Tag: Physics

📚 Feynman Diagrams: Understanding and Applications in IB Physics | 费曼图的理解与应用

Feynman diagrams are a visual language used in particle physics to represent how particles interact by exchanging virtual bosons. They were introduced by Richard Feynman in the 1940s and have since become one of the most powerful tools in the Standard Model. For IB Physics HL students, Feynman diagrams appear in the particle physics topic and are used to explain interactions such as electron scattering, beta decay and quark interactions.

费曼图是粒子物理学中一种可视化的“语言”，用来表示粒子如何通过交换虚玻色子发生相互作用。它由理查德·费曼于20世纪40年代提出，从此成为标准模型中最有力的工具之一。对IB物理HL学生而言，费曼图出现在粒子物理部分，用于解释电子散射、β衰变和夸克相互作用等过程。

1. What Are Feynman Diagrams? | 什么是费曼图？

A Feynman diagram is a spacetime diagram showing the initial particles on the left, the final particles on the right, and the interaction taking place in between. Each line represents the worldline of a particle; each vertex represents an interaction. In advanced physics, each diagram can be converted into a mathematical expression that contributes to the probability amplitude of a process.

费曼图是一种时空图：初态粒子画在左侧，末态粒子画在右侧，相互作用发生在中间。每一条线代表一个粒子的“世界线”，每一个顶点代表一次基本相互作用。在高等物理中，每个费曼图都可以转换为一个数学表达式，从而贡献到某个过程的概率幅。

In IB Physics, you do not need to perform full calculations. Instead, you should be able to draw simple diagrams, identify the exchange particle, and use conservation laws to check whether a diagram is physically possible.

在IB物理中，你不需要进行完整的计算，而是应该会画出简单的费曼图、识别交换粒子，并能利用守恒定律判断一个图在物理上是否可能。

2. The Building Blocks: Lines and Vertices | 基本构件：线、顶点与交换粒子

Every Feynman diagram is built from a small number of elements. The most common types of lines are shown below.

每一个费曼图都由有限的几种基本元素构成。最常见的线段类型如下。

• Fermion lines (solid straight lines with arrows): quarks and leptons. If time flows left to right, an arrow pointing right means a particle; an arrow pointing left means an antiparticle.
• Photon and weak bosons (wavy or dashed lines): γ, W⁺, W⁻ and Z⁰ are exchange particles.
• Gluon lines (coiled or loop lines): exchange particles of the strong interaction.
• Vertices: points where one fermion line emits/absorbs a boson, or where a boson converts into a pair.

费米子线：实直线加箭头。若时间轴从左向右，箭头向右表示粒子，箭头向左表示反粒子。光子与弱玻色子用波浪线或虚线表示，如γ、W⁺、W⁻、Z⁰。胶子线通常用螺旋线表示。顶点是费米子发射或吸收玻色子的点，也可以是一个玻色子转化为正反粒子对的点。

3. Reading Time and Space Direction | 时间轴与空间方向

By convention in IB Physics, time flows from left to right. Therefore particles entering from the left are the initial state, and particles leaving to the right are the final state. The vertical axis represents space, but it is not drawn to scale.

按照IB物理中的约定，时间从左向右流动。因此，从左侧进入的粒子是初态，向右离开的粒子是末态。纵轴代表空间，但图中并不按比例绘制。

An external line touches the edge of the diagram and represents a real, observable particle. An internal line, drawn between two vertices, represents a virtual particle that exists only during the interaction. Virtual particles cannot be detected directly because according to the energy-time uncertainty relation, they exist for too short a time.

外线伸到图的最外侧，代表真实可观测的粒子。内线连接两个顶点，代表只在相互作用过程中存在的虚粒子。虚粒子无法被直接探测到，因为根据能量-时间不确定关系，它们存在的时间极短。

ΔE Δt ≈ ħ/2

This relation allows a particle to be “off mass shell” for a very short period, meaning it can temporarily have an energy/momentum relationship different from a real particle.

这意味着虚粒子允许在极短时间内“离开质壳”，即其能量和动量之间的关系暂时不符合真实粒子的要求。

4. Electromagnetic Interactions and Virtual Photons | 电磁相互作用与虚光子

The simplest electromagnetic process is the scattering of two charged particles. For example, two electrons can interact by exchanging a virtual photon. One electron emits the photon, the other electron absorbs it, and both electrons continue with different directions and speeds.

最简单的电磁过程是两个带电粒子的散射。例如，两个电子可以通过交换虚光子发生相互作用：一个电子发射虚光子，另一个电子吸收虚光子，之后两者继续运动，方向与速率发生改变。

In a Feynman diagram, this is drawn as two straight electron lines with arrow pointing to the right, joined by a wavy photon line between two vertices. At each vertex, electric charge is conserved: the electron has charge −1 before and after the vertex, and the photon has charge 0.

在费曼图中，这个过程画为两条箭头向右的电子线，中间由一条波浪形的光子线连接两个顶点。在每个顶点处，电荷守恒：电子在顶点前后的电荷都是−1，光子电荷为0。

The strength of electromagnetic interactions is determined by the fine-structure constant:

电磁相互作用的强度由精细结构常数决定：

α = e² / 4πε₀ħc ≈ 1/137

Since α is much smaller than 1, the probability of a single-photon exchange is relatively small, and diagrams with more vertices are increasingly less important.

由于α远小于1，单光子交换的概率相对较小，因此顶点数越多的图贡献会越来越小。

5. Weak Interactions: Beta Decay and Quark Flavour Change | 弱相互作用：β衰变与夸克味变

The weak interaction is the only fundamental force that can change quark flavour. This makes it responsible for beta decay and the production of neutrinos. The charged weak bosons are W⁺ and W⁻; the neutral weak boson is Z⁰.

弱相互作用是唯一能够改变夸克味的基本力。正因如此，它造成了β衰变和中微子的产生。带弱相互作用玻色子为W⁺、W⁻，中性弱玻色子为Z⁰。

Beta-minus decay is: n → p + e⁻

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📚 The Photoelectric Effect and Its Experimental Laws | IB物理：光电效应及其实验规律

The photoelectric effect is one of the most pivotal phenomena in modern physics, providing direct evidence for the quantum nature of light. This article systematically reviews the experimental observations, the failure of classical wave theory, and Einstein’s photon explanation, all of which are essential for IB Physics HL/SL examinations.

光电效应是现代物理学中最具关键意义的现象之一，它直接证明了光的量子本质。本文系统梳理光电效应的实验规律、经典波动理论的困境以及爱因斯坦的光子解释，这些内容皆为IB物理HL/SL考试的核心考点。

1. What Is the Photoelectric Effect? | 什么是光电效应？

When electromagnetic radiation with sufficient frequency shines on a metal surface, electrons can be emitted from that surface. These emitted electrons are called photoelectrons, and the phenomenon itself is termed the photoelectric effect.

当足够频率的电磁波照射金属表面时，金属表面会释放出电子。这些被释放的电子被称为光电子，这一现象本身则被称为光电效应。

It is important to distinguish photoelectrons from other types of emitted electrons: photoelectrons are specifically those liberated from a material due to the absorption of electromagnetic radiation, typically visible or ultraviolet light.

需要将光电子与其他类型的逸出电子区分开来：光电子特指因吸收电磁辐射（通常是可见光或紫外线）而从材料中释放出来的电子。

2. Experimental Apparatus | 实验装置

The classic photoelectric effect experiment uses an evacuated glass tube containing two electrodes: a photocathode (often caesium or zinc) and an anode. A variable voltage source is connected across the tube, along with a sensitive ammeter and a voltmeter.

经典的光电效应实验使用一个真空玻璃管，内部包含两个电极：一个光阴极（常用铯或锌制成）和一个阳极。可调电压源连接在管的两端，并配有灵敏的电流表和电压表。

When monochromatic light of a chosen frequency is incident on the cathode, photoelectrons are ejected. Some of these electrons travel to the anode, producing a measurable photocurrent. By adjusting the magnitude and direction of the applied voltage, the kinetic energy of the emitted electrons can be investigated.

当选定频率的单色光照射到阴极时，光电子被发射出来。其中部分电子飞向阳极，形成可测量的光电流。通过调节外加电压的大小和方向，可以研究逸出电子的动能特性。

Photocurrent I = n × e × v_d | 光电流 I = n × e × v_d

where n is the number of photoelectrons emitted per unit time reaching the anode, e is the elementary charge (1.60 × 10⁻¹⁹ C), and v_d is the effective drift speed of the electron population.

其中 n 是单位时间内到达阳极的光电子数，e 是元电荷（1.60 × 10⁻¹⁹ C），v_d 是电子群的有效漂移速度。

3. Experimental Law 1: The Existence of a Threshold Frequency | 实验规律一：截止频率的存在

For any given metal surface, there exists a minimum frequency of incident light, (f₀), below which no photoelectrons are emitted at all, regardless of the light intensity or the duration of illumination. This minimum frequency is called the threshold frequency.

对于任何给定的金属表面，存在一个最小入射光频率 (f₀)，低于此频率时无论光强多大、照射时间多长，都不会发射任何光电子。这个最小频率被称为截止频率（也称极限频率）。

f₀ = φ / h

where φ is the work function of the metal (minimum energy required to liberate an electron from the surface) and h is Planck’s constant (6.63 × 10⁻³⁴ J·s).

其中 φ 是金属的逸出功（从表面释放一个电子所需的最小能量），h 是普朗克常数（6.63 × 10⁻³⁴ J·s）。

For example, zinc has a threshold frequency of approximately 1.04 × 10¹⁵ Hz, corresponding to ultraviolet light. Visible light does not cause emission from clean zinc under normal conditions.

例如，锌的截止频率约为 1.04 × 10¹⁵ Hz，对应紫外线波段。在普通条件下，可见光无法使洁净的锌产生光电发射。

4. Experimental Law 2: Instantaneous Emission | 实验规律二：发射的瞬时性

When light of a frequency above the threshold frequency strikes the metal, photoelectrons are emitted instantaneously, within less than 10⁻⁹ seconds of illumination. There is no measurable time delay for the accumulation of energy.

当频率高于截止频率的光照射金属时，光电子瞬时被发射，延迟时间小于 10⁻⁹ 秒。不存在可测量的能量积累时间延迟。

This is a remarkable result because classical wave theory predicts that sufficient energy from a low-intensity wave would need to be accumulated over time before emission could occur. For a typical metal illuminated by faint light, this predicted delay would be several seconds or even minutes — yet no such delay is observed.

这是一个非常显著的结果，因为经典波动理论预测：对于低强度波，需要经过一段时间的能量积累后才可能发射电子。对于微弱光照下的典型金属，这一理论预测的延迟时间可能长达数秒甚至数分钟——然而实际观测中不存在任何这样的延迟。

5. Experimental Law 3: Maximum Kinetic Energy Depends on Frequency, Not Intensity | 实验规律三：最大动能取决于频率而非光强

The maximum kinetic energy of emitted photoelectrons, (K_{text{max}}), increases linearly with the frequency of the incident light, but is independent of the light intensity.

发射光电子的最大动能 (K_{text{max}}) 随入射光频率的增加而线性增大，但与光强无关。

Experimentally, this is determined using a reverse (stopping) voltage. By applying a potential difference that opposes the motion of photoelectrons, the current can be reduced to zero. The reverse voltage at which the photocurrent becomes zero is called the stopping potential, (V_s).

在实验上，这是通过反向截止电压来测量的。施加一个阻碍光电子运动的电压，可使光电流降为零。使光电流恰好为零的反向电压被称为遏止电压 (V_s)。

(K_{text{max}}) = e × (V_s)

where e is the elementary charge. Since the stopping potential can be measured precisely, the maximum kinetic energy is directly determined.

其中 e 是元电荷。由于遏止电压可以被精确测量，最大动能也因此被直接确定。

6. Experimental Law 4: Photocurrent Is Proportional to Intensity | 实验规律四：光电流与光强成正比

When the frequency of the incident light is held constant above the threshold frequency, the saturation photocurrent (the maximum photocurrent when all emitted electrons are collected) is directly proportional to the intensity of the incident light.

当入射光的频率保持恒定且高于截止频率时，饱和光电流（所有发射电子均被收集时的最大光电流）与入射光的强度成正比。

Doubling the light intensity at a fixed frequency doubles the number of photoelectrons emitted per second, and hence doubles the saturation current. However, the stopping potential (V_s) remains unchanged.

在固定频率下将光强加倍，每秒发射的光电子数量加倍，因而饱和光电流也加倍。但遏止电压 (V_s) 保持不变。

This indicates that a greater light intensity means more photons per second, each carrying the same quantum of energy hf — not photons of greater energy.

这表明更大的光强意味着每秒钟有更多光子，而每个光子携带相同的能量量子 hf——并不是单个光子的能量更大。

7. The Failure of Classical Wave Theory | 经典波动理论的失败

Classical wave theory treats light as a continuous electromagnetic wave whose energy depends on its amplitude (intensity). This theory fails to explain any of the four experimental laws above.

经典波动理论将光视为连续的电磁波，其能量取决于振幅（强度）。该理论无法解释上述任何一条实验规律。

• Threshold frequency unexplained: In wave theory, any frequency could transfer energy to electrons; given enough time, even low-frequency light should cause emission. However, experiments show that below (f₀), emission never occurs.

• 截止频率无法解释：根据波动理论，任何频率的光都能向电子传递能量；只要有足够时间，即使是低频光也应引发发射。然而实验显示，低于 (f₀) 时永远不会发生发射。

• Time delay contradiction: Classical theory predicts a measurable delay for low-intensity light, but photoelectric emission is instantaneous.

• 延迟时间矛盾：经典理论预测低强度光需要可测量的延迟，但光电发射是瞬时的。

• Kinetic energy problem: Classical theory predicts that greater intensity should yield electrons with greater kinetic energy, but experiments show (K_{text{max}}) is independent of intensity.

• 动能问题：经典理论预测更强的光应使电子获得更大的动能，但实验表明 (K_{text{max}}) 与光强无关。

Therefore, the wave model was fundamentally incompatible with the photoelectric effect — a crisis that demanded a revolutionary new viewpoint.

因此，波动模型与光电效应在根本上不兼容——这一困境呼唤着一个革命性的新视角。

8. Einstein’s Photon Hypothesis | 爱因斯坦的光子假说

In 1905, Albert Einstein proposed that electromagnetic radiation is quantised: light consists of discrete packets of energy called photons. Each photon carries energy.

1905年，阿尔伯特·爱因斯坦提出电磁辐射是量子化的：光由称为光子的离散能量包组成。每个光子携带的能量为

(E) = hf = hc / λ

where h is Planck’s constant, f is the frequency, c is the speed of light (3.00 × 10⁸ m/s), and λ is the wavelength of the radiation.

其中 h 是普朗克常数，f 是频率，c 是光速（3.00 × 10⁸ m/s），λ 是辐射波长。

The key insight is that an electron interacts with light by absorbing one whole photon at a time, not by accumulating energy from many waves. If a single photon has energy greater than or equal to the work function, the electron is liberated; otherwise, absorption simply does not occur, regardless of how many photons strike the surface.

关键的洞见在于：电子与光的交互是一次性吸收一个完整光子，而非从许多波中积累能量。如果单个光子的能量大于或等于逸出功，电子就被释放；否则吸收根本不会发生，无论有多少光子照射表面。

9. The Photoelectric Equation | 爱因斯坦光电方程

Applying the principle of conservation of energy to the single-photon absorption process yields the famous Einstein photoelectric equation:

将能量守恒原理应用于单光子吸收过程，我们得到著名的爱因斯坦光电方程：

hf = φ + (K_{text{max}})

or equivalently:

等价地：

(K_{text{max}}) = hf − φ

Here, hf is the energy of the incident photon, φ is the work function of the metal, and (K_{text{max}}) is the maximum kinetic energy of the emitted electron.

其中 hf 是入射光子的能量，φ 是金属的逸出功，(K_{text{max}}) 是发射电子的最大动能。

The work function itself is related to the threshold frequency by φ = hf₀. Substituting into the photoelectric equation gives another useful form:

逸出功本身与截止频率的关系为 φ = hf₀。代入光电方程可得到另一个有用的形式：

(K_{text{max}}) = h(f − f₀)

This equation directly predicts that (K_{text{max}}) is a linear function of frequency with slope h, consistent with experimental observations.

该方程直接预测 (K_{text{max}}) 是频率的线性函数，斜率为 h，与实验观测完全一致。

10. Graphical Analysis and the Determination of Planck’s Constant | 图像分析与普朗克常数的测定

The relationship between the stopping potential (V_s) and the frequency of incident light is a straight line. Using (e × V_s) = hf − φ, we obtain:

遏止电压 (V_s) 与入射光频率之间的关系是一条直线。利用 (e × V_s) = hf − φ，我们得到：

(V_s) = (h/e) × f − (φ/e)

Plotting (V_s) on the vertical axis and f on the horizontal axis gives a straight line with slope h/e and y-intercept −φ/e. The x-intercept equals the threshold frequency f₀.

以 (V_s) 为纵轴、f 为横轴作图，得到一条直线，其斜率为 h/e，纵轴截距为 −φ/e，横轴截距等于截止频率 f₀。

Since e is known to be 1.60 × 10⁻¹⁹ C, the slope of the (V_s)-f graph allows physicists to measure Planck’s constant with high precision — a powerful experimental confirmation of quantum theory.

由于 e 是已知量（1.60 × 10⁻¹⁹ C），(V_s)-f 图像的斜率使物理学家能够以高精度测量普朗克常数——这是对量子理论的强有力实验证实。

11. Worked Example | 典型例题解析

Problem: Light of wavelength 250 nm is incident on a metal surface with a work function of 2.80 eV. Determine: (a) the photon energy in eV; (b) the maximum kinetic energy of the emitted electrons in eV; (c) the stopping potential.

例题：波长为250 nm的光照射在逸出功为2.80 eV的金属表面上。求：(a) 光子能量（以eV为单位）；(b) 发射电子的最大动能（以eV为单位）；(c) 遏止电压。

Solution:

解答：

(a) First convert the wavelength to frequency: f = c/λ = (3.00 × 10⁸) ÷ (250 × 10⁻⁹) = 1.20 × 10¹⁵ Hz. Then the photon energy is (E) = hf = (6.63 × 10⁻³⁴) × (1.20 × 10¹⁵) = 7.96 × 10⁻¹⁹ J. Converting to eV: 7.96 × 10⁻¹⁹ ÷ (1.60 × 10⁻¹⁹) = 4.97 eV.

(a) 先换算波长到频率：f = c/λ = (3.00 × 10⁸) ÷ (250 × 10⁻⁹) = 1.20 × 10¹⁵ Hz。则光子能量为 (E) = hf = (6.63 × 10⁻³⁴) × (1.20 × 10¹⁵) = 7.96 × 10⁻¹⁹ J。换算为eV：7.96 × 10⁻¹⁹ ÷ (1.60 × 10⁻¹⁹) = 4.97 eV。

(b) Using the photoelectric equation: (K_{text{max}}) = hf − φ = 4.97 − 2.80 = 2.17 eV.

(b) 使用光电方程：(K_{text{max}}) = hf − φ = 4.97 − 2.80 = 2.17 eV。

(c) The stopping potential is given by (V_s) = (K_{text{max}}) / e = 2.17 V.

(c) 遏止电压为 (V_s) = (K_{text{max}}) / e = 2.17 V。

12. Applications and Significance | 应用与深远意义

The photoelectric effect has numerous technological applications and played a central role in the development of quantum mechanics. Some common applications include:

光电效应具有众多技术应用，并在量子力学的发展中发挥了核心作用。一些常见的应用包括：

• Photoelectric cells (phototubes): Used in automatic lighting controls, exposure meters, and burglar alarm systems.

• 光电管：用于自动照明控制、曝光计和防盗报警系统。

• Image sensors: Digital cameras and night-vision devices rely on the conversion of light to electrical signals.

• 图像传感器：数码相机和夜视设备依赖于将光转换为电信号。

• Solar panels: Although these primarily use the photovoltaic effect, the underlying photon-electron interaction principle is closely related.

• 太阳能电池板：虽然主要利用光伏效应，但其基本的光子-电子交互原理密切相关。

Beyond applications, the photoelectric effect provided decisive evidence for the particle nature of light, complementing interference and diffraction experiments that support the wave nature. This dual nature — wave-particle duality — is a cornerstone of modern physics and a central theme in the IB curriculum.

除了应用之外，光电效应为光的粒子性提供了决定性的证据，与支持波动性的干涉和衍射实验相辅相成。这种双重性质——波粒二象性——是现代物理学的基石，也是IB课程的核心主题。

Achieving a deep conceptual understanding of the experimental laws and Einstein’s explanation is essential for solving IB exam questions on this topic. Master the graphs, memorise the key equations, and always verify units — the photoelectric effect is one of the most rewarding topics to study.

深入理解实验规律和爱因斯坦的解释，对于解答IB考试中关于该主题的题目至关重要。掌握图像、熟记关键方程，并始终检查单位——光电效应是最值得深入学习的主题之一。

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📚 IB Physics: Atoms and Nuclear Physics Knowledge Points Summary | IB物理：原子和核物理知识点汇总

Atomic and nuclear physics forms a core component of the IB Physics syllabus, connecting the microscopic world of subatomic particles to macroscopic phenomena such as energy generation and medical imaging. This article consolidates the essential knowledge points from this topic, providing a structured revision guide that aligns closely with IB assessment requirements.

原子和核物理是IB物理课程的核心组成部分，它将亚原子粒子的微观世界与能量产生、医学成像等宏观现象联系起来。本文系统梳理了该主题的关键知识点，提供一份与IB考试要求紧密对齐的结构化复习指南。

1. Atomic Structure and the Rutherford Model | 原子结构与卢瑟福模型

The modern understanding of atomic structure begins with Rutherford’s gold foil experiment in 1911. A beam of alpha particles was directed at a thin gold foil, and the scattering patterns revealed that most particles passed straight through, while a very small fraction were deflected at large angles — some even reflected back. This evidence led to the conclusion that the atom consists of a tiny, dense, positively charged nucleus surrounded by mostly empty space, with electrons orbiting at relatively large distances.

对原子结构的现代认识始于1911年的卢瑟福金箔实验。一束α粒子射向薄金箔，散射图样显示大多数粒子径直穿过，而极少数粒子发生大角度偏转——甚至有的被反弹回来。这一证据表明，原子由微小、致密、带正电的原子核构成，核外大部分是空的空间，电子在较远距离上绕核运动。

Key properties of the atomic nucleus to remember:

需要牢记的原子核关键性质如下：

• The nucleus contains protons (positive) and neutrons (neutral), collectively called nucleons.
• 原子核包含质子（带正电）和中子（电中性），统称为核子。
• The atomic number Z equals the number of protons; the mass number A equals the total number of nucleons.
• 原子序数Z等于质子数；质量数A等于核子总数。
• The nucleus has a radius on the order of 10⁻¹⁵ m, while the atom itself has a radius of about 10⁻¹⁰ m — meaning the nucleus is roughly 100,000 times smaller than the atom.
• 原子核的半径约为10⁻¹⁵ m，而原子本身半径约为10⁻¹⁰ m——这意味着原子核比整个原子小约10万倍。
• Almost all of the atom’s mass is concentrated in the nucleus due to the relatively large mass of nucleons.
• 由于核子质量较大，原子几乎全部质量都集中在原子核中。

2. Atomic Energy Levels and Spectra | 原子能级与光谱

Electrons within an atom can only occupy specific discrete energy levels, a concept central to the Bohr model. When an electron transitions from a higher energy level to a lower one, it emits a photon of energy equal to the difference between the two levels. Conversely, the absorption of a photon with exactly the right energy can excite an electron to a higher level.

原子中的电子只能占据特定的离散能级，这是玻尔模型的核心概念。当电子从高能级跃迁到低能级时，会发射一个光子，其能量等于两个能级之差。反之，吸收恰好具有相应能量的光子能使电子激发到更高能级。

ΔE = Eₕᵢᵍₕ − Eₗₒʷ = hf = hc/λ

where h is Planck’s constant (6.63 × 10⁻³⁴ J·s), f is the photon frequency, c is the speed of light, and λ is the photon wavelength. The discrete nature of these energy levels explains why atomic emission and absorption spectra consist of sharp spectral lines rather than continuous bands — each line corresponds to a specific electronic transition.

其中h是普朗克常数（6.63 × 10⁻³⁴ J·s），f是光子频率，c是光速，λ是光子波长。能级的离散性解释了为什么原子发射光谱和吸收光谱由尖锐的谱线而非连续谱带构成——每条谱线对应一次特定的电子跃迁。

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

对于氢原子，第n能级的能量由下式给出：

Eₙ = −13.6 eV / n² (n = 1, 2, 3, …)

The ground state (n = 1) has energy −13.6 eV; the negative sign indicates that the electron is bound to the nucleus. The ionization energy of hydrogen from its ground state is therefore +13.6 eV.

基态（n = 1）的能量为−13.6 eV；负号表示电子被束缚在原子核周围。因此，氢原子从基态电离所需的电离能为+13.6 eV。

3. Wave–Particle Duality and de Broglie Wavelength | 波粒二象性与德布罗意波长

Matter exhibits wave-like properties, as proposed by Louis de Broglie in 1924. Every particle with momentum p has an associated wavelength. This concept is essential for understanding why electron energies in atoms are quantized: the electron behaves as a standing wave around the nucleus, and only certain orbital circumferences can accommodate a whole number of wavelengths.

1924年，路易·德布罗意提出物质具有波动性。每个具有动量p的粒子都伴随一个特征波长。这一概念对于理解原子中电子能量为何量子化至关重要：电子表现为围绕原子核的驻波，只有某些满足整数倍波长的轨道周长才被允许。

λ = h/p = h/(mv)

where m is the particle mass and v is its velocity. Electrons accelerated through a potential difference V acquire kinetic energy eV; their de Broglie wavelength is then given by λ = h/√(2meV). Electron diffraction experiments confirm this wave-like behaviour. The IB syllabus often asks students to calculate the de Broglie wavelength of electrons or protons and to relate it to the scale of atomic spacing.

其中m为粒子质量，v为其速度。电子经电势差V加速后获得的动能为eV；其德布罗意波长为λ = h/√(2meV)。电子衍射实验证实了这种波动行为。IB大纲常要求学生计算电子或质子的德布罗意波长，并将其与原子间距的尺度联系起来。

4. Radioactive Decay: Types and Properties | 放射性衰变：类型与性质

Radioactive decay is a random, spontaneous process in which an unstable nucleus transforms into a more stable configuration. The three primary types of decay are alpha (α), beta (β), and gamma (γ), each with distinct characteristics.

放射性衰变是原子核自发、随机地转变为更稳定状态的过程。三种主要衰变类型为α衰变、β衰变和γ衰变，各自具有不同的特征。

The alpha particle has the largest mass and charge, causing intense ionization along a short path. Beta particles are lighter and penetrate farther. Gamma rays are the most penetrating but ionize least effectively.

α粒子质量和电荷最大，在短路径上产生强烈电离。β粒子较轻，穿透更深。γ射线穿透力最强，但电离效率最低。

5. Nuclear Decay Equations | 核衰变方程

Writing balanced nuclear equations is a fundamental skill tested in IB Physics. In any decay equation, both the total mass number A and the total atomic number Z must be conserved.

书写配平的核衰变方程是IB物理考查的基本技能。在任何衰变方程中，总质量数A和总原子序数Z都必须守恒。

Alpha decay: The parent nucleus loses 2 protons and 2 neutrons.

α衰变：母核失去2个质子和2个中子。

ᵤᵃX → ᵤ₋₂ᵃ⁻⁴Y + ₂⁴He

Example: Uranium-238 undergoes alpha decay to thorium-234:

示例：铀-238发生α衰变生成钍-234：

₉₂²³⁸U → ₉₀²³⁴Th + ₂⁴He

Beta-minus decay: A neutron converts into a proton, emitting an electron and an antineutrino.

β⁻衰变：一个中子转化为质子，同时发射一个电子和一个反中微子。

ₙ⁰ → ₚ⁺ + e⁻ + ν̄ₑ

Beta-plus decay: A proton converts into a neutron, emitting a positron and a neutrino.

β⁺衰变：一个质子转化为中子，同时发射一个正电子和一个中微子。

ₚ⁺ → ₙ⁰ + e⁺ + νₑ

Gamma emission usually accompanies other decay modes, releasing excess energy from the daughter nucleus without changing A or Z.

γ发射通常伴随其他衰变模式，释放子核的多余能量，但不改变A或Z。

6. Half-Life and Radioactive Decay Law | 半衰期与放射性衰变定律

Radioactive decay follows first-order kinetics. The number of undecayed nuclei N at time t is related to the initial number N₀ by an exponential decay law. The half-life T₁/₂ is the time required for half of the original nuclei to decay, and it is a characteristic constant for each radioactive isotope, independent of external conditions such as temperature or pressure.

放射性衰变遵循一级动力学规律。t时刻未衰变的原子核数N与初始核数N₀之间满足指数衰变定律。半衰期T₁/₂是原始原子核衰变一半所需的时间，它是每种放射性同位素的特征常数，与温度、压强等外部条件无关。

N = N₀ (1/2)^(t/T₁/₂)

Equivalent forms using the decay constant λ:

使用衰变常数λ的等价形式：

N = N₀e^(−λt) and T₁/₂ = ln 2 / λ

The rate of decay, or activity A, is defined as A = λN and is measured in becquerels (Bq), where 1 Bq = 1 decay per second. Students should be comfortable using these equations to determine the age of archaeological samples (carbon dating) or to calculate remaining activity after a given time.

衰变速率，即活度A，定义为A = λN，单位为贝可勒尔（Bq），1 Bq = 每秒1次衰变。学生应熟练掌握利用这些方程确定考古样品的年龄（碳定年法）或计算经过给定时间后剩余的活度。

7. Nuclear Binding Energy and Mass Defect | 核结合能与质量亏损

The mass of a nucleus is always less than the sum of the masses of its individual nucleons. This difference, called the mass defect Δm, is converted into the binding energy that holds the nucleus together. Einstein’s mass–energy equivalence relates the two quantities.

原子核的质量总是小于其组成核子各自的的质量之和。这个差值称为质量亏损Δm，它转化为将原子核束缚在一起的结合能。爱因斯坦的质能等价关系将两者联系起来。

ΔE = Δmc²

where c = 3.00 × 10⁸ m/s. In nuclear physics, masses are often expressed in atomic mass units (u), where 1 u = 1.66 × 10⁻²⁷ kg ≈ 931.5 MeV/c².

其中c = 3.00 × 10⁸ m/s。在核物理中，质量常用原子质量单位（u）表示，1 u = 1.66 × 10⁻²⁷ kg ≈ 931.5 MeV/c²。

For a nucleus ᵤᵃX with Z protons and (A − Z) neutrons:

对于具有Z个质子和(A − Z)个中子的原子核ᵤᵃX：

Δm = Z·mₚ + (A − Z)·mₙ − mₙᵤcₗₑᵤₛ

The binding energy per nucleon, ΔE/A, is a measure of nuclear stability. The binding energy per nucleon curve peaks around iron-56 (approximately 8.8 MeV per nucleon), indicating that iron is the most stable nucleus. This curve explains why energy is released in both nuclear fission (splitting heavy nuclei) and nuclear fusion (combining light nuclei): both processes move the products toward the region of maximum binding energy per nucleon.

每个核子的结合能ΔE/A是衡量原子核稳定性的指标。每个核子的结合能曲线在铁-56附近达到峰值（约8.8 MeV/核子），表明铁是最稳定的原子核。这条曲线解释了为什么核裂变（重核分裂）和核聚变（轻核聚合）都能释放能量：两种过程都将产物推向每核子结合能最大的区域。

8. Nuclear Fission and Fusion | 核裂变与核聚变

Nuclear fission occurs when a heavy nucleus (such as uranium-235 or plutonium-239) absorbs a slow neutron and splits into two smaller nuclei, emitting two or three additional neutrons and releasing a large amount of energy. A typical fission reaction of uranium-235 is:

核裂变发生在重原子核（如铀-235或钚-239）吸收一个慢中子后分裂成两个较小的核，同时发射两到三个额外中子并释放大量能量。铀-235的典型裂变反应为：

₉₂²³⁵U + ₀¹n → ₅₆¹⁴⁴Ba + ₃₆⁸⁹Kr + 3₀¹n + Energy

The emitted neutrons can trigger further fission events, producing a chain reaction. In a nuclear reactor, control rods absorb excess neutrons to maintain a steady rate of fission; the energy released heats a coolant, which drives turbines to generate electricity.

发射的中子可以触发进一步的裂变事件，产生链式反应。在核反应堆中，控制棒吸收多余中子以维持稳定的裂变速率；释放的能量加热冷却剂，驱动涡轮机发电。

Nuclear fusion is the process in which two light nuclei combine to form a heavier nucleus. Fusion of hydrogen isotopes — deuterium (²H) and tritium (³H) — is the most promising for energy production:

核聚变是两个轻原子核结合形成一个较重原子核的过程。氢同位素——氘（²H）和氚（³H）的聚变是能量生产中最有前景的反应：

₁²H + ₁³H → ₂⁴He + ₀¹n + 17.6 MeV

Fusion requires extremely high temperatures (about 10⁸ K) to overcome the electrostatic repulsion between positively charged nuclei. Unlike fission, fusion produces no long-lived radioactive waste, making it an attractive — though technically challenging — energy source. In the Sun, gravitational pressure and high temperatures sustain the fusion of hydrogen into helium.

聚变需要极高的温度（约10⁸ K）以克服带正电原子核之间的静电排斥力。与裂变不同，聚变不产生长寿命放射性废物，因此是一种很有吸引力——尽管在技术上极具挑战性——的能源。在太阳内部，引力压强和高温维持着氢聚变为氦的过程。

9. Applications and Risks of Nuclear Radiation | 核辐射的应用与风险

Radioactive isotopes are widely used across medicine, industry, and research. The same radiation that provides these benefits also poses health risks, making a thorough understanding of both aspects essential for the IB candidate.

放射性同位素广泛应用于医学、工业和研究领域。带来这些益处的辐射也同时构成健康风险，因此全面理解利弊两方面对IB考生至关重要。

Common applications:

常见应用：

• Medical imaging: Technetium-99m emits gamma rays and is used as a tracer in organ imaging (e.g., bone scans).
• 医学成像：锝-99m发射γ射线，用作器官成像（如骨扫描）中的示踪剂。
• Cancer treatment: Cobalt-60 produces gamma radiation that damages and destroys tumour tissue.
• 癌症治疗：钴-60产生γ辐射，破坏和杀死肿瘤组织。
• Carbon dating: The ratio of carbon-14 to carbon-12 in organic material indicates its age up to about 50,000 years.
• 碳定年法：有机材料中碳-14与碳-12的比例可指示其年龄，最长约5万年。
• Industrial gauging: Beta sources measure the thickness of paper, plastic, or metal sheets during production.
• 工业测量：β源在生产过程中测量纸张、塑料或金属薄板的厚度。

Health risks and protection: Ionizing radiation can remove electrons from atoms in living tissue, causing DNA damage that may lead to cancer or cell death. The three fundamental principles of radiation protection are time (minimize exposure duration), distance (maximize distance from the source), and shielding (use appropriate absorbing materials between the source and the body).

健康风险与防护：电离辐射可以从活体组织中的原子中移除电子，造成DNA损伤，可能导致癌症或细胞死亡。辐射防护的三条基本原则是：时间（尽量缩短照射时间）、距离（尽量远离辐射源）和屏蔽（在辐射源与人体之间使用合适的吸收材料）。

The absorbed dose is measured in grays (Gy), where 1 Gy = 1 J/kg. The equivalent dose accounts for the biological effect of different radiation types, with the weighting factor being 1 for gamma and beta radiation and up to 20 for alpha particles due to their high ionizing power.

吸收剂量以戈瑞（Gy）为单位，1 Gy = 1 J/kg。当量剂量则考虑了不同类型辐射的生物效应，其中γ和β辐射的权重因子为1，而α粒子因其强电离能力权重因子可达20。

This comprehensive overview of atomic and nuclear physics covers the core concepts and equations required for IB Physics Paper 1 and Paper 2. Practice writing balanced nuclear equations, interpreting binding energy curves, and applying the exponential decay law to quantitative problems — these are the most frequently tested skills in this topic. A confident grasp of these fundamentals will serve you well in the examination and in understanding the physics that shapes modern technology.

本综述覆盖了IB物理试卷1和试卷2所需的原子和核物理核心概念与方程。练习书写配平的核方程、解读结合能曲线以及应用指数衰变定律解决定量问题——这些是本主题中最常考查的技能。扎实掌握这些基础知识，将在考试中以及理解塑造现代技术的物理原理时为你提供有力支持。

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📚 IB Physics: Nuclear Reaction Equations and Energy Changes | IB物理：核反应方程式与能量变化

In nuclear physics, reactions are described by nuclear equations that conserve both nucleon number and charge. These equations allow us to calculate the energy released or absorbed during a transformation using Einstein’s mass-energy equivalence.

在核物理中，核反应通过核反应方程式来描述，这些方程式同时遵守核子数守恒和电荷守恒。借助爱因斯坦的质能关系，我们可以计算核转变过程中释放或吸收的能量。

1. Nuclear Reaction Equations Basics | 核反应方程式基础

A nuclear equation represents the transformation of one nucleus into another. It must balance the total mass number (A) and the total atomic number (Z) on both sides of the arrow. This reflects the conservation of nucleons and charge.

核反应方程式表示一个原子核转变为另一个原子核的过程。方程两边必须满足总质量数(A)和总电荷数(Z)守恒，这体现了核子数和电荷守恒定律。

For example, alpha decay of uranium-238 can be written as:

例如，铀-238的α衰变可写为：

²³⁸U → ²³⁴Th + ⁴He

Here, the mass numbers (238 = 234 + 4) and atomic numbers (92 = 90 + 2) balance exactly. In beta-minus decay, a neutron converts into a proton, and an electron and antineutrino are emitted.

这里，质量数（238 = 234 + 4）和原子序数（92 = 90 + 2）完全相等。在β⁻衰变中，一个中子转化为一个质子，同时释放一个电子和反中微子。

2. Mass–Energy Equivalence and Energy Changes | 质能等价与能量变化

Every nuclear reaction changes the total rest mass of the particles. According to Einstein’s relation, E = mc², a change in mass Δm corresponds to a change in energy ΔE given by:

每个核反应都会改变粒子的总静止质量。根据爱因斯坦关系式 E = mc²，质量变化Δm对应能量变化ΔE：

ΔE = Δm c²

If the final mass is smaller than the initial mass, Δm is negative, and energy is released. If the final mass is larger, energy must be supplied from outside.

若末态质量小于初态质量，Δm为负，反应释放能量；若末态质量更大，则需要外界提供能量。

In nuclear reactions, Δm is often called the mass defect, and the released energy is the Q-value of the reaction.

在核反应中，Δm常被称为质量亏损，释放的能量称为反应的Q值。

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

The mass of a stable nucleus is always slightly less than the sum of the masses of its individual protons and neutrons. This difference is the mass defect, and the equivalent energy is the binding energy of the nucleus.

稳定原子核的质量总是略小于其组成质子与中子质量之和。这个差值就是质量亏损，其对应的能量即为原子核的结合能。

Binding energy is the energy required to separate a nucleus into its individual nucleons. A higher average binding energy per nucleon implies a more stable nucleus. The binding-energy-per-nucleon curve peaks around iron-56.

结合能是将原子核拆分为独立核子所需的能量。每个核子的平均结合能越高，原子核越稳定。核子平均结合能曲线在铁-56附近达到峰值。

The mass defect can be calculated as:

质量亏损可计算为：

Δm = (Z m_p + N m_n) − m_nucleus

where m_p is the proton mass, m_n the neutron mass, and Z and N are the numbers of protons and neutrons.

其中m_p为质子质量，m_n为中子质量，Z和N分别为质子数和中子数。

4. Calculating Q-values | 计算Q值

The Q-value of a nuclear reaction is the total energy released or absorbed. It is defined as:

核反应的Q值是反应释放或吸收的总能量，定义为：

Q = (m_initial − m_final) c²

If Q > 0, the reaction is exothermic (energy released). If Q < 0, the reaction is endothermic (energy absorbed).

若Q > 0，反应放热（释放能量）；若Q < 0，反应吸热（吸收能量）。

In practice, masses are often given in atomic mass units (u). The conversion factor is:

实际计算中，质量通常以原子质量单位(u)给出，换算关系为：

1 u = 931.5 MeV/c²

Thus, Q can be computed in MeV by multiplying the mass defect (in u) by 931.5.

因此，将质量亏损（以u为单位）乘以931.5即可得到以MeV为单位的Q值。

5. Nuclear Fission | 核裂变

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

核裂变是指重核（如铀-235）吸收一个中子后分裂成较轻的核，同时释放能量和更多中子的过程。一个典型的裂变反应为：

²³⁵U + ¹n → ¹⁴¹Ba + ⁹²Kr + 3 ¹n

The total mass of the products is less than the total mass of the reactants; the missing mass is converted into kinetic energy of the fragments and neutrons. For uranium-235, the energy released is about 200 MeV per fission.

生成物总质量小于反应物总质量，亏损的质量转化为碎片和中子的动能。对铀-235而言，每次裂变释放约200 MeV能量。

The emitted neutrons can go on to cause further fissions, leading to a self-sustaining chain reaction, which is the basis of nuclear reactors and weapons.

释放的中子可继续引发更多裂变，形成自持链式反应，这是核反应堆和核武器的基础。

6. Nuclear Fusion | 核聚变

Nuclear fusion combines light nuclei into a heavier nucleus, releasing energy due to the strong force. An important example is the fusion of deuterium and tritium:

核聚变将轻核结合成较重的核，由强相互作用释放能量。一个重要的例子是氘和氚的聚变：

²H + ³H → ⁴He + ¹n + 17.6 MeV

This reaction releases 17.6 MeV of energy, which is much larger per nucleon than fission. Fusion requires extremely high temperatures (about 10⁸ K) so that nuclei have enough kinetic energy to overcome electrostatic repulsion.

该反应释放17.6 MeV能量，比裂变每个核子的能量大得多。聚变需要极高的温度（约10⁸ K），使核子获得足够动能克服静电斥力。

Fusion is the process that powers the Sun and other stars. It produces far less radioactive waste than fission, but achieving controlled fusion on Earth remains a major challenge.

聚变是太阳及其他恒星的能量来源。它比裂变产生少得多的放射性废物，但在地球上实现受控聚变仍是一个重大挑战。

7. Radioactive Decay and Energy | 放射性衰变与能量

Radioactive decay is a spontaneous nuclear reaction. In α-decay, the nucleus emits an alpha particle (⁴He nucleus); in β-decay, it emits an electron or positron; in γ-decay, it releases excess energy as a photon.

放射性衰变是一种自发核反应。α衰变放出α粒子（⁴He核）；β衰变放出电子或正电子；γ衰变则以光子形式释放多余能量。

The energy released in alpha decay appears as the kinetic energy of the alpha particle and the recoil nucleus. In beta decay, the energy is shared between the electron, the antineutrino, and the recoil nucleus, producing a continuous energy spectrum.

α衰变释放的能量表现为α粒子和反冲核的动能。在β衰变中，能量分布在电子、反中微子和反冲核之间，形成连续能谱。

Gamma rays have no mass or charge, so gamma emission does not change the mass number or atomic number of the nucleus.

γ射线没有质量和电荷，因此γ发射不改变原子核的质量数和原子序数。

8. Worked Example: Q-value Calculation | 例题：Q值计算

Consider the reaction ²³⁵U + ¹n → ¹⁴¹Ba + ⁹²Kr + 3 ¹n. Given that the atomic masses are:

考虑反应 ²³⁵U + ¹n → ¹⁴¹Ba + ⁹²Kr + 3 ¹n。已知原子质量：

Calculate the energy released.

计算释放的能量。

Initial mass = 235.0439 + 1.0087 = 236.0526 u
Final mass = 140.9144 + 91.9262 + 3(1.0087) = 235.8667 u
Mass defect = 236.0526 − 235.8667 = 0.1859 u

初始质量 = 235.0439 + 1.0087 = 236.0526 u
末态质量 = 140.9144 + 91.9262 + 3(1.0087) = 235.8667 u
质量亏损 = 236.0526 − 235.8667 = 0.1859 u

Q = 0.1859 × 931.5 ≈ 173 MeV.

Q = 0.1859 × 931.5 ≈ 173 MeV。

9. Common Pitfalls and Exam Tips | 常见陷阱与考点

One common mistake is forgetting to include the mass of the neutron in the initial mass when the reaction starts with a neutron. Always account for all particles on both sides.

常见错误之一是当反应物包含中子时，忘记在初始质量中加入中子质量。一定要把方程两边的所有粒子都计算在内。

Another issue is using the wrong units. If masses are given in u, convert the mass defect to energy using 1 u = 931.5 MeV/c². If masses are given in kg, use E = mc² directly with c = 3.00 × 10⁸ m/s.

另一个问题是单位使用错误。若质量以u给出，用 1 u = 931.5 MeV/c² 换算；若质量以kg给出，则直接用 E = mc²，光速c = 3.00 × 10⁸ m/s。

Remember that binding energy is always positive, while Q can be positive or negative. Also, check that nucleon number and charge are conserved in every equation.

记住：结合能均为正值，而Q值可正可负。同时，要检查每个方程中的核子数和电荷是否守恒。

10. Summary | 小结

Nuclear reaction equations are powerful tools for understanding the energy changes in nuclear processes. By applying mass-energy equivalence, we can calculate the energy released from the mass defect in fission, fusion, and radioactive decay.

核反应方程式是理解核过程中能量变化的强大工具。通过质能等价，我们可以从裂变、聚变和放射性衰变的质量亏损计算出释放的能量。

Key ideas: conservation of mass number and charge, mass defect and binding energy, Q-values, and the distinction between fission and fusion. Master these concepts to tackle IB Physics nuclear questions with confidence.

核心要点：质量数和电荷守恒、质量亏损与结合能、Q值以及裂变与聚变的区别。掌握这些概念，你就能自信地应对IB物理核能相关题目。

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📚 IB Physics: Leptons and the Weak Nuclear Force | IB物理：轻子与弱核力的相互作用

The Standard Model of particle physics classifies fundamental particles into quarks and leptons. While quarks experience all four fundamental forces, leptons interact through the weak nuclear force, which plays a crucial role in radioactive decay and nuclear reactions. This article explores the properties of leptons and the mechanisms of weak interactions, essential for IB Physics HL.

粒子物理标准模型将基本粒子分为夸克和轻子。夸克参与所有四种基本力，而轻子通过弱核力相互作用，弱核力在放射性衰变和核反应中起着至关重要的作用。本文探讨轻子的性质以及弱相互作用的机制，这对IB物理高级水平至关重要。

1. Lepton Family | 轻子家族

Leptons are spin-½ fermions that do not experience the strong nuclear force. There are six leptons arranged in three generations: electron (e⁻) and electron neutrino (νₑ); muon (μ⁻) and muon neutrino (ν_μ); tau (τ⁻) and tau neutrino (ν_τ). Each has a corresponding antiparticle with opposite charge but same mass.

轻子是自旋为½的费米子，不参与强核力。六种轻子分为三代：电子(e⁻)和电子中微子(νₑ)；μ子(μ⁻)和μ子中微子(ν_μ)；τ子(τ⁻)和τ子中微子(ν_τ)。每种轻子都有对应的反粒子，电荷相反但质量相同。

• First generation: electron and electron neutrino — stable, ordinary matter.
• Second generation: muon and muon neutrino — unstable, decay to electrons.
• Third generation: tau and tau neutrino — very heavy, decay rapidly.

• 第一代：电子和电子中微子——稳定，构成普通物质。
• 第二代：μ子和μ子中微子——不稳定，衰变为电子。
• 第三代：τ子和τ子中微子——质量很大，迅速衰变。

2. Lepton Number Conservation | 轻子数守恒

In all interactions, the total lepton number L is conserved. Furthermore, in the Standard Model, each family has its own conserved lepton number: Lₑ, L_μ, L_τ. For example, in beta-minus decay, n → p + e⁻ + ν̄ₑ, the electron lepton number changes from 0 to (+1 for e⁻) + (−1 for antineutrino) = 0, so it is conserved.

在所有相互作用中，总轻子数L守恒。此外，在标准模型中，每一代轻子都有各自的守恒量：Lₑ、L_μ、L_τ。例如，在β⁻衰变中，n → p + e⁻ + ν̄ₑ，电子轻子数从0变为(+1来自e⁻) + (−1来自反中微子) = 0，因此守恒。

Lₑ : 0 = +1 − 1 ✓

Lepton number conservation dictates which decays are allowed. A process such as μ⁻ → e⁻ + γ violates muon and electron lepton numbers separately and has never been observed.

轻子数守恒决定了哪些衰变是允许的。例如μ⁻ → e⁻ + γ分别违反μ子数和电子数守恒，从未被观测到。

3. The Weak Nuclear Force | 弱核力

The weak nuclear force is one of the four fundamental forces. It has a very short range of about 10⁻¹⁸ m and is responsible for changing quark flavour, enabling beta decay and neutrino interactions. Unlike the strong force, it affects both quarks and leptons.

弱核力是四种基本力之一。其作用距离极短，约为10⁻¹⁸ m，它能够改变夸克味，从而引起β衰变和中微子相互作用。与强力不同，弱力同时作用于夸克和轻子。

• Weak force strength: about 10⁶ times weaker than the strong force at short distances.
• It is the only force that changes flavour (e.g., d → u in beta decay).

• 弱力强度：在短距离上比强力弱约10⁶倍。
• 它是唯一能改变味量子数的力（如β衰变中d → u）。

4. Exchange Particles: W and Z Bosons | 交换粒子：W和Z玻色子

The weak interaction is mediated by massive gauge bosons: the charged W⁺ and W⁻ bosons, and the neutral Z⁰ boson. Their large masses (about 80–91 GeV/c²) explain the short range, as the uncertainty principle limits the exchange distance.

弱相互作用由有质量的规范玻色子传递：带电荷的W⁺和W⁻玻色子，以及中性的Z⁰玻色子。它们质量很大（约80–91 GeV/c²），由不确定原理限制了交换距离，因此作用程很短。

Range ≈ ℏ / (m_W c) ≈ 10⁻¹⁸ m

Charged current interactions involve W bosons and change flavour; neutral current interactions involve Z bosons and leave flavour unchanged.

带电流相互作用涉及W玻色子并改变味；中性流相互作用涉及Z玻色子，味保持不变。

5. Beta Decay as a Weak Process | 作为弱过程的β衰变

In β⁻ decay, a down quark changes to an up quark by emitting a W⁻ boson, which subsequently decays into an electron and an antineutrino. The quark-level equation is:

在β⁻衰变中，一个下夸克通过发射W⁻玻色子变为上夸克，W⁻随后衰变为电子和反中微子。夸克级方程为：

d → u + W⁻ → u + e⁻ + ν̄ₑ

In β⁺ decay, an up quark emits a W⁺ boson, becoming a down quark. The W⁺ then decays into a positron and a neutrino:

在β⁺衰变中，一个上夸克发射W⁺玻色子变为下夸克，W⁺随后衰变为正电子和中微子：

u → d + W⁺ → d + e⁺ + νₑ

6. Electron Capture | 电子俘获

Electron capture is another weak process in which a proton inside a nucleus captures an inner-shell electron and transforms into a neutron, emitting an electron neutrino:

电子俘获是另一种弱过程：原子核内的质子俘获内层电子并转化为中子，同时发射电子中微子：

p + e⁻ → n + νₑ

At the quark level, an up quark absorbs a W⁻ boson (emitted virtually by the electron) and becomes a down quark: u + W⁻ → d, with the electron becoming the neutrino.

在夸克级别，上夸克吸收（电子虚发射的）W⁻玻色子而变为下夸克：u + W⁻ → d，电子则变为中微子。

7. Neutrino Interactions | 中微子相互作用

Neutrinos are neutral leptons that interact only via the weak force and gravity. Their extremely small cross-section allows them to pass through ordinary matter almost undisturbed. For example, a neutrino can scatter off a neutron by exchanging a W⁺ boson:

中微子是电中性轻子，仅通过弱力和引力相互作用。它们极小的截面使它们几乎不受干扰地穿过普通物质。例如，中微子可以通过交换W⁺玻色子与中子发生散射：

νₑ + n → e⁻ + p

This is the inverse beta decay, a key detection mechanism for neutrinos in large detectors.

这就是逆β衰变，是大型探测器中中微子探测的关键机制。

8. Lepton Universality | 轻子普适性

The weak interaction couples identically to all three generations of leptons, a property called lepton universality. The coupling constant at the W vertex is the same for (e, νₑ), (μ, ν_μ), and (τ, ν_τ), although decays to heavier leptons may be kinematically suppressed.

弱相互作用对三代轻子的耦合方式完全相同，这种性质称为轻子普适性。在W顶点处，耦合常数对(e, νₑ)、(μ, ν_μ)和(τ, ν_τ)都一样，尽管衰变到更重的轻子可能会因运动学因素受到抑制。

• Muon decay: μ⁻ → e⁻ + ν̄ₑ + ν_μ occurs through W⁻ exchange.
• Tau decays: τ⁻ can decay into electrons, muons, or hadrons, all through weak interactions.

• μ子衰变：μ⁻ → e⁻ + ν̄ₑ + ν_μ通过W⁻交换发生。
• τ子衰变：τ⁻可以衰变为电子、μ子或强子，全部通过弱相互作用。

9. Weak Isospin and Helicity | 弱同位旋和手性

In the electroweak theory, left-handed leptons form doublets of weak isospin, while right-handed leptons are singlets. For example, the electron doublet is (νₑ, e⁻) with weak isospin components +½ and −½. The W bosons couple only to left-handed particles and right-handed antiparticles.

在电弱理论中，左手轻子形成弱同位旋二重态，而右手轻子是单重态。例如，电子二重态为(νₑ, e⁻)，弱同位旋分量分别为+½和−½。W玻色子只耦合左手粒子和右手反粒子。

Left-handed lepton doublet: (νₗ, l⁻) ; Right-handed singlet: l⁻_R

This parity violation is a unique signature of the weak force, not observed in electromagnetic or strong interactions.

这种宇称不守恒是弱力的独特标志，在电磁相互作用或强相互作用中并未观察到。

10. Weak Interactions in Stars | 恒星中的弱相互作用

The weak force is essential in stellar nucleosynthesis. In the Sun, the first step of the proton-proton chain converts two protons into a deuteron via the weak interaction:

弱力在恒星核合成中至关重要。在太阳中，质子-质子链的第一步通过弱相互作用将两个质子转化为氘核：

p + p → ²H + e⁺ + νₑ

This process converts a proton into a neutron, releasing a positron and a neutrino. Without the weak force, the Sun’s nuclear fusion would not proceed at its observed rate.

该过程将一个质子转化为中子，释放正电子和中微子。如果没有弱力，太阳的核聚变就不会以观测到的速率进行。

11. Feynman Diagrams for Weak Processes | 弱过程的费曼图

Feynman diagrams visually represent weak interactions. In beta decay, a neutron (udd) emits a W⁻ boson at one vertex, changing a d quark into a u quark; the W⁻ then decays into an e⁻ and ν̄ₑ at another vertex. Time flows left to right, and every vertex conserves charge, lepton number, and energy-momentum.

费曼图直观地表示弱相互作用。在β衰变中，中子(udd)在一个顶角发射W⁻玻色子，使d夸克变为u夸克；W⁻随后在另一个顶角衰变为e⁻和ν̄ₑ。时间从左向右流动，每个顶角都守恒电荷、轻子数和能量-动量。

• Strong vertices conserve flavour; weak charged-current vertices change flavour.
• Z⁰ exchanges leave flavour unchanged and are called neutral current diagrams.

• 强顶角保持味不变；弱带电流顶角改变味。
• Z⁰交换不改变味，称为中性流图。

12. Experimental Evidence and Significance | 实验证据与意义

Weak interactions were first observed in nuclear beta decay. Later, the discovery of weak neutral currents at CERN in 1973 and the W and Z bosons in 1983 confirmed the electroweak theory. The precise measurements of Z decay rates at LEP established that there are exactly three generations of light neutrinos.

弱相互作用首先在核β衰变中被观察到。后来，1973年在CERN发现弱中性流，1983年发现W和Z玻色子，证实了电弱理论。LEP对Z衰变速率的精确测量确定了轻中微子恰好有三代。

For IB learners, understanding leptons and weak interactions is not just about particle classification — it explains why stars shine, why nuclei decay, and how neutrinos reveal the universe. Mastery of these concepts is key to excelling in the particle physics section of IB Physics.

对IB学习者而言，理解轻子和弱相互作用不仅仅是粒子分类——它解释了恒星为何发光、原子核为何衰变以及中微子如何揭示宇宙。掌握这些概念是在IB物理粒子物理部分取得优异成绩的关键。

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📚 IB Physics: Nuclear Fission Principles & Reactions in Nuclear Power Plants | IB物理：核裂变原理与核电站中的反应

Nuclear fission is the process by which a heavy nucleus splits into two lighter nuclei, releasing a tremendous amount of energy. In this article, we explore the physics of fission, the chain reaction, and how this energy is harnessed in a nuclear power plant—an essential topic for the IB Physics curriculum.

核裂变是指重原子核分裂为两个较轻原子核并释放巨大能量的过程。本文围绕裂变物理、链式反应以及核电站如何利用这一能量展开，是IB物理课程中的核心内容。

1. The Fundamentals of Nuclear Fission | 核裂变的基本原理

When a heavy nucleus such as uranium-235 absorbs a slow (thermal) neutron, it becomes an excited compound nucleus. This nucleus oscillates and deforms until it splits into two smaller nuclei—the fission fragments—along with 2 to 3 free neutrons and a great deal of energy.

当铀-235等重原子核吸收一个慢（热）中子后，会形成高激发的复合核。该复合核发生振荡与形变，最终分裂为两个较小原子核（即裂变碎片），同时释放出2至3个自由中子和大量能量。

A typical fission reaction of uranium-235 is written as:

n + ²³⁵U → ¹⁴¹Ba + ⁹²Kr + 3n + Energy

The total mass of the products is smaller than that of the reactants. This mass difference Δm is converted into kinetic energy of the fragments and neutrons, as described by Einstein’s famous equation ΔE = Δmc².

生成物的总质量小于反应物的总质量，这一质量亏损Δm转化为碎片和中子的动能，正符合爱因斯坦著名方程ΔE = Δmc²。

An important concept behind fission is the binding energy per nucleon. Heavy nuclei such as ²³⁵U lie on the right side of the binding-energy curve, where nucleons are less tightly bound. When the nucleus splits into two mid-sized fragments, the products sit higher on the curve, meaning the released energy is the difference between these binding energies.

理解裂变的关键概念是“比结合能”。像²³⁵U这样的重核位于比结合能曲线的右侧，核子束缚较弱。当原子核分裂为两个中等质量的碎片后，生成物在曲线上位于更高处，释放的能量正是两者结合能之差。

2. Energy Release from Fission | 裂变的能量释放

For every fission event of a uranium-235 nucleus, approximately 200 MeV (3.2 × 10⁻¹¹ J) of energy is released. Most of this energy appears as kinetic energy of the fission fragments; the rest is carried by neutrons, gamma rays, and later by radioactive decay products.

每个铀-235原子核发生裂变时释放约200 MeV（即3.2 × 10⁻¹¹ J）的能量。其中大部分表现为裂变碎片的动能，其余由中子、伽马射线以及随后的放射性衰变产物带走。

The energy released can be calculated from the mass defect:

Q = (m_initial − m_final) × c²

Using the fission channel above:

m_initial = 1.008665 + 235.0439 = 236.0526 u

m_final = 140.9144 + 91.9262 + 3 × 1.008665 = 235.8666 u

Δm = 0.1860 u → Q = 0.1860 × 931.5 ≈ 173 MeV

Since 1 u = 931.5 MeV/c², the mass defect of 0.186 u corresponds to about 173 MeV for this particular channel. The average over all fission channels is about 200 MeV. In the IB exam, you may be given mass values in u or in kg and asked to find ΔE.

因为1 u = 931.5 MeV/c²，该裂变道0.186 u的质量亏损对应约173 MeV的能量。所有裂变道的平均值约为200 MeV。在IB考试中，题目可能给出以u或kg为单位的质量，要求计算ΔE。

3. The Chain Reaction | 链式反应

Each fission event releases on average 2-3 neutrons. If at least one of these neutrons goes on to induce another fission, a self-sustaining chain reaction is achieved. This gives rise to the concept of the multiplication factor k.

每次裂变平均释放2至3个中子。如果其中至少一个中子能引发下一次裂变，便可形成自持的链式反应，由此引出倍增因子k的概念。

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📚 IB Physics: Conditions for Nuclear Fusion and Energy Release | IB物理：核聚变的条件与能量释放

Nuclear fusion is the process by which two light nuclei combine to form a heavier nucleus, releasing a tremendous amount of energy. This is the fundamental reaction that powers the Sun and other stars, and it holds great promise as a future clean energy source on Earth. In this article, we will explore the conditions required for fusion to occur and analyze the energy released using the principles of mass-energy equivalence.

核聚变是两个轻原子核结合形成一个较重原子核的过程，并释放出巨大的能量。这是太阳和其他恒星动力的根本来源，也作为未来地球上清洁能源的潜在途径而受到广泛关注。本文将探讨聚变所需的条件，并运用质能等价原理来分析其释放的能量。

1. What is Nuclear Fusion? | 什么是核聚变？

Nuclear fusion involves the merging of two light nuclei, such as isotopes of hydrogen, to form a heavier nucleus. For example, deuterium (²H) and tritium (³H) can fuse to produce helium-4 (⁴He) and a neutron, along with the release of energy. The mass of the products is slightly less than the mass of the reactants, and this mass defect is converted into kinetic energy of the products according to Einstein’s equation, E = mc².

核聚变是两个轻原子核（如氢的同位素）合并形成一个较重原子核的过程。例如，氘（²H）和氚（³H）可以聚变产生氦-4（⁴He）和一个中子，同时释放能量。产物的质量略小于反应物的质量，这种质量亏损根据爱因斯坦质能方程 E = mc² 转化为产物的动能。

2. The Coulomb Barrier | 库仑势垒

For positively charged nuclei to fuse, they must overcome the electrostatic repulsion between them. This repulsion is described by Coulomb’s law, where the force is directly proportional to the product of the charges and inversely proportional to the square of the separation distance. The energy required to overcome this repulsion is called the Coulomb barrier. For typical fusion reactions, this barrier corresponds to a temperature of several hundred million kelvin (about 10⁹ K) if classical physics alone is considered.

要使带正电荷的原子核发生聚变，必须克服它们之间的静电排斥力。这种排斥力由库仑定律描述，即力与电荷的乘积成正比，与距离的平方成反比。克服这种排斥所需能量称为库仑势垒。对于典型的聚变反应，如果仅考虑经典物理，这个势垒对应的温度约为数亿开尔文（约10⁹ K）。

3. Quantum Tunneling | 量子隧穿

In reality, fusion occurs at much lower temperatures than predicted by classical physics because of quantum tunneling. Although the nuclei may not have enough kinetic energy to surmount the Coulomb barrier classically, there is a probability that they can tunnel through it. This probability is exponentially dependent on the barrier width and height. In the core of the Sun, where the temperature is about 1.5 × 10⁷ K, quantum tunneling enables the fusion of hydrogen nuclei, although the reaction rate is extremely low because of the low probability of tunneling.

实际上，由于量子隧穿效应，聚变发生的温度远低于经典物理的预测。尽管原子核可能没有足够的动能来经典地越过库仑势垒，但它们存在隧穿势垒的概率。这个概率与势垒的宽度和高度呈指数关系。在太阳的核心，温度约为 1.5 × 10⁷ K，量子隧穿使氢核聚变得以发生，尽管由于隧穿概率低，反应速率极低。

4. Conditions for Fusion: Temperature | 聚变条件：温度

For a sustained fusion reaction, the fuel must be heated to extremely high temperatures to ensure that a significant fraction of the nuclei have enough kinetic energy to either overcome or tunnel through the Coulomb barrier. This is the first condition for fusion. The required temperature is typically in the range of 10⁸ K for deuterium-tritium (D-T) fusion, which is the most promising reaction for controlled fusion on Earth. At such temperatures, matter exists in the state of plasma, where electrons are stripped from atoms, and the nuclei move freely.

为了实现持续聚变反应，燃料必须被加热到极高的温度，以确保相当比例的原子核具备足够的动能来克服或隧穿库仑势垒。这是聚变的第一个条件。对于氘-氚（D-T）聚变，这是受控聚变中最有前景的反应，所需温度通常在 10⁸ K 左右。在这种温度下，物质以等离子体状态存在，电子从原子中剥离，原子核自由运动。

5. Conditions for Fusion: Density and Confinement Time | 聚变条件：密度与约束时间

The second and third conditions are that the fuel density (n) and the confinement time (τ) must be sufficiently high. The confinement time is the average time that the thermal energy remains within the plasma before escaping. These three quantities – temperature (T), density (n), and confinement time (τ) – are combined in the Lawson criterion, which gives a measure of the feasibility of a fusion reactor. For D-T fusion, the triple product nτT must exceed a value of about 3 × 10²¹ m⁻³ s keV. In other words, for a given temperature, the product of density and confinement time must be large enough to achieve ignition, where the fusion energy released is sufficient to keep the plasma hot without external heating.

聚变的第二个和第三个条件是燃料密度（n）和约束时间（τ）必须足够高。约束时间是等离子体热能逃逸前在其中的平均停留时间。这三个量——温度（T）、密度（n）和约束时间（τ）——被结合起来构成劳森判据，用于衡量聚变反应堆的可行性。对于 D-T 聚变，三重积 nτT 必须超过约 3 × 10²¹ m⁻³ s keV。换句话说，在给定温度下，密度与约束时间的乘积必须足够大，以实现点火，即聚变释放的能量足以在没有外部加热的情况下维持等离子体温度。

6. Energy Release in Fusion: Mass Defect | 聚变中的能量释放：质量亏损

The energy released in a fusion reaction can be calculated from the difference in mass between the reactants and the products. For the D-T reaction:

聚变反应释放的能量可以通过反应物与产物之间的质量差来计算。对于 D-T 反应：

²H + ³H → ⁴He (3.5 MeV) + n (14.1 MeV)

The total mass of the reactants is 2.014102 u + 3.016049 u = 5.030151 u. The total mass of the products is 4.002603 u + 1.008665 u = 5.011268 u. The mass defect is Δm = 0.018883 u. Using the conversion 1 u = 931.5 MeV/c², the energy released per reaction is approximately 17.6 MeV, which is shared between the helium-4 nucleus and the neutron.

反应物的总质量为 2.014102 u + 3.016049 u = 5.030151 u。产物的总质量为 4.002603 u + 1.008665 u = 5.011268 u。质量亏损 Δm = 0.018883 u。利用转换系数 1 u = 931.5 MeV/c²，每次反应释放的能量约为 17.6 MeV，这些能量分配在氦-4 核与中子之间。

7. Binding Energy and Nuclear Stability | 结合能与核稳定性

The energy release in fusion is directly related to the binding energy per nucleon. The binding energy per nucleon is the energy required to remove a single nucleon from a nucleus. For very light nuclei, such as hydrogen and helium, the binding energy per nucleon is relatively low. As nuclei fuse, they form heavier nuclei with higher binding energy per nucleon, which means that the final nucleus is more stable. The increase in stability is accompanied by the release of energy. The maximum binding energy per nucleon occurs around iron-56, and for elements lighter than iron, fusion releases energy.

聚变释放的能量与每个核子的结合能直接相关。每个核子的结合能是将一个核子从原子核中移出所需的能量。对于非常轻的原子核，如氢和氦，每个核子的结合能相对较低。当原子核聚变时，它们形成具有更高每个核子结合能的较重原子核，这意味着最终原子核更稳定。稳定性的增加伴随着能量的释放。每个核子结合能在铁-56 附近达到最大值，对于比铁轻的元素，聚变释放能量。

8. Fusion in Stars: The Proton-Proton Chain | 恒星中的聚变：质子-质子链

In stars like the Sun, the main fusion process is the proton-proton chain. This sequence of reactions converts four hydrogen nuclei (protons) into one helium-4 nucleus, releasing two positrons, two neutrinos, and a total energy of about 26.7 MeV. The first step involves two protons fusing to form deuterium, a positron, and a neutrino. The deuterium then fuses with another proton to form helium-3, and finally two helium-3 nuclei combine to form helium-4 and two protons. This chain is the primary energy source for low-mass stars.

在像太阳这样的恒星中，主要的聚变过程是质子-质子链。这一系列反应将四个氢核（质子）转化为一个氦-4 核，释放出两个正电子、两个中微子以及约 26.7 MeV 的总能量。第一步是两个质子聚变形成氘、一个正电子和一个中微子。接着氘与另一个质子聚变形成氦-3，最后两个氦-3 核结合形成氦-4 和两个质子。该链是低质量恒星的主要能量来源。

9. Fusion on Earth: Magnetic and Inertial Confinement | 地球上的聚变：磁约束与惯性约束

To achieve controlled fusion on Earth, two main approaches are being pursued. The first is magnetic confinement fusion, where a strong magnetic field is used to confine the hot plasma in a toroidal (donut-shaped) chamber. The most well-known device is the tokamak. The second approach is inertial confinement fusion, where tiny fuel pellets are compressed and heated rapidly by intense laser or ion beams, causing the fuel to fuse before it can disperse. Both methods aim to satisfy the Lawson criterion and achieve a net energy gain.

为了实现地球上受控聚变，主要研究两种方法。第一种是磁约束聚变，利用强磁场将高温等离子体约束在环形（甜甜圈形状）腔室中。最著名的装置是托卡马克。第二种方法是惯性约束聚变，通过强激光或离子束快速压缩和加热微型燃料靶丸，使燃料在扩散之前完成聚变。两种方法都旨在满足劳森判据并实现净能量增益。

10. Comparing Fusion and Fission | 聚变与裂变的比较

Fusion and fission are two different nuclear reactions that release energy, but they have distinct advantages and disadvantages. Fusion produces no long-lived radioactive waste, unlike fission, which creates radioactive byproducts that require safe storage for thousands of years. Fusion fuel is abundant in seawater (deuterium) and can be bred from lithium (tritium), whereas fission fuel (uranium) is finite and requires mining. However, fusion is technically very challenging because it requires sustaining extreme temperatures and pressures, while fission has been commercially used since the mid-20th century.

聚变和裂变是两种不同的核反应，都能释放能量，但它们的优缺点各异。聚变不产生长寿命放射性废物，而裂变会产生需要安全储存数千年的放射性副产品。聚变燃料在海水中丰富（氘），可以从锂中增殖（氚），而裂变燃料（铀）是有限的，需要开采。然而，聚变在技术上极具挑战性，因为它需要维持极端的温度和压力，而裂变自20世纪中期以来已用于商业发电。

11. Challenges and Prospects for Fusion Energy | 聚变能的挑战与前景

Several significant challenges remain before fusion can become a practical energy source. Materials must withstand high-energy neutron bombardment and extremely high temperatures. Maintaining the stability of the plasma is difficult due to instabilities that can disrupt the confinement. Moreover, tritium is radioactive and scarce, although it can be bred from lithium in the fusion reactor itself. Despite these obstacles, major international projects like ITER are making steady progress, and private companies are also investing in alternative fusion concepts. If successful, fusion could provide nearly limitless, clean, and safe energy.

在聚变成为实用能源之前，仍面临几个重大挑战。材料必须能够承受高能中子轰击和极高的温度。由于可能破坏约束的不稳定性，维持等离子体的稳定性十分困难。此外，氚具有放射性且稀缺，尽管可以在聚变反应堆中从锂中增殖。尽管存在这些障碍，国际热核聚变实验堆（ITER）等重大国际项目正在稳步推进，私营公司也在投资替代性聚变概念。如果成功，聚变可以提供近乎无限、清洁且安全的能源。

12. Key Equations and IB Exam Tips | 关键方程与IB考试提示

For IB Physics, it is essential to know the mass-energy equivalence equation E = mc² and how to apply it to calculate the energy released in fusion. Practice converting atomic mass units (u) to energy units (MeV) using the conversion factor 1 u = 931.5 MeV/c². Also, be familiar with the Lawson criterion and the conditions for fusion: high temperature, high density, and sufficient confinement time. Discussing the advantages and challenges of fusion compared to fission is a common exam question, so prepare to explain these points clearly. Remember, fusion is the opposite of fission: fusing light nuclei releases energy, while splitting heavy nuclei also releases energy, each due to the binding energy curve.

对于IB物理，必须掌握质能方程 E = mc² 及其在计算聚变释放能量中的应用。练习使用转换系数 1 u = 931.5 MeV/c² 将原子质量单位（u）转换为能量单位（MeV）。还要熟悉劳森判据以及聚变的条件：高温、高密度和足够的约束时间。与裂变相比，聚变优势与挑战的讨论是常见的考试问题，因此请准备好清楚阐述这些要点。记住，聚变与裂变相反：轻核聚变释放能量，重核裂变也释放能量，两者都源于结合能曲线的特点。

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📚 IB Physics: Circuit Analysis and Ohm’s Law | IB物理：电路分析与欧姆定律

Circuit analysis forms the backbone of IB Physics Topic 5 (Electricity and Magnetism). Mastering Ohm’s law, Kirchhoff’s rules, and their applications to series and parallel networks is essential for both Paper 1 and Paper 2 success. This guide walks through the key concepts, common pitfalls, and exam-style strategies to help you secure full marks.

电路分析是IB物理Topic 5（电与磁）的核心内容。熟练掌握欧姆定律、基尔霍夫定律及其在串联和并联电路中的应用，对Paper 1和Paper 2都至关重要。本指南将梳理关键概念、常见误区及应试策略，助你冲击满分。

1. Ohm’s Law and Resistance | 欧姆定律与电阻

Ohm’s law states that the current through a metallic conductor is directly proportional to the potential difference across it, provided the physical conditions (temperature, strain, etc.) remain constant. Mathematically, ( V = IR ), where ( V ) is the potential difference in volts (V), ( I ) is the current in amperes (A), and ( R ) is the resistance in ohms (Ω). This linear relationship only holds for ohmic conductors.

欧姆定律指出：在物理条件（温度、形变等）保持恒定时，通过金属导体的电流与导体两端的电势差成正比。数学表达式为 ( V = IR )，其中 ( V ) 为电势差（单位：伏特 V），( I ) 为电流（单位：安培 A），( R ) 为电阻（单位：欧姆 Ω）。这一线性关系仅适用于欧姆导体。

Resistance is defined as the ratio of potential difference to current: ( R = frac{V}{I} ). It quantifies how strongly a component opposes the flow of charge. For a uniform wire, resistance depends on length ( L ), cross-sectional area ( A ), and resistivity ( rho ) of the material: ( R = rhofrac{L}{A} ).

电阻定义为电势差与电流之比：( R = frac{V}{I} )。它衡量元件对电荷流动的阻碍程度。对于均匀导线，电阻取决于长度 ( L )、横截面积 ( A ) 和材料的电阻率 ( rho )：( R = rhofrac{L}{A} )。

V = IR

R = ρL/A

2. Resistivity and Temperature | 电阻率与温度

Resistivity ( rho ) is an intrinsic property of a material, independent of its shape or size. For metals, resistivity increases with temperature because lattice vibrations scatter conduction electrons more frequently. This results in a positive temperature coefficient of resistance.

电阻率 ( rho ) 是材料的固有属性，与形状和尺寸无关。对于金属，电阻率随温度升高而增大，因为晶格振动更频繁地散射传导电子。这导致金属具有正的温度电阻系数。

For semiconductors (e.g., silicon, germanium) and carbon, resistivity decreases as temperature rises. Higher temperatures liberate more charge carriers, reducing resistance. This negative temperature coefficient is exploited in thermistors, which are often used in temperature-sensing circuits.

对于半导体（如硅、锗）和碳，电阻率随温度升高而减小。温度升高释放更多载流子，从而降低电阻。负温度系数被应用于热敏电阻，常用于温度传感电路。

In IB exams, you may be asked to sketch or interpret ( I-V ) graphs for different components. A straight line through the origin indicates an ohmic conductor. A curve that flattens (for a filament lamp) or steepens (for a thermistor) indicates non-ohmic behaviour.

IB考试中，你可能会被要求绘制或解读不同元件的 ( I-V ) 图像。过原点的直线表示欧姆导体；曲线趋于平缓（白炽灯）或变陡（热敏电阻）则表示非欧姆行为。

3. Series Circuits | 串联电路

In a series circuit, components are connected end-to-end, forming a single path for current. The current is identical at every point in the circuit: ( I = I_1 = I_2 = cdots ). The total potential difference across the battery is equal to the sum of potential differences across each component: ( V_{text{total}} = V_1 + V_2 + cdots ).

在串联电路中，元件首尾相连，形成单一电流路径。电路中各点电流相同：( I = I_1 = I_2 = cdots )。电池两端的总电势差等于各元件两端电势差之和：( V_{text{total}} = V_1 + V_2 + cdots )。

The equivalent resistance of resistors in series is simply the sum: ( R_{text{eq}} = R_1 + R_2 + cdots ). This is because each resistor adds its own opposition to the flow of current. For ( n ) identical resistors of value ( R ), the equivalent resistance is ( nR ).

串联电阻的等效电阻等于各电阻之和：( R_{text{eq}} = R_1 + R_2 + cdots )。因为每个电阻都增加了对电流的阻碍。对于 ( n ) 个阻值均为 ( R ) 的相同电阻，等效电阻为 ( nR )。

V_total = V₁ + V₂ + …

R_eq = R₁ + R₂ + …

A common exam question involves a voltage divider in series: a potential divider consisting of two resistors connected across a battery. The output voltage ( V_{text{out}} ) across one resistor ( R_2 ) is given by ( V_{text{out}} = frac{R_2}{R_1 + R_2}V_{text{in}} ).

常见考题涉及串联分压：由两个电阻构成的电位分压器跨接在电池两端。电阻 ( R_2 ) 上的输出电压 ( V_{text{out}} ) 为 ( V_{text{out}} = frac{R_2}{R_1 + R_2}V_{text{in}} )。

4. Parallel Circuits | 并联电路

In a parallel circuit, components are connected across the same two nodes, providing multiple paths for current. The potential difference across each branch is equal to the applied voltage: ( V = V_1 = V_2 = cdots ). The total current from the source splits among branches: ( I_{text{total}} = I_1 + I_2 + cdots ).

在并联电路中，元件连接在同一对节点之间，为电流提供多条路径。每条支路两端的电势差都等于外加电压：( V = V_1 = V_2 = cdots )。电源提供的总电流在各支路中分流：( I_{text{total}} = I_1 + I_2 + cdots )。

The equivalent resistance of parallel resistors is found by:

并联电阻的等效电阻由下式计算：

1/R_eq = 1/R₁ + 1/R₂ + …

For two resistors in parallel, this simplifies to ( R_{text{eq}} = frac{R_1R_2}{R_1 + R_2} ). The equivalent resistance is always less than the smallest individual resistance. Adding more parallel branches always reduces the total resistance.

对于两个并联电阻，简化为 ( R_{text{eq}} = frac{R_1R_2}{R_1 + R_2} )。等效电阻总是小于其中最小的电阻值。增加并联支路会降低总电阻。

A common question: two resistors 6 Ω and 3 Ω are connected in parallel. The equivalent resistance is ( frac{6 times 3}{6 + 3} = 2 Omega ). If a 12 V battery drives this combination, the total current is ( I = frac{12}{2} = 6 text{A} ), with 4 A through the 3 Ω resistor and 2 A through the 6 Ω resistor.

典型问题：两个电阻6 Ω和3 Ω并联，等效电阻为 ( frac{6 times 3}{6 + 3} = 2 Omega )。若由12 V电池供电，总电流为 ( I = frac{12}{2} = 6 text{A} )，其中通过3 Ω电阻的电流为4 A，通过6 Ω电阻的电流为2 A。

5. Kirchhoff’s Laws | 基尔霍夫定律

Kirchhoff’s Current Law (KCL) states that the total current entering a junction equals the total current leaving it. This reflects conservation of charge. For example, if 5 A enters a junction and splits into 3 A and 2 A, KCL is satisfied.

基尔霍夫电流定律（KCL）指出：流入节点的总电流等于流出节点的总电流。这体现了电荷守恒。例如，5 A流入一个节点并分成3 A和2 A，KCL成立。

Kirchhoff’s Voltage Law (KVL) states that the sum of the electromotive forces (emf) and potential differences around any closed loop is zero. When traversing a loop, a rise in potential is positive, and a drop is negative. This reflects conservation of energy.

基尔霍夫电压定律（KVL）指出：沿任意闭合回路，电动势与电势差之和为零。沿回路行进时，电势升高为正，电势降低为负。这体现了能量守恒。

ΣI_enter = ΣI_leave

ΣV_rise = ΣV_drop

When applying KVL, first assign a direction to each loop (clockwise or counterclockwise). For each resistor, if the loop direction matches the assumed current direction, the voltage drop is ( IR ); otherwise it is ( -IR ). For a battery, if the loop goes from negative to positive terminal, the emf is positive; if from positive to negative, it is negative.

应用KVL时，先为每个回路指定方向（顺时针或逆时针）。对于每个电阻，若回路方向与假设电流方向一致，则电压降为 ( IR )；否则为 ( -IR )。对于电池，若回路从负极端到正极端，电动势为正；若从正极端到负极端，则为负。

A typical IB problem involves two loops with one or two batteries. Set up simultaneous equations and solve for unknown currents. Always check that the final currents are consistent with KCL at junctions.

典型IB题目包含两个回路、一个或两个电池。列出联立方程并求解未知电流，最后务必用KCL检查节点处电流是否一致。

6. Potential Dividers | 电位分压器

A potential divider is a circuit that produces a fraction of the input voltage. It consists of two or more resistors in series. The output voltage across one resistor is proportional to its resistance relative to the total resistance.

电位分压器是一种产生输入电压一部分的电路，由两个或多个串联电阻组成。某个电阻上的输出电压与其电阻占总电阻的比例成正比。

For a divider with resistors ( R_1 ) and ( R_2 ), the output voltage ( V_{text{out}} ) across ( R_2 ) is:

对于由 ( R_1 ) 和 ( R_2 ) 构成的分压器，( R_2 ) 上的输出电压 ( V_{text{out}} ) 为：

V_out = V_in × R₂/(R₁+R₂)

Potential dividers are widely used in sensors. A light-dependent resistor (LDR) in a divider changes its resistance with light intensity, producing a variable output voltage that can trigger a transistor or comparator. Similarly, a thermistor in a divider is used in temperature control systems.

电位分压器广泛应用于传感器。光敏电阻（LDR）在分压器中的阻值随光照强度变化，产生可变的输出电压，从而触发晶体管或比较器。类似地，热敏电阻在分压器中用于温度控制系统。

Exam tip: When a load is connected across ( R_2 ), the effective resistance of that branch drops, and the output voltage changes. You must recalculate using the parallel combination of ( R_2 ) and the load.

考试提示：当在 ( R_2 ) 两端接入负载时，该支路的等效电阻会下降，输出电压随之改变。你必须用 ( R_2 ) 与负载的并联组合重新计算。

7. Internal Resistance and EMF | 内电阻与电动势

A real battery is modelled as an ideal emf source ( mathcal{E} ) in series with an internal resistance ( r ). When current ( I ) flows through the external circuit (load resistance ( R )), the terminal voltage ( V_{text{terminal}} ) is the emf minus the voltage drop across the internal resistance:

真实电池可建模为理想电动势源 ( mathcal{E} ) 与内电阻 ( r ) 的串联。当电流 ( I ) 流过外部电路（负载电阻 ( R )）时，端电压 ( V_{text{terminal}} ) 等于电动势减去内电阻上的电压降：

V_terminal = ℰ − Ir

When no current flows (open circuit), the terminal voltage equals the emf. As current increases, the internal resistance causes the terminal voltage to decrease. If a battery is short-circuited (( R = 0 )), the maximum current is ( I_{text{short}} = mathcal{E}/r ).

无电流时（开路），端电压等于电动势。随着电流增大，内电阻导致端电压下降。若电池短路（( R = 0 )），最大电流为 ( I_{text{short}} = mathcal{E}/r )。

In the lab, plotting ( V_{text{terminal}} ) against ( I ) yields a straight line with intercept ( mathcal{E} ) on the voltage axis and slope ( -r ). This is a standard IB Data-based question: students read the emf from the y-intercept and the internal resistance from the (negative) gradient.

在实验中，绘制 ( V_{text{terminal}} ) 与 ( I ) 的图像，得到一条直线，电压轴截距为 ( mathcal{E} )，斜率为 ( -r )。这是IB数据型题目的标准考点：从y截距读电动势，从（负）斜率读内电阻。

Energy consideration: the total power supplied by the emf is ( P_{text{total}} = mathcal{E}I ). The power dissipated in the internal resistance is ( P_{text{internal}} = I^2r ), and the useful power delivered to the load is ( P_{text{load}} = I^2R ). Maximum power transfer occurs when ( R = r ).

能量角度：电动势提供的总功率为 ( P_{text{total}} = mathcal{E}I )。内电阻消耗的功率为 ( P_{text{internal}} = I^2r )，传递给负载的有用功率为 ( P_{text{load}} = I^2R )。最大功率传输发生在 ( R = r ) 时。

8. Power and Energy in Circuits | 电路中的功率与能量

The power dissipated by a component is the rate at which electrical energy is converted to other forms (heat, light, sound, etc.). Three equivalent expressions are:

元件消耗的功率是电能转化为其他形式能量（热、光、声等）的速率。三个等价表达式为：

P = VI = I²R = V²/R

Use the form that is most convenient for the given known quantities. For example, if current and resistance are given, use ( P = I^2R ). If voltage and resistance are given, use ( P = V^2/R ).

根据已知量选择最方便的表达式。例如，已知电流和电阻时用 ( P = I^2R )；已知电压和电阻时用 ( P = V^2/R )。

Electrical energy ( E ) is given by ( E = Pt = VIt = I^2Rt ). The unit is the joule (J), but the kilowatt-hour (kWh) is commonly used for household energy consumption: 1 kWh = 3.6 × 10⁶ J.

电能 ( E ) 由 ( E = Pt = VIt = I^2Rt ) 给出，单位是焦耳（J）。但家庭用电中常用千瓦时（kWh）：1 kWh = 3.6 × 10⁶ J。

A classic IB question: a 12 V battery drives a current of 2 A through a resistor for 5 minutes. The energy dissipated is ( E = VIt = 12 times 2 times 300 = 7200 text{J} ). If the resistor is 4 Ω, the power is ( P = I^2R = 2^2 times 4 = 16 text{W} ), which also equals ( VI = 12 times 2 = 24 text{W} ) only if the battery is ideal; otherwise the difference is the internal heat loss.

经典IB题：12 V电池在5分钟内驱动2 A电流通过电阻。消耗的能量为 ( E = VIt = 12 times 2 times 300 = 7200 text{J} )。若电阻为4 Ω，功率为 ( P = I^2R = 2^2 times 4 = 16 text{W} )，这与 ( VI = 12 times 2 = 24 text{W} ) 相等仅在理想电池时成立；否则差值即为内阻热损耗。

9. Ammeters and Voltmeters | 安培计与伏特计

An ideal ammeter has zero resistance and is connected in series with the component whose current is being measured. In practice, ammeters have a small but non-zero resistance, which slightly reduces the current in the circuit. IB questions often ask you to explain why ammeters are connected in series and why their resistance should be very low.

理想安培计内阻为零，应串联在被测电流的支路中。实际安培计具有很小但非零的电阻，会略微减小电路电流。IB题目常要求解释安培计为何串联连接以及内阻为何要非常小。

An ideal voltmeter has infinite resistance and is connected in parallel with the component whose voltage is being measured. A real voltmeter draws a small current, which can affect the circuit, especially if the circuit resistance is high. Therefore, voltmeters should have as high a resistance as possible.

理想伏特计内阻为无穷大，应并联在被测电压的元件两端。真实伏特计会分走微小电流，尤其在电路电阻较大时会影响测量结果。因此，伏特计内阻应尽可能大。

Common exam trap: Placing a voltmeter in series or an ammeter in parallel will produce misleading readings. A voltmeter in series breaks the circuit, and an ammeter in parallel creates a short circuit path for current.

常见考试陷阱：伏特计串联或安培计并联会产生误导读数。伏特计串联会断开电路，安培计并联会造成电流短路。

10. Non-Ohmic Components | 非线性元件

Non-ohmic components do not obey Ohm’s law; their resistance changes with the applied voltage or current. The ( I-V ) graph is not a straight line. Key examples include filament lamps, diodes, thermistors, and LDRs.

非线性元件不满足欧姆定律；其电阻随外加电压或电流变化。( I-V ) 图像不是直线。典型例子包括白炽灯、二极管、热敏电阻和光敏电阻。

A filament lamp’s resistance increases as the filament heats up. At low voltages, the graph is nearly linear; at higher voltages, the curve flattens because resistance grows. The dynamic resistance is found from the gradient of the tangent at a given point.

白炽灯的电阻随灯丝温度升高而增大。低电压时图像近似线性；高电压时曲线变得平缓，因为电阻增大。动态电阻由某点切线的斜率给出。

A diode conducts current in one direction only (forward bias) and blocks current in the reverse direction (reverse bias) until breakdown. Its ( I-V ) graph shows a sharp rise in current after the forward threshold voltage (about 0.7 V for silicon). In IB questions, you may be asked to interpret such graphs and explain the shape.

二极管只允许单向导通（正向偏置），反向截止（反向偏置）直至击穿。其 ( I-V ) 图像在正向阈值电压（硅管约0.7 V）之后电流急剧上升。IB题目中可能要求解读此类图像并解释形状。

For a thermistor or LDR, the ( I-V ) curve is symmetrical (no polarity) but non-linear. The resistance at a given voltage can be read from ( V/I ), while the dynamic resistance is the local slope ( dV/dI ).

对于热敏电阻或光敏电阻，( I-V ) 曲线是对称的（无极性）但非线性。某电压下的电阻可由 ( V/I ) 读出，而动态电阻是局部斜率 ( dV/dI )。

11. Analysing Complex Circuits | 复杂电路分析

When faced with a complex circuit, follow a systematic approach: (1) Simplify any obvious series or parallel combinations; (2) Redraw the circuit to make the structure clearer; (3) Apply Ohm’s law to find branch currents; (4) Use KVL and KCL for remaining unknowns.

面对复杂电路时，采用系统化步骤：(1) 简化明显的串联或并联组合；(2) 重绘电路使结构更清晰；(3) 用欧姆定律求支路电流；(4) 对剩余未知量用KVL和KCL求解。

Wheatstone bridge circuits are common in IB extended-response questions. The bridge balances when the potential difference across the central galvanometer (or voltmeter) is zero, which occurs when ( frac{R_1}{R_2} = frac{R_3}{R_4} ). At balance, no current flows through the bridge, so the branches can be treated as independent series pairs.

惠斯通电桥常见于IB扩展答题中。当中央检流计（或伏特计）两端电势差为零时，电桥平衡，此时 ( frac{R_1}{R_2} = frac{R_3}{R_4} )。平衡时桥路无电流，各支路可视为独立的串联对。

Another recurring task: determining the current through a specific resistor when multiple batteries are present. Draw labelled currents for each branch, then write KVL equations for two independent loops. Solve the linear system algebraically. If a computed current is negative, it simply means the actual direction is opposite to the assumed one.

另一类常见任务：当存在多个电池时求某个电阻中的电流。为各支路标出电流方向，然后对两个独立回路写出KVL方程，联立求解线性方程组。若计算的电流为负值，仅表示实际方向与假设方向相反。

12. Exam Strategy and Common Errors | 应试策略与常见错误

Error 1: Forgetting unit conversions. Always convert mA to A, kJ to J, and Ω to kΩ as needed. A current of 200 mA must be written as 0.2 A in ( V = IR ).

错误1：忘记单位换算。 始终将mA换算为A，kJ换算为J，Ω与kΩ按需转换。200 mA必须写成0.2 A才能代入 ( V = IR )。

Error 2: Confusing series and parallel formulas. Resistances add in series but reciprocals add in parallel. A common mistake is using ( R_{text{eq}} = frac{1}{R_1} + frac{1}{R_2} ) directly without taking the reciprocal of the sum.

错误2：混淆串联与并联公式。 串联电阻相加，并联电阻倒数相加。常见错误是直接写 ( R_{text{eq}} = frac{1}{R_1} + frac{1}{R_2} ) 而忘记对总和取倒数。

Error 3: Wrong sign convention in KVL. When moving through a battery, always check the direction from negative to positive (gain) or positive to negative (loss). Inconsistent sign conventions are the leading cause of Kirchhoff errors.

错误3：KVL符号约定错误。 通过电池时，务必判断是从负极到正极（电势升）还是从正极到负极（电势降）。符号约定不一致是基尔霍夫解题错误的主要原因。

Error 4: Ignoring internal resistance. In any real battery question, remember the terminal voltage is ( mathcal{E} – Ir ), not simply ( mathcal{E} ). The total resistance in the circuit includes the internal resistance ( r ).

错误4：忽略内电阻。 在真实电池问题中，端电压是 ( mathcal{E} – Ir )，而不只是 ( mathcal{E} )。电路总电阻包含内阻 ( r )。

For Paper 2 calculations, show every step with units, and state the direction of currents explicitly. For data-based questions, always read the intercept and gradient from the graph carefully, including the units on both axes.

Paper 2计算题中，每一步都要写单位，并明确指出电流方向。数据型题目中，仔细从图像读取截距和斜率，注意两个坐标轴的单位。

Strategy: In multiple-choice questions, estimate before calculating. If the answer is about 5 Ω, you can quickly eliminate options that are orders of magnitude different. This saves time and catches arithmetic slips.

策略： 选择题中，先估算再计算。如果答案约为5 Ω，可迅速排除数量级偏差过大的选项。这既节省时间又能发现运算失误。

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📚 IB Physics: Radioactive Decay Law and Half-Life Calculations | IB物理：放射性衰变规律与半衰期计算

The study of radioactive decay is central to IB Physics. It explains how unstable nuclei transform, how we quantify the rate of decay, and how half-life serves as a practical measure for dating and medical applications. This article covers the decay law, decay constant, half-life calculations, and common IB exam pitfalls.

放射性衰变是IB物理课程中的核心内容之一，涉及不稳定原子核的转变、衰变速率的量化，以及半衰期在考古测年和医学中的应用。本文将系统讲解衰变规律、衰变常数、半衰期计算，并帮助同学们避开IB考试中的常见陷阱。

1. The Nature of Radioactive Decay | 放射性衰变的本质

Radioactive decay is a random and spontaneous process. Each unstable nucleus has a certain probability of decaying per unit time, but we cannot predict exactly when a particular nucleus will decay. This randomness is key to understanding the statistical nature of decay.

放射性衰变是一个随机且自发的过程。每个不稳定原子核在单位时间内都有一定的衰变概率，但我们无法精确预测某个核在何时衰变。这种随机性是理解衰变统计规律的关键。

• Random: The time of decay of any single nucleus cannot be predicted.

  随机性：无法预测单个原子核的衰变时刻。

• Spontaneous: Decay is not affected by external factors such as temperature, pressure, or chemical state.

  自发性：衰变不受温度、压力或化学状态等外部因素影响。

• Statistical: With a large number of nuclei, the average behaviour is predictable.

  统计性：对于大量原子核，其平均行为是可预测的。

The number of decays per unit time is proportional to the number of undecayed nuclei present. This leads directly to the radioactive decay law.

单位时间内的衰变数与当前未衰变的原子核数目成正比，由此可直接导出放射性衰变规律。

2. The Decay Law: N = N₀e^(−λt) | 衰变规律：N = N₀e^(−λt)

The decay law states that the rate of decay is proportional to the number of radioactive nuclei present. Mathematically, we write the differential equation:

衰变规律表明衰变速率与当前放射性原子核的数目成正比。用微分方程表示为：

dN/dt = −λN

where N is the number of undecayed nuclei, t is time, and λ (lambda) is the decay constant, which has units of s⁻¹ or a⁻¹. The negative sign indicates that N decreases over time.

其中N为未衰变的原子核数目，t为时间，λ（lambda）为衰变常数，单位是s⁻¹或a⁻¹。负号表示N随时间减少。

Solving this differential equation gives the exponential decay law:

解该微分方程得到指数衰减规律：

N = N₀e^(−λt)

where N₀ is the initial number of radioactive nuclei at t = 0. The same equation applies to activity A, with A = A₀e^(−λt), and to mass m, with m = m₀e^(−λt).

其中N₀是t = 0时初始放射性原子核数。该方程同样适用于活度A：A = A₀e^(−λt)，以及质量m：m = m₀e^(−λt)。

Be careful: the decay constant λ is different from half-life T₁/₂, though they are related. Do not confuse λ with the wavelength symbol.

注意：衰变常数λ与半衰期T₁/₂不同，但两者密切相关。切勿将λ与波长符号混淆。

3. Half-Life: T₁/₂ = ln2 / λ | 半衰期：T₁/₂ = ln2 / λ

Half-life (T₁/₂) is the time taken for the number of radioactive nuclei to reduce to half of its original value. It is a constant for a given isotope under all normal conditions.

半衰期（T₁/₂）是放射性原子核数目减少到原来一半所需的时间。在通常条件下，特定同位素的半衰期是一个常量。

From the decay law, when N = N₀/2, we have:

由衰变规律，当N = N₀/2时：

N₀/2 = N₀e^(−λT₁/₂)

Taking natural logs on both sides:

两边取自然对数：

T₁/₂ = ln2 / λ

Since ln2 ≈ 0.693, an alternative form is T₁/₂ = 0.693/λ. This relationship is essential for converting between half-life and decay constant.

因为ln2 ≈ 0.693，所以也可写成T₁/₂ = 0.693/λ。该关系在计算中经常用于半衰期与衰变常数之间的转换。

Worked example: The half-life of iodine-131 is 8.02 days. What is its decay constant in s⁻¹?

例题：碘-131的半衰期为8.02天，求其衰变常数（单位s⁻¹）。

λ = 0.693 / (8.02 × 24 × 3600) = 9.99 × 10⁻⁷ s⁻¹

4. Activity and Its Units | 活度及其单位

Activity A is the rate at which decays occur, defined as A = |dN/dt| = λN. The SI unit of activity is the becquerel (Bq), where 1 Bq = 1 decay per second.

活度A是单位时间内发生衰变的次数，定义为A = |dN/dt| = λN。活度的SI单位是贝克勒尔（Bq），1 Bq = 1次衰变每秒。

Since N decreases exponentially, activity also decreases exponentially:

由于N呈指数减少，活度也呈指数减小：

A = A₀e^(−λt)

where A₀ = λN₀.

其中A₀ = λN₀。

Worked example: A sample of strontium-90 has an initial activity of 500 Bq and a decay constant of 7.85 × 10⁻¹⁰ s⁻¹. How many strontium-90 nuclei are initially present?

例题：某锶-90样品初始活度为500 Bq，衰变常数为7.85 × 10⁻¹⁰ s⁻¹，求初始核数。

N₀ = A₀/λ = 500 / (7.85 × 10⁻¹⁰) = 6.37 × 10¹¹

Always check units: if A is in Bq and λ in s⁻¹, then N is a pure count.

务必检查单位：若A以Bq为单位，λ以s⁻¹为单位，则N为纯计数个数，无量纲。

5. Graphical Representation | 图形表示

Exponential decay is represented by a curve that falls steeply at first, then approaches zero asymptotically. On a graph of N versus t, the half-life is the horizontal distance between points where N is halved.

指数衰减曲线先急剧下降，然后逐渐趋近于零。在N-t图中，半衰期是使N减半的相邻两点之间的时间间隔。

There is a very useful property: if you plot the natural logarithm of N (or A) against t, you get a straight line with slope −λ.

有一个非常有用的性质：若绘制ln N（或ln A）与t的关系图，可得一条斜率为−λ的直线。

ln N = ln N₀ − λt

This linear form is often used in exam data analysis questions. The y-intercept is ln N₀ and the slope is −λ.

该线性形式常用于数据分析题。y轴截距为ln N₀，斜率为−λ。

In an IB exam, you may be asked to determine the half-life from a graph. Read the time corresponding to half the initial count rate from the curve. For a logarithmic graph, use the slope to find λ, then T₁/₂ = 0.693/λ.

在IB考试中，可能要求从图中确定半衰期。从曲线中读取计数率降为初始值一半所对应的时间。对于对数坐标图，则利用斜率求λ，再用T₁/₂ = 0.693/λ。

6. Calculating with Whole Numbers of Half-Lives | 用整数倍半衰期计算

When the elapsed time is an exact multiple of the half-life, calculations are simplest. After n half-lives, the remaining fraction is (1/2)ⁿ.

当经过时间为半衰期的整数倍时，计算最为简单。经过n个半衰期后，剩余比例为(1/2)ⁿ。

Remaining fraction = (1/2)ⁿ, where n = t / T₁/₂.

剩余比例 = (1/2)ⁿ，其中n = t / T₁/₂。

For example, after 3 half-lives, the remaining fraction is (1/2)³ = 1/8. After 5 half-lives, it is 1/32.

例如，经过3个半衰期后，剩余比例为(1/2)³ = 1/8。经过5个半衰期后，则为1/32。

Worked example: A sample contains 2.4 × 10¹⁰ radioactive nuclei with a half-life of 6 hours. How many remain after 24 hours?

例题：某样品含2.4 × 10¹⁰个放射性原子核，半衰期为6小时。24小时后还剩多少？

n = 24/6 = 4, so N = 2.4 × 10¹⁰ × (1/2)⁴ = 1.5 × 10⁹

This method avoids the exponential calculation entirely, but it only works for whole-number multiples. For arbitrary times, use the decay law directly.

这种方法完全避免了指数计算，但只能用于整数倍的情况。对于任意时间，应直接使用衰变规律。

7. Worked IB-Style Problems | IB典型例题精讲

Problem 1: The activity of a radioactive sample falls from 800 Bq to 100 Bq in 9 days. Find the half-life.

例题1：某放射性样品活度从800 Bq降为100 Bq用了9天，求半衰期。

Solution: 800 → 400 → 200 → 100 is three half-lives in 9 days. Therefore T₁/₂ = 9/3 = 3 days.

解：800 → 400 → 200 → 100 经历了3个半衰期，共9天。因此T₁/₂ = 9/3 = 3天。

Problem 2: The half-life of carbon-14 is 5730 years. A sample of living wood has an activity of 0.25 Bq per gram. An ancient wooden artifact shows an activity of 0.0625 Bq per gram. Estimate the age of the artifact.

例题2：碳-14的半衰期为5730年。活木材每克活度为0.25 Bq，某古代木制文物每克活度为0.0625 Bq，估算文物的年代。

Solution: 0.25 → 0.125 → 0.0625 is two half-lives. Age = 2 × 5730 = 11460 years.

解：0.25 → 0.125 → 0.0625 经历了两个半衰期。年代 = 2 × 5730 = 11460年。

Problem 3: A radioactive source contains 4.0 × 10¹⁵ nuclei with a decay constant of 1.2 × 10⁻⁸ s⁻¹. Calculate the initial activity and the activity after 24 hours.

例题3：某放射源含4.0 × 10¹⁵个原子核，衰变常数为1.2 × 10⁻⁸ s⁻¹，求初始活度及24小时后的活度。

Initial activity: A₀ = λN₀ = 1.2 × 10⁻⁸ × 4.0 × 10¹⁵ = 4.8 × 10⁷ Bq.

初始活度：A₀ = λN₀ = 1.2 × 10⁻⁸ × 4.0 × 10¹⁵ = 4.8 × 10⁷ Bq。

After 24 h: A = 4.8 × 10⁷ × e^(−1.2 × 10⁻⁸ × 86400) = 4.8 × 10⁷ × e^(−0.0010368) ≈ 4.795 × 10⁷ Bq.

24小时后：A = 4.8 × 10⁷ × e^(−1.2 × 10⁻⁸ × 86400) = 4.8 × 10⁷ × e^(−0.0010368) ≈ 4.795 × 10⁷ Bq。

Note that the activity

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📚 IB Physics: Circular Motion Force Analysis and Critical Problems | IB物理：圆周运动受力分析与临界问题

In IB Physics, circular motion is a rich application of Newton’s laws. Students must identify the real forces acting on an object, resolve them along the radial direction, and recognise when the required centripetal force exceeds what the forces can supply. This article gives a complete framework for force analysis in uniform circular motion and solves the critical speed problems that often appear in IB Paper 2 and internal assessments.

在IB物理中，圆周运动是牛顿定律的综合应用。学生需要找出物体受到的真实力，沿径向进行分解，并判断当所需向心力超过系统能提供的最大值时会出现什么情况。本文为匀速圆周运动的受力分析提供完整框架，并系统讲解IB Paper 2和内部评估中常见的临界速度问题。

1. Key Concepts: Angular Velocity and Centripetal Acceleration | 关键概念：角速度与向心加速度

For an object moving in a circle of radius r, the linear speed v and angular velocity ω are related by v = ωr. The angular velocity is also related to the period T and frequency f by ω = 2π/T = 2πf.

对于半径为r的圆周运动，线速度v与角速度ω满足v = ωr。角速度还与周期T和频率f有关：ω = 2π/T = 2πf。

Even when the speed is constant, the direction of the velocity changes continuously. This change in direction corresponds to a centripetal acceleration directed toward the centre of the circle.

即使速度大小不变，速度方向也在连续改变。这种方向变化对应着指向圆心的向心加速度。

a_c = v² / r = ω² r

The centripetal acceleration is never zero in circular motion, because the velocity vector is never parallel to the acceleration vector except in the limiting radial sense of the change in direction.

在圆周运动中向心加速度永远不会为零，因为速度方向始终在改变，而向心加速度正是描述这种方向变化快慢的物理量。

2. Centripetal Force Is a Net Force | 向心力是合力

Centripetal force is not a new physical force. It is the name given to the net force component that points toward the centre of the circle. Newton’s second law along the radial direction gives:

向心力并不是一种新的相互作用力，而是指指向圆心的合力分量。沿径向应用牛顿第二定律得到：

F_c = m a_c = m v² / r = m ω² r

This equation is always true for uniform circular motion, but it does not tell us which real force provides the centripetal force. In every question, you must first draw a free-body diagram and then determine whether the real forces can combine to produce the required F_c.

这个方程对匀速圆周运动总是成立，但它并不告诉我们哪一个真实力提供向心力。做每道题时，必须先画受力分析图，再判断真实力能否合成为所需的F_c。

A very common error is to add “centrifugal force” as an extra force on the diagram. Centrifugal force is an inertial pseudoforce that appears only in a rotating reference frame; in an inertial frame it does not exist.

一个非常常见的错误是在受力图中额外加上“离心力”。离心力只在转动参考系中才作为惯性力出现；在惯性参考系中它并不存在。

3. Sources of Centripetal Force | 向心力的常见来源

Different physical situations use different real forces as the centripetal force. A single force may act alone, or several forces may combine to point toward the centre.

不同物理情境中，充当向心力的真实力各不相同。可以是某个力单独起作用，也可以是几个力合成后指向圆心。

When identifying the source, always ask: which real force or combination of real forces has a component toward the centre? The answer is the centripetal force.

在判断来源时，要问：哪一个真实力或哪些真实力的合力具有指向圆心的分量？这个答案就是向心力。

4. Horizontal Circular Motion: Tension and Friction | 水平圆周运动：拉力与摩擦力

Consider a mass attached to a string moving on a horizontal frictionless table in a circle of radius r. Vertically, the normal force N balances the weight mg, so N = mg. Horizontally, the only force is the tension T, which points toward the centre.

考虑一个系在绳子上的物体在水平光滑桌面上做半径为r的圆周运动。竖直方向支持力N与重力mg平衡，因此N = mg。水平方向唯一受到的力是拉力T，方向指向圆心。

T = m v² / r

If the same object is placed on a rotating turntable, the static friction f_s provides the centripetal force. Static friction can increase up to a maximum value f_s,max = μ_s N = μ_s mg.

如果同一个物体放在旋转转盘上，则由静摩擦力f_s提供向心力。静摩擦力最大只能达到f_s,max = μ_s N = μ_s mg。

m v² / r ≤ μ_s m g

This inequality immediately gives the maximum speed before slipping. You will see the same structure in many critical speed problems.

这个不等式直接给出了物体开始滑动前的最大速度。很多临界速度问题都具有相同结构。

5. Vertical Circular Motion: Top and Bottom Forces | 竖直圆周运动：最高点与最低点受力

A mass attached to a string moving in a vertical circle has a changing speed: it slows down going up and speeds up going down. The forces are still radial, but gravity has a different direction relative to the radius at each position.

用绳子系住物体在竖直平面内做圆周运动时，速度大小不断变化：上升时减速，下降时加速。受力仍然沿径向，但重力在每一位置与半径方向的关系不同。

At the highest point, tension T and gravitational force mg both point downward, so Newton’s second law gives:

在最高点，拉力T和重力mg都向下，因此牛顿第二定律给出：

T_top + m g = m v_top² / r

At the lowest point, tension points upward while gravity points downward, so:

在最低点，拉力向上，重力向下，因此：

T_bottom − m g = m v_bottom² / r

Notice that the tension at the bottom is always greater than at the top for the same mass and radius, because it must do more work against gravity and provide the centripetal force.

注意到对于同一质量和半径，最低点的拉力总是大于最高点，因为它既要克服重力，又要提供向心力。

6. Critical Speed at the Highest Point | 最高点的临界速度

For a ball attached to a string, tension cannot push; it can only pull. If the required centripetal force at the top is larger than mg, the string must pull downward. If it is smaller, gravity alone is too large, and the string would go slack before the ball reaches the top.

对于绳子系住的小球，拉力只能拉，不能推。如果最高点所需的向心力大于mg，绳子需要向下拉；如果所需向心力小于mg，重力本身过大，小球还没到达最高点绳子就会松弛。

The critical condition is that tension becomes exactly zero at the highest point. Then gravity alone provides the centripetal force:

临界条件是在最高点拉力恰好为零。此时重力单独提供向心力：

m g = m v_c² / r ⇒ v_c = √(g r)

This is the minimum speed at the top for a string or for any object that can only be pulled inward while in contact with the track, such as a ball rolling inside an inverted U-shaped track without a retaining rail.

这是绳子模型或只能被向内拉而不能被向外推的轨道模型在最高点的最小速度，例如没有内侧导轨、小球沿外轨内侧滚动的情况。

If the mass is attached to a rigid rod, the rod can push outward at the top. In that case, the minimum speed at the top can be zero, because the rod can support the ball against gravity.

如果物体连接在一根刚性杆上，杆在最高点可以向外推。此时最高点最小速度可以为零，因为杆可以支撑物体对抗重力。

7. Energy and the Minimum Speed at the Bottom | 能量关系与最低点所需最小速度

To complete a full vertical circle with a string, the speed at the bottom must be large enough to still leave enough speed at the top. Using conservation of mechanical energy between the lowest point and the highest point, with height difference 2r:

要用绳子完成整个竖直圆周运动，最低点的速度必须足够大，以保证到达最高点时仍有足够速度。利用最低点和最高点之间的机械能守恒，高度差为2r：

½ m v_bottom² = ½ m v_top² + m g (2r)

At the critical limit v_top = √(gr), so:

在临界极限下v_top = √(gr)，因此：

v_bottom = √(v_top² + 4 g r) = √(g r + 4 g r) = √(5 g r)

This famous result, v_bottom = √(5gr), is often tested in loop-the-loop questions. It shows that the bottom speed must be about 2.24 times the minimum top speed for a taut-string vertical circle.

这个著名结论v_bottom = √(5gr)经常出现在“竖直圆环”问题中。它说明在绳子绷紧的竖直圆周运动中，最低点速度大约是最高点最小速度的2.24倍。

8. Conical Pendulum | 圆锥摆

A conical pendulum is a mass moving in a horizontal circle while the string traces out a cone. The string makes an angle θ with the vertical. The mass moves with constant speed but its acceleration is horizontal, directed toward the centre of the horizontal circle.

圆锥摆是指小球在水平面内做圆周运动，同时绳子扫过一个圆锥面。绳子与竖直方向夹角为θ。小球速度大小不变，但加速度水平指向水平圆的圆心。

The vertical component of tension balances gravity, while the horizontal component provides the centripetal force:

拉力的竖直分量与重力平衡，水平分量提供向心力：

T cos θ = m g
T sin θ = m v² / r

Dividing the second equation by the first gives a direct expression for the angle:

将第二个方程除以第一个方程，得到夹角的正切表达式：

tan θ = v² / (r g)

Using r = L sin θ, where L is the string length, the angular velocity satisfies ω² = g/(L cos θ). As θ increases, the pendulum must spin faster.

利用r = L sin θ（L为绳长），可得角速度满足ω² = g/(L cos θ)。θ越大，圆锥摆转动越快。

9. Banked Curves | 倾斜弯道

On an ideally banked curve, no friction is needed if the road is inclined at the correct angle for a particular speed. The normal force N is inclined, so it has both vertical and horizontal components.

在理想倾斜弯道上，如果路面倾斜角与速度相匹配，就不需要摩擦力。支持力N是倾斜的，因此同时具有竖直和水平分量。

N cos θ = m g
N sin θ = m v² / r

Dividing the radial equation by the vertical equation gives:

用径向方程除以竖直方程得到：

tan θ = v² / (r g)

This equation is identical in form to the conical pendulum result. It gives the ideal banking angle for a given speed and radius. If the actual speed is higher, additional static friction is required; if lower, friction acts up the slope to prevent sliding down.

这个方程在形式上与圆锥摆相同。它给出了给定速度和半径下的理想倾斜角。如果实际速度更大，则需要额外的静摩擦力；如果实际速度更小，摩擦力会沿斜面向上，防止物体向下滑动。

10. Flat Curve with Friction | 平路弯道与摩擦力

On a flat curve, friction is the only horizontal force available to turn the car. The maximum centripetal force is limited by the maximum static friction:

在平路弯道上，水平方向只有摩擦力能让汽车转弯。最大向心力受最大静摩擦力限制：

f_s,max = μ_s m g

For a car of mass m moving at speed v on a turn of radius r, the requirement is:

对于质量m、速度v、转弯半径r的汽车，要求为：

m v² / r ≤ μ_s m g

Thus the maximum safe speed is:

因此最大安全速度为：

v_max = √(μ_s g r)

If the car exceeds this speed, it will skid outward because static friction is no longer large enough. On ice, μ_s is much smaller, so v_max decreases dramatically.

如果汽车超过这个速度，就会因为静摩擦力不足而向外侧滑。在冰面上μ_s大大减小，因此v_max也会急剧降低。

11. Problem-Solving Strategy | 解题策略

Follow these steps for any circular motion problem in IB Physics:

在IB物理中解决圆周运动问题时，请按以下步骤操作：

• Identify the object and draw a full free-body diagram showing all real forces.

  确定研究对象并画出完整受力图，标出所有真实力。

• Choose the radial direction as pointing toward the centre of the circle, and the perpendicular direction vertically or tangentially as appropriate.

  选择指向圆心的方向为径向，必要时选择竖直或切向为垂直方向。

• Resolve forces along the radial direction and set the net radial force equal to mv²/r.

  沿径向分解力，并令径向合力等于mv²/r。

• Identify the physical limit: tension cannot be negative, normal force cannot be negative, static friction has a maximum value.

  找出物理极限：拉力不能为负，支持力不能为负，静摩擦力存在最大值。

• If v at different heights is involved, use conservation of mechanical energy to connect the speeds.

  如果涉及不同高度的速度，使用机械能守恒来联系各位置的速度。

This procedure changes a seemingly complicated problem into a series of simple algebraic steps.

这个方法可以把看似复杂的问题简化成一系列代数步骤。

12. Worked Example and Common Mistakes | 例题精讲与常见错误

Worked example: A small block slides from rest along a frictionless track and enters a vertical loop of radius R. Find the minimum release height h required for the block to just complete the loop without leaving the track.

例题：一个小滑块从静止开始沿光滑轨道滑下，进入半径为R的竖直圆环。求滑块刚好能完成整个圆环而不脱离轨道时的最小释放高度h。

At the top of the loop, the critical condition is that the normal force N from the track is zero. Gravity alone provides the centripetal force:

在圆环最高点，临界条件是轨道对滑块的支持力N为零。这时重力单独提供向心力：

m g = m v_top² / R ⇒ v_top² = g R

Using energy conservation from the release point to the top of the loop, where the block has risen a vertical distance 2R:

从释放点到最高点应用能量守恒，释放点到最高点的竖直高度为2R：

m g h = m g (2R) + ½ m v_top²

m g h = 2 m g R +