移，ω 是角频率。该式表明加速度始终与位移方向相反，并且大小与位移成正比。

For a physical system to undergo SHM, two conditions must be met: there must be a restoring force following Hooke’s law, and the system must have inertia that carries it past equilibrium. Common examples are a mass on a spring and a small-angle pendulum.

物理系统要做简谐运动，必须满足两个条件：存在遵循胡克定律的回复力，并且系统具有使它能越过平衡位置的惯性。常见例子有弹簧振子和小角度单摆。

• Equilibrium position is the point where the net force is zero.

  平衡位置是合力为零的位置。

• Amplitude A is the maximum displacement from equilibrium.

  振幅 A 是离开平衡位置的最大位移。

• Period T is the time taken for one complete cycle of oscillation.

  周期 T 是完成一次全振动所需的时间。

• Frequency f is the number of oscillations per second, with f = 1/T.

  频率 f 是每秒振动的次数，f = 1/T。

2. Kinematic Equations | 运动学方程

Taking displacement as x = A cos(ωt + φ), where φ is the phase constant, the velocity and acceleration are obtained by differentiation: v = -Aω sin(ωt + φ) and a = -Aω² cos(ωt + φ).

若位移表达式为 x = A cos(ωt + φ)，其中 φ 是初相位，则对时间求导可得速度 v = -Aω sin(ωt + φ) 和加速度 a = -Aω² cos(ωt + φ)。

The phase angle (ωt + φ) determines the instantaneous state of motion, such as position, velocity direction, and acceleration direction. Two SHMs with the same frequency can have a constant phase difference.

相位角 (ωt + φ) 决定了运动的瞬时状态，包括位置、速度方向和加速度方向。两个同频率的简谐运动可以具有恒定的相位差。

At the equilibrium position, x = 0 and speed is maximum: v_max = Aω. At the extreme positions, x = ±A, speed is zero and acceleration magnitude is maximum: a_max = Aω².

在平衡位置，x = 0，速度最大：v_max = Aω；在极端位置，x = ±A，速度为零，而加速度大小最大：a_max = Aω²。

Initial conditions are used to find A and φ. For example, if the oscillator starts from maximum displacement, then φ = 0; if it starts from equilibrium moving in the positive direction, then φ = -π/2.

初始条件用于确定 A 和 φ。例如，若振子从最大位移处开始，则 φ = 0；若从平衡位置向正方向运动开始，则 φ = -π/2。

3. Dynamics and Restoring Force | 动力学与回复力

For a mass attached to a spring, Hooke’s law gives F = -kx. Applying Newton’s second law, ma = -kx, so a = -(k/m)x. Comparing with a = -ω²x gives ω² = k/m.

对于连接在弹簧上的物体，胡克定律给出 F = -kx。由牛顿第二定律 ma = -kx，可得 a = -(k/m)x。与 a = -ω²x 比较，得到 ω² = k/m。

The direction of the restoring force is always towards equilibrium. Its magnitude grows linearly with displacement, which is the dynamic hallmark of SHM.

回复力的方向始终指向平衡位置。它的大小随位移线性增大，这是简谐运动的动力学特征。

In the vertical spring-mass system, gravity shifts the equilibrium position to x₀ = mg/k. The force law relative to this new equilibrium is still F = -kx’, where x’ is displacement from the new equilibrium.

在竖直弹簧振子中，重力使平衡位置移动到 x₀ = mg/k。相对这个新平衡位置，力的规律依然是 F = -kx’，其中 x’ 是相对新平衡位置的位移。

4. Period and Frequency | 周期与频率

The angular frequency is related to the period by ω = 2π/T. For a spring-block system, the period is T = 2π√(m/k), independent of amplitude.

角频率与周期的关系为 ω = 2π/T。对于弹簧振子，周期 T = 2π√(m/k)，与振幅无关。

For a simple pendulum with small amplitude, the period is T = 2π√(L/g), where L is the length of the pendulum and g is gravitational field strength.

对于小角度单摆，周期 T = 2π√(L/g)，其中 L 是摆长，g 是重力场强度。

Key facts to remember: the spring-mass period increases with mass and decreases with spring constant; the pendulum period increases with length and decreases with g, but does not depend on mass or amplitude (for small angles).

需要记住的关键结论：弹簧振子的周期随质量增大而增大，随劲度系数增大而减小；单摆周期随摆长增大而增大，随 g 增大而减小，但与质量和小振幅无关。

In experiments, the period is often determined by timing multiple oscillations, say 20 cycles, then dividing by the number of cycles to reduce timing error.

在实验中，通常通过计时多个振荡周期（例如20次全振动）再除以次数来测量周期，以减小计时误差。

5. Energy in SHM | 简谐运动的能量

The total mechanical energy in SHM is constant: E = ½ kA² = ½ mω²A². It is conserved if no damping exists.

简谐运动中的总机械能守恒：E = ½ kA² = ½ mω²A²。在没有阻尼的情况下保持不变。

At any displacement x, the kinetic energy is K = ½ mω²(A² – x²) and the potential energy is U = ½ kx². The sum always equals E.

在任意位移 x 处，动能 K = ½ mω²(A² – x²)，势能 U = ½ kx²。两者之和恒等于 E。

At the equilibrium position, all energy is kinetic: K_max = ½ mω²A². At the extremes, all energy is potential: U_max = ½ kA².

在平衡位置，能量全部为动能：K_max = ½ mω²A²；在极端位置，能量全部为势能：U_max = ½ kA²。

Energy conversion can also be visualized on a graph of K and U versus time: both oscillate at twice the frequency of the motion, while the total energy remains a horizontal line.

动能和势能随时间变化的图像显示它们以运动频率的两倍振荡，而总能量则保持水平直线。

6. Graphical Representation | 图像分析

The x-t graph is a cosine curve, the v-t graph is a negative sine curve, and the a-t graph is a negative cosine curve. Velocity leads displacement by π/2, and acceleration is in antiphase with displacement.

x-t 图像是余弦曲线，v-t 图像是负正弦曲线，a-t 图像是负余弦曲线。速度比位移超前 π/2，加速度与位移反相。

On these graphs, the slope of x-t gives velocity, and the slope of v-t gives acceleration. Make sure to check units and scale before reading any values.

在图像中，x-t 图的斜率表示速度，v-t 图的斜率表示加速度。读取数据前务必确认单位与标度。

Energy-time graphs show K and U oscillating between 0 and E, each completing two cycles of oscillation during one displacement cycle. The frequency of energy oscillation is 2f.

能量-时间图像显示 K 和 U 在 0 与 E 之间振荡，每个位移周期内各完成两个循环。能量振荡频率为 2f。

7. Typical Models: Spring and Pendulum | 典型模型：弹簧振子与单摆

The horizontal spring-mass model is the simplest: the equilibrium is at the natural length of the spring, and motion is purely horizontal. All SHM equations apply directly.

水平弹簧振子模型最简单：平衡位置在弹簧原长处，运动纯水平，所有简谐运动方程直接适用。

The vertical spring-mass model has gravity shifting the equilibrium downward by x₀ = mg/k. If you measure displacement from this new equilibrium, the motion is identical to the horizontal case.

竖直弹簧振子模型中，重力使平衡位置下移 x₀ = mg/k。若从新平衡位置测量位移，其运动与水平情形完全一致。

The simple pendulum is SHM only for small angular displacements, where θ < about 10° (0.17 rad). The restoring force is the tangential component of weight: F = -mg sinθ ≈ -mgθ = -(mg/L)x.

单摆仅在小角度（θ 约小于10°，即0.17 rad）下才做简谐运动。回复力是重力的切向分量：F = -mg sinθ ≈ -mgθ = -(mg/L)x。

For a pendulum, the mass of the bob and the amplitude do not affect the period, but the length does. This is exploited in pendulum clocks and in measuring g.

对于单摆，摆球质量和振幅不影响周期，但摆长影响周期。这被用于摆钟设计和测量 g 的实验。

8. Damping, Forced Oscillations and Resonance | 阻尼、受迫振动与共振

Damping is the loss of energy from an oscillating system due to resistive forces such as friction and air resistance. As a result, the amplitude decays exponentially with time.

阻尼是由摩擦、空气阻力等阻力导致系统能量损失的现象。因此，振幅随时间呈指数衰减。

In lightly damped systems, the period is slightly larger than the undamped period, but for small damping the change is often negligible. The total energy decreases continuously, converting into thermal energy.

在弱阻尼系统中，周期略大于无阻尼周期，但小阻尼时变化常可忽略。总能量不断减少，转化为内能。

A forced oscillation occurs when an external periodic driving force is applied. The system then oscillates at the driving frequency, which may differ from its natural frequency.

受迫振动发生在施加周期性外部驱动力时。此时系统以驱动频率振动，该频率可能与固有频率不同。

Resonance is the condition when the driving frequency equals the natural frequency. The amplitude reaches a maximum, and energy transfer into the system is most efficient.

共振是指驱动频率等于系统固有频率的状态。此时振幅达到最大，能量输入效率也最高。

Resonance can be useful, such as in tuning a radio or in microwave heating, but it can also be destructive, as in the famous Tacoma Narrows bridge collapse. Understanding resonance helps in designing safe structures.

共振可以是有益的，如收音机

Published by TutorHao | IB Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

📚 IB Physics: Complete Guide to Gas Laws | IB物理：气体定律全解析

Gas laws are a core topic in the IB Physics Thermal Physics section. They connect the macroscopic variables of pressure, volume, temperature, and amount of gas, and they also link to the microscopic motion of molecules through kinetic theory. This guide covers every gas law you need for your IB exams, including the ideal gas equation, kinetic theory calculations, and real-gas limitations.

气体定律是 IB 物理热学部分的核心内容。它将气体的宏观变量——压强、体积、温度和物质的量——联系起来，并通过气体动理论将宏观规律与分子的微观运动相结合。本指南涵盖你在 IB 考试中需要掌握的所有气体定律，包括理想气体方程、动理论计算以及真实气体的局限性。

1. The Ideal Gas Model | 理想气体模型

The ideal gas is a theoretical model built on several assumptions: gas molecules have negligible volume compared to the container; there are no intermolecular forces except during brief elastic collisions; all collisions are perfectly elastic; the molecules are in constant random motion; and the time spent in collisions is negligible compared with the time between collisions.

理想气体是一种理论模型，它建立在若干假定之上：与容器的体积相比，气体分子本身的体积可以忽略；除短暂弹性碰撞外，分子间没有相互作用力；所有碰撞都是完全弹性的；分子处于永不停息的无规则运动中；碰撞的持续时间远小于两次碰撞之间的时间。

These assumptions simplify the mathematics greatly and give accurate predictions for real gases at high temperature and low pressure. Under other conditions, real gases deviate from ideal behaviour, as discussed in Section 10.

这些假定极大地简化了数学推导，并且能在高温低压条件下对真实气体作出准确的预测。在其他条件下，真实气体则会偏离理想行为，这将在第 10 节中讨论。

A real gas behaves most ideally when its temperature is high and its pressure is low. Raising the temperature increases molecular speeds and weakens the effect of intermolecular attractions; lowering the pressure makes the molecular volume even more negligible relative to the container.

当温度较高、压强较低时，真实气体的行为最接近理想气体。升高温度会使分子速度增大，从而削弱分子间引力的影响；降低压强则使分子本身体积相对于容器更加可以忽略。

2. Boyle’s Law | 玻意耳定律

Boyle’s law states that for a fixed mass of gas at constant temperature, the pressure is inversely proportional to the volume. The mathematical form is p₁V₁ = p₂V₂, where p is pressure and V is volume.

玻意耳定律指出：对于质量一定的气体，在温度不变时，其压强与体积成反比。其数学形式为 p₁V₁ = p₂V₂，其中 p 是压强，V 是体积。

Halving the volume doubles the concentration of molecules, so the frequency of collisions with the container walls doubles, which doubles the pressure. On a p-V graph, each constant-temperature curve is a hyperbola called an isotherm. A graph of p against 1/V is instead a straight line through the origin.

体积减半会使分子浓度加倍，分子与容器壁碰撞的频率翻倍，从而使压强也加倍。在 p-V 图上，每一条等温曲线都是双曲线，称为等温线；而 p 对 1/V 的图像则是一条过原点的直线。

Boyle’s law applies only to isothermal processes. In IB exams, you may be asked to sketch two isotherms; a higher temperature corresponds to a curve further from the origin because the product pV increases with temperature.

玻意耳定律只适用于等温过程。在 IB 考试中，你可能会被要求画出两条等温线；温度越高，对应的曲线越远离原点，因为 pV 的乘积随温度升高而增大。

3. Charles’s Law | 查理定律

Charles’s law states that for a fixed mass of gas at constant pressure, the volume is directly proportional to the absolute temperature: V₁/T₁ = V₂/T₂. The temperature must always be in kelvin.

查理定律指出：对于质量一定的气体，在压强不变时，体积与绝对温度成正比：V₁/T₁ = V₂/T₂。温度必须始终使用开尔文单位。

Using Celsius degrees in this ratio gives incorrect results. If the absolute temperature doubles from 300 K to 600 K, the volume doubles; but if the Celsius temperature changes from 27 °C to 327 °C, the ratio is not 1:2.

在此比例式中使用摄氏度会得到错误结果。若绝对温度从 300 K 升高到 600 K，体积会加倍；但若摄氏温度从 27 °C 变为 327 °C，其比值并不是 1:2。

On a V-T graph plotted in kelvin, the line passes through the origin. At constant pressure, heating a gas increases the average kinetic energy of its molecules, so they push the piston outward and the volume expands.

在以开尔文为单位的 V-T 图上，直线经过原点。在等压条件下加热气体，分子平均动能增大，分子推着活塞向外运动，体积膨胀。

4. Gay-Lussac’s Law | 盖-吕萨克定律

Gay-Lussac’s law, sometimes called the pressure-temperature law, states that for a fixed mass of gas at constant volume, the pressure is directly proportional to the absolute temperature: p₁/T₁ = p₂/T₂.

盖-吕萨克定律又称压强-温度定律，它指出：对于质量一定的气体，在体积不变时，压强与绝对温度成正比：p₁/T₁ = p₂/T₂。

When the temperature rises, the average speed of the molecules increases, so they strike the walls more often and with greater average force. Both effects increase the pressure the gas exerts on the container.

当温度升高时，分子的平均速率增大，它们撞击器壁更频繁，每次撞击的平均作用力也更大。这两种效应都会增大气体对容器施加的压强。

This law explains why sealed aerosol cans may explode when heated: the volume is approximately fixed, so the pressure rises dangerously with temperature. In IB questions, this law is often combined with the ideal gas equation to find missing quantities.

该定律解释了密封气雾罐受热后可能爆炸的原因：体积近似不变，因此压强随温度升高而危险地增大。在 IB 题目中，这条定律常与理想气体方程结合，用来求解未知量。

5. Avogadro’s Law and the Mole | 阿伏伽德罗定律与摩尔

Avogadro’s law states that equal volumes of all gases at the same temperature and pressure contain the same number of molecules. One mole of any ideal gas contains N_A = 6.02 × 10²³ particles, known as the Avogadro constant.

阿伏伽德罗定律指出：在相同的温度和压强下，相同体积的任何气体都含有相同数量的分子。任何理想气体的 1 摩尔所含的粒子数为 N_A = 6.02 × 10²³，称为阿伏伽德罗常数。

At standard temperature and pressure (273 K and 1.01 × 10⁵ Pa), one mole of an ideal gas occupies 2.24 × 10⁻² m³, which is 22.4 L. This molar volume is a useful conversion factor in stoichiometry problems.

在标准温度和压强下（273 K 和 1.01 × 10⁵ Pa），1 摩尔理想气体占据的体积为 2.24 × 10⁻² m³，即 22.4 L。这个摩尔体积是化学计量问题中常用的换算因子。

In calculations, the number of moles n can be found from n = N/N_A or n = m/M, where N is the number of molecules, m is the mass of the gas, and M is its molar mass in kilograms per mole. Master these conversions because they appear in most ideal gas questions.

在计算中，物质的量 n 可用 n = N/N_A 或 n = m/M 求得，其中 N 是分子数，m 是气体质量，M 是以千克每摩尔为单位的摩尔质量。掌握这些换算很重要，因为大多数理想气体问题都会用到它们。

6. The Ideal Gas Equation | 理想气体方程

The ideal gas equation combines Boyle’s law, Charles’s law, Gay-Lussac’s law, and Avogadro’s law into one expression:

理想气体方程把玻意耳定律、查理定律、盖-吕萨克定律和阿伏伽德罗定律合并为一个表达式：

pV = nRT

Here, p is the pressure in pascals, V is the volume in cubic metres, n is the number of moles, T is the absolute temperature in kelvin, and R = 8.31 J mol⁻¹ K⁻¹ is the molar gas constant.

其中 p 是以帕斯卡为单位的压强，V 是以立方米为单位的体积，n 是物质的量，T 是以开尔文为单位的绝对温度，R = 8.31 J mol⁻¹ K⁻¹ 是摩尔气体常量。

An equivalent form uses N, the total number of molecules: pV = NkT, where k = 1.38 × 10⁻²³ J K⁻¹ is the Boltzmann constant. The two constants are related by R = N_A k. Use the second form when quantities involve individual molecules rather than moles.

另一个等价形式使用分子总数 N：pV = NkT，其中 k = 1.38 × 10⁻²³ J K⁻¹ 是玻尔兹曼常量。两个常量之间的关系为 R = N_A k。当问题涉及单个分子而非摩尔时，使用第二种形式更方便。

Worked example: A cylinder of volume 2.0 × 10⁻³ m³ contains 0.10 mol of an ideal gas at 300 K. The pressure is p = nRT/V = 0.10 × 8.31 × 300 / (2.0 × 10⁻³) = 1.25 × 10⁵ Pa. Notice how all quantities were already in SI units.

例题：一个体积为 2.0 × 10⁻³ m³ 的气缸装有 0.10 mol 的理想气体，温度为 300 K。压强为 p = nRT/V = 0.10 × 8.31 × 300 / (2.0 × 10⁻³) = 1.25 × 10⁵ Pa。注意所有物理量都已使用国际制单位。

7. Kinetic Theory of Gases | 气体动理论

Kinetic theory connects the macroscopic pressure of a gas to the microscopic motion of its molecules. Starting from Newton’s laws and averaging over all molecules, the pressure satisfies the relation:

气体动理论将气体的宏观压强与分子微观运动联系起来。从牛顿定律出发并对所有分子求平均，压强满足如下关系：

pV = ⅓Nm⟨v²⟩

In this equation, m is the mass of one molecule, N is the number of molecules, and ⟨v²⟩ is the mean square speed of the molecules. The angle brackets denote an average over all molecules.

在这个方程中，m 是一个分子的质量，N 是分子总数，⟨v²⟩ 是分子的均方速率

Published by TutorHao | IB Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）