Tag: Physics

📚 A-Level Physics: Experimental Methods for Measuring Magnetic Flux Density | A-Level物理：测量磁通量密度的实验方法

In A-Level Physics, the measurement of magnetic flux density (B) is a fundamental practical skill that appears frequently in the CIE examination syllabus. This article systematically introduces several experimental methods used to determine the magnitude of a magnetic field, with emphasis on the underlying principles, apparatus setup, procedural steps, and error analysis.

在 A-Level 物理中，磁通量密度（B）的测量是一项基础实验技能，在 CIE 考试大纲中频繁出现。本文系统介绍几种用于测定磁场大小的实验方法，重点阐述其基本原理、装置布置、操作步骤及误差分析。

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1. Understanding Magnetic Flux Density | 理解磁通量密度

Magnetic flux density B is defined as the magnetic flux Φ passing through a unit area A perpendicular to the field direction. Its SI unit is the tesla (T), where 1 T = 1 Wb/m². The relationship is expressed as Φ = BA for a uniform field perpendicular to the area.

磁通量密度 B 定义为垂直于磁场方向上通过单位面积 A 的磁通量 Φ。其国际单位是特斯拉（T），1 T = 1 Wb/m²。对于均匀且垂直于面积的磁场，关系式表示为 Φ = BA。

Φ = BA ⇒ B = Φ / A

When the field is not perpendicular, the general expression becomes Φ = BA cos θ, where θ is the angle between the field lines and the normal to the area.

当磁场不垂直于面积时，一般表达式为 Φ = BA cos θ，其中 θ 是磁感线与面积法线之间的夹角。

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2. The Search Coil Method | 搜索线圈法

A search coil is a small flat coil of known area A with N turns, connected to a ballistic galvanometer or a fluxmeter. When the coil is placed in the magnetic field and then suddenly removed, the change in magnetic flux induces an electromotive force (EMF) according to Faraday’s law.

搜索线圈是一个已知面积 A 并绕有 N 匝的小型扁平线圈，连接至冲击电流计或磁通计。当线圈置于磁场中然后迅速移出时，根据法拉第定律，磁通量的变化会感应出电动势（EMF）。

ε = -N (ΔΦ / Δt)

The total charge Q that flows through the galvanometer is proportional to the total change in flux linkage, so B can be determined from the measured deflection.

流过电流计的总电荷 Q 与磁链的总变化量成正比，因此可通过测得的偏转来确定 B。

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3. Using a Ballistic Galvanometer | 使用冲击电流计

A ballistic galvanometer measures the total charge passing through it in a short time interval. The first deflection of the needle is proportional to the charge Q = ∫ I dt. For a search coil of area A and N turns, the charge is related to the change in magnetic flux by Q = N ΔΦ / R, where R is the total circuit resistance.

冲击电流计用于测量短时间内通过它的总电荷量。指针的首次偏转与电荷 Q = ∫ I dt 成正比。对于面积为 A、匝数为 N 的搜索线圈，电荷与磁通量变化的关系为 Q = N ΔΦ / R，其中 R 是回路总电阻。

B = Q R / (N A)

The coil is aligned with its plane perpendicular to the magnetic field, placed at the desired point, and then quickly pulled out to a region of zero field. The deflection gives Q, from which B is calculated.

将线圈平面垂直于磁场放置于待测点，然后快速拉出至零磁场区域。偏转给出 Q，由此计算 B。

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4. The Hall Effect Method | 霍尔效应法

The Hall effect provides a direct and convenient method for measuring magnetic flux density. A semiconductor strip (Hall probe) carrying a current I is placed in the magnetic field perpendicular to both the current direction and the probe’s flat surface. The magnetic field exerts a Lorentz force on the charge carriers, producing a transverse voltage V_H known as the Hall voltage.

霍尔效应为测量磁通量密度提供了一种直接且便捷的方法。承载电流 I 的半导体薄片（霍尔探头）置于磁场中，磁场方向同时垂直于电流方向和探头的扁平表面。磁场对载流子施加洛伦兹力，产生横向电压 V_H，称为霍尔电压。

V_H = k I B / d

Here k is the Hall coefficient, I is the current, and d is the thickness of the semiconductor. If V_H, I, and d are known, B can be obtained directly.

其中 k 是霍尔系数，I 是电流，d 是半导体的厚度。若 V_H、I 和 d 已知，则可以直接求出 B。

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5. Hall Probe Calibration | 霍尔探头的校准

In practice, the Hall probe must be calibrated against a known magnetic field before use. This is typically done inside a solenoid or Helmholtz coil where the field can be calculated precisely from the current and geometry. During calibration, a graph of V_H against B is plotted, which should yield a straight line passing through the origin.

在实际应用中，霍尔探头在使用前必须针对已知磁场进行校准。这通常在螺线管或亥姆霍兹线圈内完成，在这些装置中，磁场可以根据电流和几何尺寸精确计算。校准时绘制 V_H 对 B 的图线，应得到一条过原点的直线。

B (mT)	0	2.5	5.0	7.5	10.0
V_H (mV)	0	0.8	1.6	2.4	3.2

The gradient of this calibration graph gives the sensitivity of the probe in mV/mT, allowing unknown fields to be measured reliably.

校准图线的斜率给出探头的灵敏度（单位为 mV/mT），从而可以可靠地测量未知磁场。

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6. Advantages of the Hall Probe | 霍尔探头的优点

The Hall probe offers several significant advantages over the search coil method. It measures the field directly at a point without disturbing the field, and it responds to static (DC) fields as well as time-varying fields. The probe is compact and can access confined spaces where search coils cannot fit.

霍尔探头相比搜索线圈方法具有多项显著优势。它能在某一点直接测量磁场而不干扰原磁场，并且既能响应静态（直流）磁场，也能响应时变磁场。探头体积小巧，可以进入搜索线圈无法到达的狭窄空间。

• Direct measurement of B in real time | 实时直接测量B

• High spatial resolution due to small sensing area | 感应面积小，空间分辨能力高

• Suitable for both DC and AC fields | 适用于直流和交流磁场

The main disadvantage is that the probe is sensitive to temperature changes, requiring compensation circuits for accurate measurements in varying thermal conditions.

主要缺点在于探头对温度变化敏感，在热环境变化时需借助补偿电路来保证测量准确度。

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7. Measuring B in a Solenoid | 测量螺线管内的磁场

A solenoid provides a convenient way to produce a uniform magnetic field inside its core. The theoretical value of the field inside an ideal long solenoid is B = μ₀nI, where μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current.

螺线管可以在其内部产生均匀磁场，是一种便捷的磁场来源。理想长螺线管内部磁场的理论值为 B = μ₀nI，其中 μ₀ 是真空磁导率，n 是单位长度匝数，I 是电流强度。

B = μ₀ n I

Note that μ₀ = 4π × 10⁻⁷ T·m/A. To verify this experimentally, a Hall probe is inserted along the axis of the solenoid, and B is measured at various positions to confirm the uniformity of the field in the central region.

注意 μ₀ = 4π × 10⁻⁷ T·m/A。为从实验上验证该关系，可将霍尔探头沿螺线管轴线插入，测量不同位置处的 B 值，从而确认中心区域的磁场均匀性。

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8. Error Analysis and Precision | 误差分析与精密度

A thorough error analysis is essential for the A-Level practical examination. For the search coil method, the main sources of error include: the finite time taken to remove the coil, which may cause the deflection to be non-ballistic; the difficulty in ensuring the coil is exactly perpendicular to the field; and the uncertainty in measuring the deflection angle.

在 A-Level 实验考试中，全面的误差分析至关重要。对于搜索线圈法，主要误差来源包括：移出线圈所需的时间有限，可能导致偏转不完全呈冲击性；难以确保线圈完全垂直于磁场；以及偏转角度的测量不确定度。

For the Hall probe method, errors arise from the calibration procedure, temperature drift, and the fact that the probe has a finite sensing area, which averages the field over a small region. The accuracy of the current measurement also contributes to the overall uncertainty.

对于霍尔探头法，误差来自校准过程、温度漂移以及探头具有有限的感应面积（会在小区域内对磁场取平均）。电流测量的准确度也会影响总体不确定度。

ΔB/B = ΔV_H/V_H + ΔI/I + Δd/d

The fractional uncertainty in B is the sum of the fractional uncertainties in each directly measured quantity. Reducing systematic errors requires careful calibration and maintaining a constant temperature throughout the experiment.

B 的相对不确定度等于各直接测量量相对不确定度之和。减少系统误差需要仔细校准，并在整个实验过程中保持温度恒定。

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9. Computational Uncertainties | 不确定度的计算

When repeated measurements are taken, the mean value of B and its standard deviation should be calculated. For a set of n measurements, the standard error of the mean is given by σ/√n. The final result should be quoted with an appropriate number of significant figures and a stated uncertainty.

当进行重复测量时，应计算 B 的平均值及其标准偏差。对于 n 次测量，平均值的标准误差为 σ/√n。最终结果应保留适当的有效数字，并注明不确定度。

B = B̄ ± ΔB

For example, if the mean value is 12.4 mT and the uncertainty is 0.3 mT, the result should be recorded as (12.4 ± 0.3) mT. The percentage uncertainty is then (0.3/12.4) × 100% ≈ 2.4%.

例如，若平均值为 12.4 mT，不确定度为 0.3 mT，则结果应记录为 (12.4 ± 0.3) mT。此时百分比不确定度为 (0.3/12.4) × 100% ≈ 2.4%。

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10. Practical Considerations | 实验注意事项

Several practical factors can affect the reliability of magnetic flux density measurements. First, the Earth’s magnetic field (approximately 0.05 mT) should be considered when measuring weak fields. This can be eliminated by taking readings in both orientations and subtracting the background value.

多个实际因素会影响磁通量密度测量的可靠性。首先，当测量弱磁场时，应考虑到地磁场（约 0.05 mT）。可以通过在两个方向上进行读数并扣除背景值来消除其影响。

• Ensure all connections are tight and free from corrosion | 确保所有连接牢固且无腐蚀

• Orient the probe perpendicular to the field direction for maximum reading | 将探头垂直于磁场方向以获得最大读数

• Allow the equipment to warm up before taking measurements | 测量前让设备预热

• Keep ferromagnetic materials away from the measurement area | 使铁磁材料远离测量区域

These precautions help to minimise systematic errors and improve the consistency of the results.

这些预防措施有助于减小系统误差，提高结果的一致性。

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11. Safety Considerations | 安全注意事项

When measuring magnetic flux density, especially using electromagnets or high-current solenoids, safety must be a priority. Strong magnetic fields can interfere with pacemakers and cause metal objects to become projectiles. Always ensure that the current does not exceed the rated value of the apparatus to avoid overheating.

测量磁通量密度时，尤其是在使用电磁铁或大电流螺线管的情况下，必须将安全放在首位。强磁场可能干扰心脏起搏器，并使金属物体变成抛射体。务必确保电流不超过设备额定值，以避免过热。

For the Hall probe, handle the semiconductor element with care as it is fragile and sensitive to electrostatic discharge. When using a ballistic galvanometer, the instrument should be placed on a stable surface and levelled properly before readings are taken.

对于霍尔探头，应小心取放半导体元件，因其易碎且对静电放电敏感。使用冲击电流计时，仪器应放置在稳定平面上，并在读数前进行水平调节。

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12. Summary and Exam Tips | 总结与考试要点

In summary, the two most important methods for measuring magnetic flux density at A-Level are the search coil method combined with a ballistic galvanometer, and the Hall probe method. The former relies on the electromagnetic induction principle and is suitable for measuring changes in flux, while the latter depends on the Hall effect and provides direct point measurements of B.

总而言之，A-Level 阶段测量磁通量密度的两种最重要方法是搜索线圈配合冲击电流计法，以及霍尔探头法。前者依赖电磁感应原理，适用于测量磁通量的变化；后者依赖霍尔效应，可提供 B 的直接点测量。

For the examination, candidates should be able to describe the experimental setup, explain the underlying physics, identify sources of error, and suggest improvements. Also remember to quote the formulae ε = -N ΔΦ/Δt and V_H = kIB/d accurately, with all symbols clearly defined.

考试中，考生应能够描述实验装置、解释相关物理原理、识别误差来源并提出改进方案。同时，要准确写出公式 ε = -N ΔΦ/Δt 和 V_H = kIB/d，并清晰说明各符号的含义。

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📚 Observing and Demonstrating Magnetic Force in A-Level Physics | A-Level 物理：磁场力的观察与演示实验

Magnetic force is a fundamental interaction in A-Level physics, and seeing it in action is far more convincing than memorising formulas. This article walks through classic observations and demonstration experiments — from permanent magnets to current-carrying wires and charged particles in a magnetic field.

磁场力是 A-Level 物理中的基本相互作用。亲眼看它起作用，远比死记公式更有说服力。本文带你逐一梳理经典观察现象与演示实验——从永磁体、载流导线到磁场中的带电粒子。

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1. The Magnetic Force Between Two Permanent Magnets | 永磁体之间的磁场力

The simplest observation of magnetic force involves two bar magnets. When unlike poles are brought close, they attract; when like poles are brought close, they repel. The force is a non-contact force: it acts through space without any physical connection.

磁场力最简单的观察来自两根条形磁铁。当异名磁极靠近时相互吸引；同名磁极靠近时相互排斥。磁场力是一种非接触力：它不需要任何实物连接，就能在空间中起作用。

To demonstrate this in class, suspend one bar magnet from a thread so that it can rotate freely. Bring another magnet slowly toward it. The suspended magnet will swing to show both attraction and repulsion, clearly indicating the direction of the magnetic force.

课堂演示时，可用细线悬挂一根条形磁铁，让它能自由转动。手持另一根磁铁慢慢靠近，悬挂的磁铁便会摆动，既能展示吸引也能展示排斥，清晰说明磁场力的方向。

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2. Plotting Magnetic Field Lines | 描绘磁感线

Magnetic field lines are a visual tool used to represent the direction and strength of a magnetic field. Around a bar magnet, they emerge from the north pole and enter the south pole. The spacing of the lines indicates the field strength: closer lines mean a stronger field.

磁感线是表示磁场方向与强弱的可视化工具。在条形磁铁周围，磁感线从 N 极发出，进入 S 极。线的疏密反映磁场的强弱：越密则场越强。

A simple demonstration uses iron filings sprinkled on a sheet of paper placed over a bar magnet. Gently tap the paper; the filings align along the field lines. This works because each iron filing becomes a tiny temporary magnet and aligns with the local magnetic field direction.

一个简单的演示是把铁粉撒在覆盖条形磁铁的纸面上，轻轻敲击纸面，铁粉会沿磁感线排列。这是因为每个铁屑都变成微小临时磁体，沿当地磁场方向排列。

Alternatively, use a small plotting compass. Place the compass at one point, mark the direction of its needle, then move the compass so that the tail of the needle follows the previous mark. Repeating this traces one continuous field line.

另一种方法是使用小型罗盘磁针。在一点标记磁针方向，然后移动罗盘使磁针尾部对准前一个标记，重复操作即可画出一条连续的磁感线。

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3. Force on a Current-Carrying Wire in a Magnetic Field | 载流导线在磁场中受力

A current-carrying wire placed in a magnetic field experiences a magnetic force. This is the basis of electric motors and is described by the equation F = BIL sin θ, where B is the magnetic flux density, I is the current, L is the length of the wire in the field, and θ is the angle between the wire and the magnetic field.

置于磁场中的载流导体会受到磁场力。这是电动机的基础，其公式为 F = BIL sin θ，其中 B 是磁通密度，I 是电流，L 是处于磁场中的导线长度，θ 是导线与磁场方向的夹角。

F = BIL sin θ

For the maximum force, the wire must be perpendicular to the field (θ = 90°). If the wire is parallel to the field, the force is zero.

当导线与磁场垂直（θ = 90°）时力最大；若导线与磁场平行，则力为零。

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4. The Classic Cookbook Demonstration | 经典“天平”演示实验

One well-known classroom demonstration is the “cookbook” setup: a stiff copper wire is placed horizontally between the poles of a strong U-shaped magnet. A large current is passed through the wire. When the current is switched on, the wire jumps upward or downward, depending on the direction of the current.

一个著名的课堂演示是“天平”装置：将一根硬铜线水平置于强 U 形磁铁的两极之间，通以大电流。当电流接通时，铜线会向上或向下跳动，具体方向取决于电流方向。

This experiment clearly shows that the force is perpendicular to both the current direction and the magnetic field direction. Reversing the current reverses the force, which is a direct confirmation of Fleming’s left-hand rule.

该实验清楚表明，力的方向同时垂直于电流方向和磁场方向。改变电流方向会改变受力方向，这直接验证了弗莱明左手定则。

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5. Fleming’s Left-Hand Rule | 弗莱明左手定则

To predict the direction of the magnetic force on a current-carrying conductor, use Fleming’s left-hand rule: hold the thumb, first finger, and second finger of your left hand mutually at right angles. The First finger points in the direction of the magnetic Field, the seCond finger points in the direction of the Current, and the thuMb points in the direction of the Motion (force).

要判断载流导体所受磁场力的方向，可用弗莱明左手定则：左手拇指、食指与中指相互垂直。食指指向磁场方向（Field），中指指向电流方向（Current），拇指指向运动（受力）方向（Motion）。

A memorable way to demonstrate this is with a simple wire suspended between two vertical magnets. When current flows, the wire deflects sideways. Students can verify the rule by predicting the direction before switching on the current.

一个便于记忆的演示方式是用一根导线悬挂在两块竖直磁铁之间，电流通过时导线会向侧面偏转。学生可先预测受力方向，再接通电流验证定则。

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6. Turning Effect on a Coil: The Electric Motor Effect | 线圈的转动效应：电动机原理

If a rectangular coil is placed in a magnetic field and current is passed through it, two opposite sides of the coil experience forces in opposite directions. This creates a couple that rotates the coil. This is the electric motor effect.

若将矩形线圈置于磁场中并通入电流，线圈两条对边会受到方向相反的力，形成力偶使线圈转动。这就是电动机效应。

In the demonstration, a coil of wire is mounted on an axle between the poles of a magnet. When current flows, the coil rotates until it reaches the vertical position. A split-ring commutator reverses the current direction every half turn, so the coil continues to rotate.

演示时，将线圈装在转轴上，置于磁铁两极之间。通电后线圈转动，直到竖直位置。换向器（半环）每半圈改变一次电流方向，使线圈持续转动。

For A-Level, you should be able to calculate the maximum torque acting on a coil: τ = BANI, where A is the area of the coil and N is the number of turns.

在 A-Level 中，你需要会计算线圈所受的最大转矩：τ = BANI，其中 A 是线圈面积，N 是匝数。

τ = BANI

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7. Demonstrating the Force with a Cathode-Ray Tube | 用阴极射线管观察磁场力

A beam of electrons is invisible, but when it strikes a fluorescent screen it produces a bright spot. If a bar magnet is brought near the tube, the spot moves, showing that a magnetic field exerts a force on moving charged particles.

电子束本身不可见，但打在荧光屏上会形成亮点。当磁铁靠近阴极射线管时，亮点会移动，说明磁场对运动的带电粒子施加了力。

The direction of the force can be predicted using Fleming’s right-hand rule, since electrons move opposite to the conventional current direction. This experiment beautifully links electric current to the motion of charges.

受力方向可用弗莱明右手定则判断，因为电子运动方向与常规电流方向相反。该实验巧妙地将电流与电荷运动联系起来。

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8. Circular Motion of Charged Particles in a Magnetic Field | 带电粒子在磁场中的圆周运动

When a charged particle, such as an electron, enters a uniform magnetic field perpendicular to its velocity, the magnetic force acts as a centripetal force. The particle moves in a circle of radius r given by r = mv/(Bq), where m is the mass, q is the charge, B is the flux density, and v is the speed.

当带电粒子（如电子）垂直于磁场方向进入匀强磁场时，磁场力充当向心力，粒子做圆周运动。轨道半径 r = mv/(Bq)，其中 m 是质量，q 是电荷量，B 是磁通密度，v 是速度。

r = mv / (Bq)

A low-pressure gas tube can be used to make the electron beam visible. With Helmholtz coils producing a known magnetic field, the circular path of the beam is clearly seen. Varying B changes r inversely, confirming the mathematical relationship.

使用低压气体管可使电子束的轨迹可见。利用亥姆霍兹线圈产生已知磁场，能清晰看到电子束的圆形轨迹。改变 B 时 r 随之成反比变化，印证了上述数学关系。

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9. The Force Between Two Parallel Current-Carrying Wires | 两根平行载流导线之间的力

Two parallel wires carrying currents exert a magnetic force on each other. If the currents are in the same direction, the wires attract; if they are in opposite directions, they repel. This is because each wire creates a magnetic field that acts on the other wire.

两相互平行的载流导线之间会产生磁场力。当电流方向相同时，导线相互吸引；电流方向相反时，导线相互排斥。这是因为每根导线产生的磁场会对另一根导线施加作用力。

To demonstrate, suspend two flexible strips of metal foil vertically with a small separation. Connect them in series or parallel to control the current direction. When a large current passes, the strips visibly move together or apart. This effect is the definition of the ampere in SI units.

演示时，将两条柔性金属箔竖直悬挂，间隔很小。通过串联或并联来控制电流方向。当大电流通过时，箔条会明显靠近或分开。这一效应正是国际单位制中安培的定义基础。

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10. Using a Search Coil and CRO to Observe Induced Magnetic Effects | 用探测线圈与示波器观察感生磁场效应

Although magnetic force is the focus of this article, observing induction with a search coil can also illustrate magnetic field changes. When a magnet is moved into and out of a coil, the galvanometer deflects in opposite directions. The faster the motion, the larger the deflection.

虽然本文以磁场力为主线，但用探测线圈观察电磁感应也能帮助理解磁场的变化。当磁铁插入或拔出线圈时，电流计指针会向相反方向偏转；运动越快，偏转越大。

This demonstration introduces Faraday’s law and Lenz’s law, both of which link magnetic fields to forces on charges. A-level students are expected to explain the direction of the induced current using energy conservation.

这一实验引入法拉第定律和楞次定律，两者都将磁场与电荷受力联系起来。A-Level 学生应会用能量守恒解释感应电流的方向。

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📚 A-Level Physics: Interaction Forces Between Parallel Currents | A-Level 物理：平行电流间的相互作用力

When two straight conductors carrying electric currents are placed parallel to each other, they exert a force on each other. This force is not due to direct contact but arises from the magnetic fields each current creates and the interaction of those fields with the other current.

当两根通有电流的直导线平行放置时，它们之间会相互施力。这种力并非来自直接接触，而是源于每根导线中电流所产生的磁场，以及该磁场与另一根导线中电流的相互作用。

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1. Magnetic Field Around a Long Straight Wire | 长直导线周围的磁场

A steady current I flowing through a long straight wire generates a magnetic field with circular field lines centred on the wire. At a perpendicular distance r from the wire, the magnetic flux density (magnetic field strength) B is given by:

恒定电流 I 流过一根长直导线时，会在导线周围产生以导线为中心的圆形磁场线。在距导线垂直距离 r 处，磁通密度（磁感应强度）B 的表达式为：

B = μ₀I / (2πr)

Here μ₀ is the permeability of free space, μ₀ = 4π × 10⁻⁷ T·m·A⁻¹. The direction of the field is given by the right-hand grip rule: if the thumb points along the current, the curled fingers show the direction of the magnetic field lines.

其中 μ₀ 是真空磁导率，μ₀ = 4π × 10⁻⁷ T·m·A⁻¹。磁场方向由右手螺旋定则判断：拇指指向电流方向，弯曲的四指即表示磁场线方向。

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2. Force on a Current-Carrying Conductor in a Magnetic Field | 磁场中载流导线所受的力

A conductor of length L carrying current I placed in a uniform magnetic field B experiences a magnetic force given by:

长度为 L、通有电流 I 的导体置于匀强磁场 B 中时，受到的磁场力为：

F = BIL sinθ

where θ is the angle between the direction of the current and the magnetic field. When the current is perpendicular to the magnetic field (θ = 90°), the force is maximum: F = BIL. The direction of the force is determined by Fleming’s left-hand rule.

其中 θ 是电流方向与磁场方向之间的夹角。当电流垂直于磁场时（θ = 90°），力最大：F = BIL。力的方向用弗莱明左手定则判断。

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3. Magnetic Field at One Wire Due to the Other | 一根导线在另一根导线处产生的磁场

Consider two long parallel wires separated by distance d. Wire 1 carries current I₁ and wire 2 carries current I₂. At the location of wire 2, wire 1 produces a magnetic field B₁ perpendicular to the plane containing the two wires, with magnitude:

考虑两根相距为 d 的长平行导线。导线 1 通有电流 I₁，导线 2 通有电流 I₂。在导线 2 所在位置，导线 1 产生的磁场 B₁ 垂直于两根导线所在的平面，其大小为：

B₁ = μ₀I₁ / (2πd)

This field is perpendicular to the current I₂ in wire 2, so the force on a length L of wire 2 is:

该磁场垂直于导线 2 中的电流 I₂，因此长度为 L 的导线 2 所受的力为：

F = B₁ I₂ L = (μ₀ I₁ I₂ L) / (2πd)

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4. Force per Unit Length Between Two Parallel Wires | 两平行导线间单位长度的作用力

Rearranging the expression above gives the force per unit length between the two parallel wires:

整理上式可得两平行导线间单位长度的作用力：

F / L = μ₀ I₁ I₂ / (2πd)

This is the standard formula for the magnetic force between two long, straight, parallel currents. The force is attractive when the currents are in the same direction and repulsive when they are in opposite directions.

这就是两根长直平行电流之间磁力的标准公式。当电流方向相同时，力表现为吸引力；当电流方向相反时，力表现为排斥力。

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5. Direction of the Force: Same Direction Attracts | 力的方向：同向相吸

Use the right-hand grip rule to find the magnetic field due to wire 1 at wire 2. Then use Fleming’s left-hand rule on wire 2: point the first finger along the field and the middle finger along the current I₂; the thumb gives the direction of the magnetic force. For parallel currents in the same direction, the force on wire 2 points toward wire 1. Similarly, the force on wire 1 points toward wire 2, so the wires attract.

用右手螺旋定则找出导线 1 在导线 2 处产生的磁场方向，然后对导线 2 应用弗莱明左手定则：食指指向磁场方向，中指指向电流 I₂ 方向，拇指即为磁场力方向。当两电流同向时，导线 2 所受的力指向导线 1；同理，导线 1 所受的力指向导线 2，因此两根导线相互吸引。

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6. Direction of the Force: Opposite Directions Repel | 力的方向：反向相斥

If the currents flow in opposite directions, the same analysis shows that the magnetic field due to wire 1 at wire 2 is reversed, and the force on wire 2 points away from wire 1. The force on wire 1 also points away from wire 2, so the wires repel each other. This is consistent with Newton’s third law: the magnitudes of the forces on the two wires are always equal.

如果电流方向相反，同样的分析表明，导线 1 在导线 2 处产生的磁场方向反转，导线 2 所受的力指向远离导线 1 的方向；导线 1 所受的力也指向远离导线 2 的方向，因此两根导线相互排斥。这与牛顿第三定律一致：两根导线所受的力大小总是相等。

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7. Definition of the Ampere | 安培的定义

The force between parallel currents provides the operational definition of the ampere, the SI base unit of electric current. One ampere is defined as the constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, placed one metre apart in vacuum, would produce between these conductors a force equal to 2 × 10⁻⁷ newtons per metre of length.

平行电流之间的作用力为安培提供了操作定义。安培是国际单位制中电流的基本单位。安培定义为：真空中相距 1 米的两根无限长、圆截面可忽略的平行直导线内通以等量恒定电流时，若导线间相互作用力在每米长度上为 2 × 10⁻⁷ 牛顿，则此恒定电流的大小为 1 安培。

Substituting I₁ = I₂ = 1 A and d = 1 m into the formula gives F/L = (4π × 10⁻⁷ × 1 × 1) / (2π × 1) = 2 × 10⁻⁷ N·m⁻¹, confirming the definition.

将 I₁ = I₂ = 1 A、d = 1 m 代入公式，可得 F/L = (4π × 10⁻⁷ × 1 × 1) / (2π × 1) = 2 × 10⁻⁷ N·m⁻¹，与定义相符。

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8. Worked Example: Numerical Calculation | 例题：数值计算

Two long parallel wires are separated by 5.0 cm. Wire A carries a current of 3.0 A and wire B carries a current of 4.0 A in the same direction. Calculate the force per unit length between them and state whether it is attractive or repulsive.

两根长平行导线相距 5.0 cm。导线 A 通有 3.0 A 电流，导线 B 通有 4.0 A 电流，方向相同。计算它们之间的单位长度作用力，并说明是吸引力还是排斥力。

Using the formula with d = 0.050 m:

将 d = 0.050 m 代入公式：

F / L = (4π × 10⁻⁷ × 3.0 × 4.0) / (2π × 0.050)

= (4.8π × 10⁻⁶) / (0.100π) = 4.8 × 10⁻⁵ N·m⁻¹

Since the currents are in the same direction, the force is attractive.

由于电流方向相同，该力为吸引力。

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9. Common Pitfalls and Exam Tips | 常见易错点与考试提示

• Remember to use the perpendicular distance between the wires, not the length of the wire, in the denominator.

  注意公式中分母使用的是导线间的垂直距离，而不是导线本身的长度。

• The force per unit length formula already includes L in the denominator; do not multiply by L again unless asked for the total force.

  单位长度力的公式已经包含 L 在分母中；除非题目要求总力，否则不要再乘以 L。

• The force is magnetic, not electrostatic, even though the wires carry currents. Charges are not static.

  这种力是磁力而非静电力，即使导线中载有电流——电荷并非静止。

• For non-parallel wires, the force is not given by this simple formula; the magnetic field varies along the wire and the angle between B and I must be considered.

  对于不平行导线，此简单公式不适用；磁场沿导线变化，且必须考虑 B 与 I 之间的夹角。

• Always state the direction (attractive/repulsive) in your answer if the question asks for it.

  若题目要求说明方向（吸引/排斥），务必在答案中写出。

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10. Experimental Demonstration: Current Balance | 实验演示：电流天平

The force between parallel currents can be measured using a current balance. Two horizontal parallel conductors are arranged so that the upper one is balanced on a pivot. When currents flow in the same direction, the attractive force adds to the weight of the upper conductor; when opposite, the repulsive force reduces the apparent weight. By calibrating the balance, the value of F/L can be determined and compared with the theoretical formula.

平行电流间的作用力可通过电流天平进行测量。将两根水平平行导体布置成上导体可在支点上平衡的结构。当电流同向流动时，吸引力叠加到上导体的重量上；当电流反向时，排斥力减小表观重量。通过校准天平，可以测定 F/L 的值并与理论公式进行比较。

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11. Relation to Magnetic Flux Density | 与磁通密度的关系

From the force per unit length expression, we can also define the magnetic flux density B produced by a long straight wire. Since F/L = B₁ I₂, and B₁ = μ₀ I₁ / (2πd), the formula is consistent. This shows that the concept of magnetic field strength is a convenient intermediate step in understanding the force between currents.

由单位长度力的表达式，我们也可以定义长直导线产生的磁通密度 B。因为 F/L = B₁ I₂，且 B₁ = μ₀ I₁ / (2πd)，公式自洽。这表明磁感应强度的概念是理解电流间作用力的一个便捷中间步骤。

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12. Summary | 总结

Two parallel currents exert equal and opposite forces on each other. The magnitude of the force per unit length is given by F/L = μ₀ I₁ I₂ / (2πd). Parallel currents in the same direction attract; opposite currents repel. This principle is fundamental to the definition of the ampere and has numerous practical applications, including electromagnetic switches, railguns, and the forces between wires in electrical circuits.

两平行电流对彼此施加等大反向的作用力。单位长度力的大小为 F/L = μ₀ I₁ I₂ / (2πd)。同向平行电流相吸，反向平行电流相斥。这一原理是安培定义的基础，并具有众多实际应用，包括电磁开关、电磁轨道炮以及电路中导线之间的作用力。

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📚 A-Level Physics | 磁场的产生与图示方法

Magnetic fields are a fundamental part of the electromagnetic theory covered in A-Level physics. This article explains how magnetic fields are produced by permanent magnets and by electric currents, and how they can be represented using field lines, flux density and magnetic flux.

磁场是 A-Level 物理电磁学理论中的基础内容。本文将讲解磁场如何由永磁体和电流产生，以及如何用磁感线、磁通密度和磁通量来图示磁场。

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1. What Is a Magnetic Field? | 什么是磁场？

A magnetic field is a region of space in which a magnetic force is experienced by a moving electric charge or by a magnetic material such as iron. It is a vector field, meaning that at every point it has both a direction and a magnitude.

磁场是空间中这样一个区域：在该区域内，运动的电荷或铁等磁性材料会受到磁力。磁场是矢量场，即空间每一点都有方向和大小。

Magnetic fields are produced fundamentally by moving charge. This includes currents in wires and the internal electron motion that gives permanent magnets their properties.

磁场从根本上由运动电荷产生，包括导线中的电流，以及使永磁体具有磁性的内部电子运动。

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2. Production by Permanent Magnets | 永磁体产生的磁场

Permanent magnets contain many small regions called magnetic domains. Within each domain, the atomic magnetic moments are aligned in the same direction. When a large number of domains point the same way, the material produces a net external magnetic field.

永磁体内部包含许多称为磁畴的小区域。在每个磁畴内，原子磁矩都沿同一方向排列。当大量磁畴方向一致时，材料就会产生净的外部磁场。

Every magnet has a north pole (N) and a south pole (S). Outside the magnet, field lines leave the north pole and enter the south pole; inside the magnet, they continue from the south pole back to the north pole, forming closed loops.

每个磁体都有北极（N）和南极（S）。在磁体外部，磁感线从北极出发进入南极；在磁体内部，磁感线从南极回到北极，形成闭合回路。

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3. Magnetic Field Around a Straight Current-Carrying Conductor | 直线载流导线周围的磁场

When a direct current flows through a straight wire, a magnetic field is produced around the wire. The magnetic field lines are concentric circles centred on the wire, lying in planes perpendicular to the wire.

当直流电通过直导线时，导线周围会产生磁场。磁感线是以导线为圆心的同心圆，位于与导线垂直的平面内。

The direction of the field is given by the right-hand grip rule: if the right thumb points in the direction of conventional current, the curled fingers show the direction of the magnetic field.

磁场方向由右手螺旋定则确定：若右手拇指指向电流方向，弯曲的四指所指方向即为磁场方向。

For a long straight wire in a vacuum, the magnetic flux density B at a perpendicular distance r from the wire is

在真空中，长直导线外距导线垂直距离 r 处的磁通密度 B 为

B = μ₀I / (2πr)

where I is the current and μ₀ is the permeability of free space. Thus the field strength is proportional to the current and inversely proportional to the distance from the wire.

式中 I 为电流，μ₀ 为真空磁导率。因此磁场强度与电流成正比，与到导线的距离成反比。

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4. Magnetic Field of a Flat Circular Coil | 平面圆形线圈的磁场

A single circular loop of wire carrying a current produces a magnetic field that is concentrated around the loop. Field lines pass through the centre of the coil, curve around outside, and return to the other side.

一个载流圆形线圈产生的磁场集中在线圈周围。磁感线穿过线圈中心，在外侧弯曲，再回到另一侧。

When viewed from one face, if the current flows anticlockwise, that face acts as a north pole; if the current flows clockwise, that face acts as a south pole.

从一侧面观察时，若电流沿逆时针方向流动，则该面等效为北极；若电流沿顺时针方向流动，则该面等效为南极。

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5. Magnetic Field of a Solenoid | 螺线管的磁场

A solenoid is a long coil formed by many turns of wire. Its magnetic field is very similar to that of a bar magnet. Inside the solenoid, the field lines are nearly parallel and equally spaced, so the field is approximately uniform.

螺线管是由多匝导线绕成的长线圈。它的磁场与条形磁铁的磁场非常相似。螺线管内部磁感线近似平行且间距相等，因此内部磁场近似匀强。

For an ideal solenoid, the magnetic flux density inside the solenoid is

理想螺线管内部的磁通密度为

B = μ₀nI

where n is the number of turns per unit length. The right-hand grip rule also applies: curl the fingers in the direction of the current, and the thumb points toward the north-pole end of the solenoid.

式中 n 为单位长度上的匝数。右手螺旋定则同样适用：四指沿电流方向弯曲，拇指指向螺线管的北极端。

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6. Representing Fields with Magnetic Field Lines | 用磁感线表示磁场

Magnetic field lines, also called flux lines, are a graphical way to describe a magnetic field. The direction of the field at any point is tangent to the line passing through that point.

磁感线，又称

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📚 A-Level Physics: Definition and Units of Magnetic Flux Density | A-Level 物理：磁通量密度的定义与单位

Magnetic flux density is one of the most important concepts in A-Level physics, especially in the CIE syllabus. It connects magnetic forces, electric currents, and the geometry of magnetic fields into a single measurable quantity. This article explains its definition, the equation used to define it, its units, and common exam applications.

磁通量密度是 A-Level 物理（尤其是 CIE 考纲）中最核心的概念之一。它将磁力、电流和磁场的几何形状联系成一个可测量的物理量。这篇文章将讲解它的定义、定义式、单位以及常见考点应用。

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1. What Is Magnetic Flux Density? | 什么是磁通量密度？

Magnetic flux density, usually given the symbol B, describes how strong and how concentrated a magnetic field is at a particular point. It is a vector quantity, meaning it has both magnitude and direction. The direction of B is the direction in which a tiny north pole would experience a force, or equivalently, the direction of the field lines.

磁通量密度通常用符号 B 表示，用来描述磁场在某一点的强弱与密集程度。它是一个矢量，既有大小也有方向。B 的方向是小磁针北极受力方向，也就是磁场线的切线方向。

Although many students first meet B in the context of the force on a current-carrying wire, the formal definition relies on that very force. The SI unit of B is the tesla (T), named after Nikola Tesla.

虽然很多同学首次接触 B 是在电流导线受力的场景中，但 B 的正式定义恰恰依赖这个力。B 的国际单位是特斯拉（T），以尼古拉·特斯拉命名。

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2. The Defining Equation: F = BIL sin θ | 定义式：F = BIL sin θ

For a straight wire of effective length L carrying a current I inside a uniform magnetic field of flux density B, the magnetic force F on the wire is given by:

对于一段有效长度为 L、通有电流 I 的直导线，处于磁通量密度为 B 的匀强磁场中时，导线受到的磁力 F 为：

F = B I L sin θ

Here, θ is the angle between the direction of the current and the direction of the magnetic field. When the wire is perpendicular to the field, θ = 90° and sin θ = 1, so the force is maximum:

其中 θ 是电流方向与磁场方向之间的夹角。当导线垂直于磁场时，θ = 90°，sin θ = 1，力达到最大值：

F = B I L

This maximum-force case is used to define B. Rearranging gives:

这个最大力的情况被用来定义 B。整理后得到：

B = F / (I L)

Thus magnetic flux density can be defined as the force per unit current per unit length acting on a straight conductor placed perpendicular to the magnetic field.

因此磁通量密度可以定义为：垂直于磁场的直导线，单位电流、单位长度上所受到的磁力。

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3. Alternative Definition: Magnetic Flux per Unit Area | 另一种定义：单位面积上的磁通量

Magnetic flux density is also related to magnetic flux Φ. Magnetic flux is the total number of magnetic field lines passing through a given area, while flux density is the concentration of those lines per unit area. For a uniform field perpendicular to a plane of area A:

磁通量密度还与磁通量 Φ 有关。磁通量是穿过某一面积的总磁场线数，而磁通量密度是单位面积上磁场线的密集程度。对于匀强磁场且方向垂直于面积为 A 的平面时：

Φ = B A

If the field is not perpendicular to the area, a cos θ factor is needed, where θ is the angle between the magnetic field direction and the normal to the area:

如果磁场不垂直于平面，则需要乘以 cos θ，其中 θ 是磁场方向与平面法线之间的夹角：

Φ = B A cos θ

Because the tesla can be expressed as a weber per square metre (Wb m⁻²), B can be thought of as “flux per unit area.” This is why it is called flux density.

因为特斯拉也可以表示为韦伯每平方米（Wb m⁻²），所以 B 可以被理解为“单位面积上的磁通量”。这就是为什么它被称为磁通量密度。

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4. Unit of Magnetic Flux Density: The Tesla | 磁通量密度的单位：特斯拉

From B = F / (I L), we can derive the tesla in base SI units. Force F has units of newtons (N = kg m s⁻²), current I has units of amperes (A), and length L has units of metres (m). Therefore:

从 B = F / (I L) 出发，我们可以推导出特斯拉用 SI 基本单位表示的形式。力 F 的单位是牛顿（N = kg m s⁻²），电流 I 的单位是安培（A），长度 L 的单位是米（m）。因此：

1 T = 1 N A⁻¹ m⁻¹ = 1 kg A⁻¹ s⁻²

Equivalently, using the flux definition Φ = B A, we have:

等效地，从磁通量定义 Φ = B A 可得：

1 T = 1 Wb m⁻²

So one tesla is the flux density that produces a force of 1 newton on a 1 metre wire carrying a current of 1 ampere, when the wire is perpendicular to the field.

因此，1 特斯拉是：当 1 米长导线通有 1 安培电流且垂直磁场放置时，受到 1 牛顿磁力所对应的磁通量密度。

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5. Magnetic Flux Density and Moving Charges | 磁通量密度与运动电荷

A charged particle moving with velocity v through a magnetic field B experiences a magnetic force. For a charge q, the magnitude of the force is:

带电粒子以速度 v 在磁场 B 中运动时，会受到磁力。对于电荷 q，力的大小为：

F = q v B sin θ

where θ is the angle between the velocity and the magnetic field direction. When the charge moves perpendicular to the field, sin θ = 1 and F = qvB. This equation is often used in circular motion problems involving charged particles in magnetic fields.

其中 θ 是速度方向与磁场方向的夹角。当电荷垂直磁场运动时，sin θ = 1，F = qvB。这个公式常用于解带电粒子在磁场中做圆周运动的题目。

This relationship is also a possible way to define B: the magnetic flux density is the force acting on a unit charge moving at unit velocity perpendicular to the field.

这一关系也可以用来定义 B：磁通量密度是单位电荷以单位速度垂直磁场运动时所受到的磁力。

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6. Magnetic Flux Density vs Magnetic Flux | 磁通量密度与磁通量的区别

Students often confuse magnetic flux density B and magnetic flux Φ. The table below summarises the key differences.

同学们经常混淆磁通量密度 B 和磁通量 Φ。下表总结了它们的核心区别。

Quantity	Magnetic Flux Density B	Magnetic Flux Φ
Nature	Vector (strength of field at a point)	Scalar (total field lines through an area)
Symbol	B	Φ
SI unit	tesla (T)	weber (Wb)
Related formula	B = F / (IL)	Φ = B A cos θ

Note that Φ depends on the area through which the field passes, while B is a local property of the field. A strong field can have a small flux if the area is tiny; a weak field can have a large flux if the area is huge.

注意：Φ 取决于磁场穿过的面积，而 B 是磁场的局部属性。场很强但面积很小时，磁通量可能很小；场很弱但面积很大时，磁通量也可能很大。

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7. Worked Example: Calculating Force from B | 例题：由 B 计算磁力

A straight wire of length 0.20 m carries a current of 4.0 A and is placed in a uniform magnetic field of flux density 0.30 T. The wire is perpendicular to the field. Calculate the magnetic force on the wire.

一根长为 0.20 m 的直导线通有 4.0 A 的电流，放在磁通量密度为 0.30 T 的匀强磁场中，导线与磁场垂直。求导线所受磁力。

Since the wire is perpendicular to the field, sin θ = 1. Using F = BIL:

因为导线垂直于磁场，sin θ = 1。利用 F = BIL：

F = 0.30 × 4.0 × 0.20 = 0.24 N

The force is 0.24 N. Use Fleming’s left-hand rule to determine the direction if required.

力为 0.24 N。如果需要确定方向，可以用弗莱明左手定则。

If the wire were at 30° to the field instead, then θ = 30° and:

如果导线与磁场夹角为 30°，则 θ = 30°，于是：

F = 0.30 × 4.0 × 0.20 × sin 30° = 0.12 N

This smaller value shows why the sin θ factor is essential in the general formula.

这个较小的值说明为什么通用公式中的 sin θ 因子必不可少。

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8. Worked Example: Determining B from Measurements | 例题：由测量值求 B

A student places a 5.0 cm wire at right angles to a magnetic field. When a current of 2.0 A flows, the wire experiences a force of 0.015 N. What is the magnetic flux density?

学生将一根 5.0 cm 的导线垂直放入磁场中，当通过 2.0 A 电流时，导线受到 0.015 N 的力。求磁通量密度。

First convert length to metres: L = 0.050 m. Since the wire is perpendicular to the field, rearrange B = F / (IL):

首先将长度换算为米：L = 0.050 m。由于导线垂直磁场，整理 B = F / (IL)：

B = 0.015 / (2.0 × 0.050) = 0.15 T

So the flux density is 0.15 T.

因此磁通量密度为 0.15 T。

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9. Common Exam Misconceptions | 常见考试误区

• Using the wrong angle: The angle θ in F = BIL sin θ is the angle between the current direction and the magnetic field direction, not the angle with the normal to the plane.

  角度用错：F = BIL sin θ 中的 θ 是电流方向与磁场方向的夹角，而不是与平面法线的夹角。

• Forgetting the sin θ factor: When the wire is parallel to the field, θ = 0° and the force is zero. Some students still use F = BIL.

  忘记 sin θ 因子：当导线平行于磁场时，θ = 0°，力为零。有些同学仍然使用 F = BIL。

• Confusing B and Φ: B has units T, Φ has units Wb. In exam questions, check whether the area is already included.

  混淆 B 与 Φ：B 的单位是 T，Φ 的单位是 Wb。做题目时注意面积是否已经包含在内。

• Not converting units: Length must be in metres, area in square metres, and force in newtons before substituting into formulas.

  未换算单位：代入公式前，长度必须用米，面积必须用平方米，力必须用牛顿。

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10. Key Points for Revision | 复习要点

• Magnetic flux density B is a vector quantity representing the strength of a magnetic field at a point.

  磁通量密度 B 是矢量，代表磁场在某一点的强弱。

• The defining equation is B = F / (IL) for a wire perpendicular to the field.

  定义式为 B = F / (IL)，适用于导线垂直磁场的情况。

• The SI unit is the tesla: 1 T = 1 N A⁻¹ m⁻¹ = 1 Wb m⁻² = 1 kg A⁻¹ s⁻².

  国际单位是特斯拉：1 T = 1 N A⁻¹ m⁻¹ = 1 Wb m⁻² = 1 kg A⁻¹ s⁻²。

• The general force equation is F = BIL sin θ; the force is maximum when the wire is perpendicular to B and zero when parallel.

  通用力公式是 F = BIL sin θ；导线垂直于 B 时力最大，平行于 B 时力为零。

• Magnetic flux density can also be related to flux by Φ = B A cos θ; therefore B = Φ / (A cos θ).

  磁通量密度也可通过 Φ = B A cos θ 与磁通量联系；因此 B = Φ / (A cos θ)。

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With these definitions and formulas, you can confidently solve CIE A-Level problems on magnetic flux density and connect them to forces on currents and moving charges.

掌握了这些定义和公式，你就能自信地解决 CIE A-Level 中关于磁通量密度的问题，并将其与电流受力和运动电荷受力联系起来。

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📚 A-Level Physics: The Nature and Calculation of Magnetic Forces | A-Level 物理：磁场力的本质与计算

Magnetic forces are one of the fundamental interactions in physics, arising from the motion of charged particles. In the CIE A-Level syllabus, students are required not only to calculate magnetic forces using quantitative relationships but also to understand the physical origin of these forces. This article explains the essential nature of magnetic forces, the key equations, and common application scenarios.

磁场力是物理学中最基本的相互作用之一，源于带电粒子的运动。在 CIE A-Level 考纲中，学生不仅需要运用定量关系计算磁场力，还需要理解这些力的物理本质。本文旨在解释磁场力的核心本质、关键公式以及常见应用场景。

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1. The Origin of Magnetic Forces | 磁场力的起源

Magnetic forces are fundamentally relativistic effects of electric forces. When a charged particle moves relative to an observer, its electric field becomes distorted, creating a magnetic field. Another moving charge interacting with this field experiences a magnetic force. In the A-Level syllabus, we treat magnetic fields as existing entities produced by moving charges or permanent magnets, without delving into special relativity.

从本质上讲，磁场力是电场力的相对论效应。当一个带电粒子相对于观察者运动时，它的电场会发生畸变，从而产生磁场。另一个运动电荷与该磁场相互作用时就会受到磁场力的作用。在 A-Level 课程中，我们将磁场视为由运动电荷或永磁体产生的客观存在，不需深入涉及狭义相对论。

Two key facts about magnetic forces: they only act on moving charges; and they do no work on a charged particle because the force is always perpendicular to the velocity.

关于磁场力有两条关键事实：它只对运动电荷起作用；并且它对带电粒子不做功，因为力的方向始终垂直于速度方向。

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2. Magnetic Flux Density B | 磁感应强度 B

The magnetic flux density B is a vector quantity that describes the strength and direction of a magnetic field. Its SI unit is the tesla (T), where 1 T = 1 N·A⁻¹·m⁻¹. One tesla is defined as the magnetic flux density that produces a force of 1 newton on a wire of length 1 metre carrying a current of 1 ampere placed perpendicular to the field.

磁感应强度 B 是描述磁场强弱和方向的矢量，其国际单位是特斯拉（T），1 T = 1 N·A⁻¹·m⁻¹。1 特斯拉定义为：在垂直于磁场方向放置的长度为 1 米、通有 1 安培电流的直导线上产生 1 牛顿力的磁感应强度。

B is distinguished from magnetic flux Φ, which is the product of B and the perpendicular area it passes through. The relationship is Φ = BA cos θ, where θ is the angle between the field direction and the normal to the area.

B 与磁通量 Φ 不同，后者是 B 与其垂直穿过的面积的乘积，关系为 Φ = BA cos θ，其中 θ 是磁场方向与面积法线方向之间的夹角。

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3. Force on a Moving Charge in a Magnetic Field | 磁场对运动电荷的作用力

When a charged particle with charge q moves with velocity v through a uniform magnetic field B, the magnetic force F is given by:

当一个带电量为 q 的粒子以速度 v 穿过匀强磁场 B 时，所受磁场力 F 为：

F = Bqv sin θ

where θ is the angle between the velocity vector and the magnetic field vector. The direction of this force is given by Fleming’s left-hand rule for positive charges. The force is maximum when θ = 90° (v perpendicular to B), and zero when θ = 0° or 180° (v parallel or anti-parallel to B).

其中 θ 是速度矢量与磁场矢量之间的夹角。力的方向由弗莱明左手定则确定（适用于正电荷）。当 θ = 90°（v 垂直于 B）时力最大；当 θ = 0° 或 180°（v 平行或反平行于 B）时力为零。

This equation is often written in vector form as F = qv × B, where × denotes the cross product. Since magnetic force is always perpendicular to velocity, it changes the direction of motion but not the speed, resulting in circular motion when the velocity is perpendicular to the field.

该方程常用矢量形式写作 F = qv × B，其中 × 表示叉积。由于磁场力始终垂直于速度，它只改变运动方向而不改变速度大小，因此当速度垂直于磁场时，带电粒子做匀速圆周运动。

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4. Circular Motion of Charged Particles | 带电粒子的圆周运动

When a charged particle enters a uniform magnetic field with velocity perpendicular to the field, the magnetic force provides the centripetal force:

当带电粒子以垂直于磁场方向的速度进入匀强磁场时，磁场力提供向心力：

Bqv = mv²/r

Rearranging gives the radius of the circular path:

整理后可得圆周运动的半径：

r = mv / (Bq)

The angular velocity ω and the period T of the circular motion are independent of the particle’s speed:

圆周运动的角速度 ω 和周期 T 与粒子速度无关：

ω = Bq/m, T = 2πm / (Bq)

This speed independence is the principle behind cyclotrons and mass spectrometers.

周期与速度无关这一特性是回旋加速器和质谱仪的工作原理基础。

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5. Force on a Current-Carrying Conductor | 磁场对通电导体的作用力

A current-carrying wire placed in a magnetic field experiences a force because the moving charges (electrons) inside the wire each experience a magnetic force, which is collectively transmitted to the wire lattice. The total force on a straight wire of length L carrying current I in a uniform magnetic field B is:

通电导线在磁场中会受到力的作用，因为导线内部运动的电荷（电子）各自受到磁场力，这些力集体传递给导线晶格。长度为 L、通有电流 I 的直导线在匀强磁场 B 中所受的合力为：

F = BIL sin θ

where θ is the angle between the wire direction and the magnetic field direction. When θ = 90°, the force is simply F = BIL. This is the equation used to define the tesla.

其中 θ 是导线方向与磁场方向之间的夹角。当 θ = 90° 时，力简化为 F = BIL。这正是定义特斯拉所用的关系式。

This force is the basis of electric motors, galvanometers, and loudspeakers. Fleming’s left-hand rule determines the direction: thumb points in the direction of the force, first finger in the field direction (N to S), and second finger in the conventional current direction.

这种力是电动机、电流计和扬声器的基础。弗莱明左手定则用于判断方向：大拇指指向力的方向，食指指向磁场方向（N 到 S），中指指向电流方向。

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6. Comparison of Magnetic and Electric Forces | 磁场力与电场力的比较

It is useful to compare the behaviour of electric and magnetic forces on moving charges:

比较运动电荷所受电场力与磁场力的行为特征很有帮助：

Property	Electric Force | 电场力	Magnetic Force | 磁场力
Acts on stationary charges 是否作用于静止电荷	Yes | 是	No | 否
Force direction | 力的方向	Parallel or anti-parallel to E | 平行或反平行于 E	Perpendicular to both v and B | 垂直于 v 和 B
Work done on particle | 对粒子做功	Can do work | 可以做功	Always zero | 始终为零
Change in kinetic energy | 动能变化	Can change | 可以改变	No change | 不改变

A magnetic field can change the direction of a charged particle’s motion but never its speed, so kinetic energy remains constant. In contrast, an electric field can accelerate or decelerate particles, changing their kinetic energy.

磁场可以改变带电粒子的运动方向，但永远不能改变其速率，因此动能保持不变。相比之下，电场可以使粒子加速或减速，从而改变其动能。

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7. Magnetic Force between Two Parallel Currents | 两平行电流之间的磁场力

Two parallel current-carrying wires exert magnetic forces on each other. Wire 1 creates a magnetic field at the location of wire 2; wire 2, carrying current, experiences a force in that field. According to Ampère’s law, the force per unit length between two long parallel wires separated by distance d and carrying currents I₁ and I₂ is:

两条平行的通电导线之间会相互施加磁场力。导线 1 在导线 2 的位置产生磁场；导线 2 通有电流，在该磁场中受力。根据安培定律，相距 d、分别通有电流 I₁ 和 I₂ 的两条长直平行导线之间单位长度的力为：

F/L = μ₀I₁I₂ / (2πd)

where μ₀ = 4π × 10⁻⁷ T·m·A⁻¹ is the permeability of free space. Currents in the same direction attract; currents in opposite directions repel. This is the operational definition of the ampere.

其中 μ₀ = 4π × 10⁻⁷ T·m·A⁻¹ 是真空磁导率。同向电流相互吸引，反向电流相互排斥。这也是安培的操作性定义。

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8. Hall Effect | 霍尔效应

The Hall effect occurs when a current-carrying conductor is placed in a perpendicular magnetic field. The magnetic force pushes the charge carriers to one side of the conductor, creating a transverse potential difference known as the Hall voltage. This voltage is given by:

霍尔效应发生在通电导体置于垂直磁场中时。磁场力将载流子推向导体的一侧，从而产生横向电势差，即霍尔电压。霍尔电压为：

V_H = B I / (n t q)

where n is the number density of charge carriers, t is the thickness of the conductor in the direction of B, and q is the charge of each carrier. The Hall effect is used to measure magnetic field strength and to determine the sign and density of charge carriers in materials.

其中 n 是载流子数密度，t 是导体沿 B 方向的厚度，q 是每个载流子的电荷量。霍尔效应可用于测量磁场强度，以及判断材料中载流子的符号和密度。

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9. Worked Example | 典型例题

Example 1: A proton (mass m = 1.67 × 10⁻²⁷ kg, charge q = 1.60 × 10⁻¹⁹ C) moves at v = 2.0 × 10⁶ m/s perpendicular to a magnetic field B = 0.50 T. Calculate the radius of its circular path.

例 1：一个质子（质量 m = 1.67 × 10⁻²⁷ kg，电荷 q = 1.60 × 10⁻¹⁹ C）以 v = 2.0 × 10⁶ m/s 的速度垂直于 B = 0.50 T 的磁场运动。求其圆周轨道半径。

Solution | 解答：

Using r = mv/(Bq):

由 r = mv/(Bq)：

r = (1.67 × 10⁻²⁷)(2.0 × 10⁶) / (0.50 × 1.60 × 10⁻¹⁹) = 0.042 m

The radius is approximately 4.2 cm.

半径约为 4.2 cm。

Example 2: A wire of length 0.20 m carries a current of 3.0 A at 30° to a uniform magnetic field of 0.40 T. Calculate the magnetic force on the wire.

例 2：一根长度为 0.20 m 的导线通有 3.0 A 的电流，与 0.40 T 匀强磁场的夹角为 30°。求导线所受磁场力。

Solution | 解答：

F = BIL sin θ = 0.40 × 3.0 × 0.20 × sin 30° = 0.12 N

The direction is given by Fleming’s left-hand rule.

方向由弗莱明左手定则确定。

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10. Common Exam Pitfalls | 常见考试误区

• Using F = BIL when the wire is not perpendicular to B. Always include sin θ.

  导线不与 B 垂直时直接套用 F = BIL。务必加上 sin θ。

• Forgetting that magnetic force does no work; it cannot increase a particle’s speed.

  忘记磁场力不做功，它不能增大粒子的速率。

• Applying Fleming’s left-hand rule for negative charges. For electrons, the current direction is opposite to the electron motion.

  对负电荷错误地应用弗莱明左手定则。对于电子，电流方向与电子运动方向相反。

• Confusing magnetic flux density B with magnetic flux Φ. B is measured in tesla, Φ in weber.

  混淆磁感应强度 B 和磁通量 Φ。B 的单位是特斯拉，Φ 的单位是韦伯。

• Forgetting that the period of circular motion in a magnetic field is independent of speed.

  忘记磁场中圆周运动的周期与速度无关。

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11. Experimental Determination of B | 测量磁感应强度的实验方法

In the laboratory, the magnetic flux density can be measured using a search coil connected to a calibrated fluxmeter, or by measuring the force on a current-carrying wire. The current balance experiment directly uses F = BIL. A wire of known length L is placed perpendicular to the magnetic field, and a current I is passed through it. The force is measured by the change in balance reading; B is then calculated as B = F/(IL).

在实验室中，可以使用探测线圈连接校准过的磁通计来测量磁感应强度，也可以通过测量通电导线所受的力来测定。电流天平实验直接利用 F = BIL。将已知长度 L 的导线垂直于磁场放置，通以电流 I，通过天平读数的变化测量力的大小，再由 B = F/(IL) 计算 B。

The Hall probe is another useful instrument. When the probe is placed in a magnetic field with its thin semiconductor layer perpendicular to B, the Hall voltage generated is proportional to B, providing a direct and convenient measurement.

霍尔探头是另一种常用仪器。将探头薄半导体层垂直于 B 放入磁场时，产生的霍尔电压与 B 成正比，从而直接方便地测量磁感应强度。

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12. Summary | 总结

Magnetic forces arise from the interaction between moving charges and magnetic fields. The fundamental equations to remember are:

磁场力源于运动电荷与磁场的相互作用。需要牢记的基本公式为：

F = Bqv sin θ (single charge) | 单个电荷

F = BIL sin θ (conductor) | 导体

r = mv/(Bq) (circular motion) | 圆周运动

The magnetic force is always perpendicular to both the velocity and the magnetic field; it changes direction but not speed. Understanding the vector nature of this force, along with Fleming’s left-hand rule, is essential for solving A-Level problems. Mastery of these concepts forms the foundation for electromagnetism topics in further study.

磁场力始终垂直于速度方向和磁场方向；它改变运动方向但不改变速率。理解该力的矢量性质，并熟练掌握弗莱明左手定则，是解决 A-Level 问题的关键。牢固掌握这些概念，将为后续深入学习电磁学奠定坚实的基础。

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📚 A-Level Physics: Communication Channel Types & Information Transmission | A-Level 物理：通信信道类型与信息传输

Communication systems are a fundamental part of modern physics and technology. This article explores the different types of communication channels, the principles of information transmission, and the key parameters that determine system performance — all essential knowledge for CIE A-Level Physics students.

通信系统是现代物理学和技术的重要组成部分。本文探讨不同类型的通信信道、信息传输的原理以及决定系统性能的关键参数——这些都是CIE A-Level物理学生必备的知识。

1. What is a Communication Channel? | 什么是通信信道？

A communication channel is the physical medium through which information is transmitted from a source (transmitter) to a destination (receiver). Information is typically carried by electromagnetic waves, electrical signals, or light pulses.

通信信道是信息从信源（发射器）传输到目的地（接收器）所经过的物理介质。信息通常由电磁波、电信号或光脉冲来承载。

The channel is part of a larger communication system consisting of:

• Transmitter: converts the original information into a form suitable for transmission.
• Channel: the medium that carries the signal.
• Receiver: decodes the transmitted signal back into usable information.

信道是更大通信系统的一部分，该系统的组成包括：

• 发射器：将原始信息转换为适合传输的形式。
• 信道：承载信号的介质。
• 接收器：将传输的信号解码恢复为可用信息。

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2. Types of Communication Channels | 通信信道的类型

Communication channels can be broadly classified into two main categories: guided (wired) media and unguided (wireless) media. Each type has distinct characteristics, advantages, and limitations.

通信信道可大致分为两类：有线（导向）介质和无线（非导向）介质。每种类型都有其独特的特性、优点和局限性。

Channel Type	Examples	Frequency Range	Typical Use
信道类型	实例	频率范围	典型用途
Guided (wired) 有线	Twisted pair 双绞线 Coaxial cable 同轴电缆 Fibre-optic cable 光纤电缆	KHz – GHz	Telephone, LAN, broadband 电话、局域网、宽带
Unguided (wireless) 无线	Radio waves 无线电波 Microwaves 微波 Infrared 红外线 Satellite links 卫星链路	KHz – THz	Broadcast, mobile, space comms 广播、移动通信、太空通信

For A-Level purposes, you should be able to compare these channels in terms of bandwidth, attenuation, noise susceptibility, and cost.

就A-Level考试而言，你需要能够从带宽、衰减、噪声敏感度和成本等方面比较这些信道。

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3. Wired Channels: Twisted Pair and Coaxial Cable | 有线信道：双绞线和同轴电缆

Twisted pair cable consists of two insulated copper wires twisted together. The twisting reduces electromagnetic interference (EMI) because the noise induced in each wire cancels out. It is cheap, flexible, and suitable for short-distance applications such as telephone lines and local area networks (LANs).

双绞线电缆由两根相互缠绕的绝缘铜线组成。缠绕可以减少电磁干扰（EMI），因为每根导线中感应的噪声相互抵消。它价格低廉、柔韧性好，适用于短距离应用，如电话线和局域网（LAN）。

Coaxial cable has a central copper conductor surrounded by a dielectric insulator, a metallic braided shield, and an outer insulating jacket. The shielding provides better protection against noise and allows higher bandwidths than twisted pair. Coaxial cables are commonly used for cable television and high-speed internet connections.

同轴电缆的中心是铜导体，周围由介电绝缘层、金属编织屏蔽层和外部绝缘护套包裹。屏蔽层提供更好的抗噪声保护，并能支持比双绞线更高的带宽。同轴电缆常用于有线电视和高速互联网连接。

Key comparison points:

关键比较要点：

• Coaxial cable has higher bandwidth than twisted pair due to better shielding.
• Twisted pair is cheaper but more susceptible to interference.
• Both suffer from attenuation — signal loses energy as it travels along the cable.

• 同轴电缆由于屏蔽效果更好，带宽高于双绞线。
• 双绞线更便宜，但更容易受到干扰。
• 两者都存在衰减——信号沿电缆传播时能量逐渐损失。

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4. Fibre-Optic Channels | 光纤信道

Optical fibre is a thin strand of glass or plastic that transmits information using pulses of light. The principle of total internal reflection (TIR) keeps the light trapped inside the core. The fibre has a higher refractive index ((n_1)) core surrounded by a lower refractive index ((n_2)) cladding.

光纤是极细的玻璃或塑料丝，利用光脉冲传输信息。全内反射（TIR）原理使光线被限制在纤芯内部。光纤由较高折射率（(n_1)）的纤芯和较低折射率（(n_2)）的包层组成。

Critical angle: sin θ_c = n₂ / n₁

临界角：sin θ_c = n₂ / n₁

For TIR to occur, the angle of incidence at the core-cladding boundary must exceed the critical angle, and the cladding must have a lower refractive index than the core.

要实现全内反射，光线在纤芯-包层界面的入射角必须大于临界角，且包层的折射率必须低于纤芯。

Advantages of optical fibre over electrical cables:

光纤相对于电缆的优势：

• Much higher bandwidth — can transmit thousands of channels simultaneously.
• Lower attenuation — signals travel further before needing amplification.
• Immune to electromagnetic interference.
• Thinner, lighter, and more secure (harder to tap).

• 带宽高得多——可同时传输数千个信道。
• 衰减更低——信号在需要放大之前能传输更远的距离。
• 不受电磁干扰的影响。
• 更细、更轻、更安全（更难窃听）。

One limitation is that optical fibres suffer from dispersion — different wavelengths (colours) of light travel at slightly different speeds, causing pulses to spread and overlap — which limits the maximum usable data rate over long distances.

一个局限性是光纤存在色散——不同波长（颜色）的光传播速度略有差异，导致脉冲展宽和重叠——这限制了长距离传输的最大可用数据速率。

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5. Wireless Channels: Radio and Microwave | 无线信道：无线电波和微波

Radio waves (frequency range roughly 3 kHz to 3 GHz) are used for broadcasting, maritime communication, and mobile phones. They can diffract around obstacles and travel long distances by reflecting off the ionosphere, making them ideal for wide-area coverage.

无线电波（频率范围约为3 kHz至3 GHz）用于广播、海事通信和移动电话。它们能绕过障碍物发生衍射，并可通过电离层反射进行远距离传播，因此非常适合广域覆盖。

Microwaves (frequency range roughly 3 GHz to 300 GHz) have shorter wavelengths and are used for point-to-point communication links (e.g., between cell towers) and satellite communication. They travel in straight lines (line-of-sight) and are attenuated by rain and atmospheric gases.

微波（频率范围约为3 GHz至300 GHz）波长较短，用于点对点通信链路（例如基站之间）和卫星通信。微波以直线传播（视线传播），且受雨水和大气气体衰减。

Key differences between radio waves and microwaves:

无线电波与微波的主要区别：

Property 属性	Radio waves 无线电波	Microwaves 微波
Bandwidth 带宽	Lower 较低	Higher 较高
Propagation 传播方式	Diffracts around obstacles; reflects off ionosphere 绕射绕过障碍物；经电离层反射	Line-of-sight only 仅视线传播
Attenuation 衰减	Moderate 中等	Higher (absorption by water vapour) 较高（水汽吸收）

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6. Satellite Communication | 卫星通信

Satellites act as microwave relay stations in space. A signal is transmitted from a ground station (uplink) to the satellite, where it is amplified and retransmitted back to another ground station (downlink).

卫星充当太空中继站。信号从地面站上行发送到卫星，在卫星上经放大后再传回另一地面站。

Uplink frequency > Downlink frequency

上行频率 > 下行频率

Using a higher frequency for the uplink reduces interference with the downlink and requires a smaller receiving antenna on the satellite for a given gain.

上行链路使用较高频率可减少与下行链路的干扰，并且在给定增益下卫星上的接收天线可以更小。

Geostationary satellites orbit at an altitude of approximately 36,000 km above the equator. They have an orbital period of 24 hours, matching the Earth’s rotation, so they appear stationary relative to a fixed point on the Earth’s surface. This allows a fixed ground antenna to maintain contact without tracking.

地球同步卫星在赤道上方约36,000公里的高度运行。它们的轨道周期为24小时，与地球自转同步，因此相对于地球表面某固定点看起来是静止的。这使得固定地面天线无需追踪即可保持联络。

Signal delay for a geostationary satellite: the round-trip distance is approximately 2 × 36,000 km = 72,000 km. At the speed of light (3 × 10⁸ m/s), the one-way delay is about 0.12 s, and the round-trip delay (such as in a telephone call) is about 0.24 s — noticeable to users.

地球同步卫星的信号延迟：往返距离约为2 × 36,000 km = 72,000 km。以光速（3 × 10⁸ m/s）计算，单向延迟约0.12秒，往返延迟（如电话通话中）约0.24秒——用户能明显感觉到。

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7. Information Transmission: Bandwidth and Data Rate | 信息传输：带宽和数据速率

Bandwidth is the range of frequencies that a communication channel can carry, measured in hertz (Hz) or more commonly in kilohertz (kHz), megahertz (MHz), or gigahertz (GHz). It represents the information-carrying capacity of the channel.

带宽是通信信道能够承载的频率范围，单位为赫兹（Hz），常用干赫兹（kHz）、兆赫兹（MHz）或吉赫兹（GHz）表示。它代表信道的信息承载能力。

Data rate (bit rate) is the number of bits transmitted per second, measured in bits per second (bps) or multiples such as kbps, Mbps, Gbps. A wider bandwidth allows a higher data rate, as described by the Shannon-Hartley theorem.

数据速率（比特率）是每秒传输的比特数，单位为比特每秒（bps）或其倍数如kbps、Mbps、Gbps。更宽的带宽允许更高的数据速率，这由香农-哈特利定理描述。

Channel capacity: C = B log₂(1 + S/N)

信道容量：C = B log₂(1 + S/N)

Where C is the maximum data rate in bps, B is the bandwidth in Hz, and S/N is the signal-to-noise power ratio (a pure number).

其中C为最大数据速率（bps），B为带宽（Hz），S/N为信噪功率比（无量纲数）。

This equation shows that to increase the data rate, you can either increase the bandwidth or improve the signal-to-noise ratio. Doubling the bandwidth doubles the maximum data rate, while improving SNR has a logarithmic (diminishing) effect.

该方程表明，要提高数据速率，可以增加带宽或改善信噪比。带宽加倍使最大数据速率加倍，而改善信噪比则是对数（递减）效应。

Note: In the CIE A-Level syllabus, you should be able to calculate channel capacity using this formula and explain the trade-off between bandwidth and noise.

注意：在CIE A-Level课程中，你需要会用此公式计算信道容量，并能解释带宽与噪声之间的权衡关系。

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8. Analogue versus Digital Signals | 模拟信号与数字信号

Analogue signals are continuous waveforms that vary smoothly over time. Their amplitude, frequency, or phase carries the information. They are susceptible to noise and attenuation, and signal degradation cannot be fully removed once it occurs.

模拟信号是随时间平滑变化的连续波形。其幅度、频率或相位承载信息。模拟信号易受噪声和衰减的影响，且一旦发生信号退化就无法完全消除。

Digital signals are discrete — they have only two states: high (1) and low (0). Advantages of digital transmission include:

数字信号是离散的——只有两个状态：高电平（1）和低电平（0）。数字传输的优势包括：

• Better immunity to noise — the receiver only needs to distinguish between two levels.
• Signals can be regenerated (repeated) without accumulating errors.
• Data compression, encryption, and error detection are possible.
• Different types of data (audio, video, text) can be combined and transmitted together.

• 抗噪声性能更好——接收器只需区分两个电平。
• 信号可以被再生（中继）而不会累积错误。
• 可进行数据压缩、加密和错误检测。
• 不同类型的数据（音频、视频、文本）可以合并一起传输。

The process of converting an analogue signal to digital form involves three steps: sampling (measuring the signal at regular intervals), quantisation (rounding the samples to discrete levels), and coding (encoding the quantised values as binary numbers).

将模拟信号转换为数字形式的过程包括三个步骤：采样（以固定时间间隔测量信号）、量化（将采样值舍入到离散电平）和编码（将量化值编码为二进制数）。

According to the Nyquist criterion, the sampling rate must be at least twice the highest frequency component of the signal to avoid aliasing.

根据奈奎斯特定理，采样频率必须至少为信号最高频率分量的两倍，以避免混叠。

Sampling rate ≥ 2 × f_max

采样率 ≥ 2 × f_max

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9. Attenuation and Amplification | 衰减与放大

Attenuation is the loss of signal power as it travels through a transmission medium. It is measured in decibels (dB):

衰减是信号在传输介质中传播时功率的损耗。它用分贝（dB）来度量：

Attenuation (dB) = 10 log₁₀(P_in / P_out)

衰减（dB）= 10 log₁₀(P_in / P_out)

Where P_in is the input (transmitted) power and P_out is the output (received) power. A positive value indicates a power loss; a negative value would indicate a gain (amplification).

其中P_in为输入（发射）功率，P_out为输出（接收）功率。正值表示功率损耗；负值表示增益（放大）。

To compensate for attenuation, amplifiers are placed at intervals along the channel. In analogue systems, amplifiers amplify both the signal and the noise — so noise accumulates over many stages of amplification. In digital systems, repeaters first recover the bit pattern (removing noise) and then regenerate a clean digital signal, so errors do not accumulate.

为了补偿衰减，放大器沿信道按一定间距设置。在模拟系统中，放大器同时放大信号和噪声——因此噪声会在多级放大后累积。在数字系统中，中继器先恢复比特模式（去除噪声），然后重新生成干净的数字信号，因此错误不会累积。

Example: If a signal of power 40 mW is transmitted through 5 km of cable with attenuation of 3 dB per km, the total attenuation is 15 dB. The received power is:

示例：如果功率为40 mW的信号通过衰减为每公里3 dB的5公里电缆传输，总衰减为15 dB。接收功率为：

P_received = P_transmitted × 10^(-15/10) = 40 × 10^(-1.5) ≈ 1.26 mW

P_接收 = P_发射 × 10^(-15/10) = 40 × 10^(-1.5) ≈ 1.26 mW

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10. Noise and Signal-to-Noise Ratio (SNR) | 噪声与信噪比（SNR）

Noise is any unwanted signal that interferes with the transmitted signal. Sources of noise include thermal noise (random electron motion in conductors), electromagnetic interference from other devices, crosstalk between adjacent cables, and atmospheric noise.

噪声是任何干扰传输信号的不需要的信号。噪声来源包括热噪声（导体中电子的随机运动）、其他设备的电磁干扰、相邻电缆之间的串扰以及大气噪声。

The signal-to-noise ratio (SNR) compares the power of the signal to the power of the noise:

信噪比（SNR）比较信号功率与噪声功率：

SNR (dB) = 10 log₁₀(P_signal / P_noise)

信噪比（dB）= 10 log₁₀(P_信号 / P_噪声)

A high SNR means the signal is much stronger than the noise, resulting in reliable communication. A low SNR means the noise is significant and may corrupt the received signal, causing errors in digital systems or audible hiss/snow in analogue systems.

高信噪比意味着信号远强于噪声，通信可靠。低信噪比意味着噪声显著，可能破坏接收信号，导致数字系统出错或模拟系统出现听得见的嘶嘶声/雪花点。

Strategies to improve SNR include:

改善信噪比的策略包括：

• Using shielded cables to reduce electromagnetic interference.
• Using optical fibre, which is immune to electrical noise.
• Increasing transmitter power (but limited by safety and energy constraints).
• Using amplifiers/repeaters at appropriate intervals.
• Choosing a channel with less external interference.

• 使用屏蔽电缆以减少电磁干扰。
• 使用光纤，它不受电噪声影响。
• 增大发射功率（但受安全和能源限制）。
• 在适当的间距处使用放大器/中继器。
• 选择外部干扰更少的信道。

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11. Modulation Techniques | 调制技术

Modulation is the process of varying a property of a high-frequency carrier wave (the carrier frequency) in proportion to the information signal. The carrier wave provides the “transport” frequency needed for efficient transmission and allows multiple users to share the same medium using different carrier frequencies (frequency-division multiplexing).

调制是根据信息信号按比例改变高频载波（载波频率）的某个属性的过程。载波提供高效传输所需的”运输”频率，并允许多个用户使用不同载波频率共享同一介质（频分复用）。

The three main types of modulation are:

三种主要的调制类型是：

Type 类型	Property varied 被改变的属性	Application 应用
Amplitude Modulation (AM) 调幅	Amplitude 幅度	MW/LW radio broadcasting 中波/长波广播
Frequency Modulation (FM) 调频	Frequency 频率	VHF radio, music broadcasting 超短波广播、音乐广播
Phase Modulation (PM) 调相	Phase 相位	Digital communication systems 数字通信系统

Why is modulation necessary? The information signal (e.g., audio at 20 Hz – 20 kHz) has a low frequency and travels poorly over long distances. By superimposing it onto a high-frequency carrier, the transmitted signal propagates efficiently through space and can be radiated using an antenna of reasonable size.

为什么需要调制？信息信号（如20 Hz – 20 kHz的音频）频率低，远距离传输效果差。通过将其叠加到高频载波上，传输信号能在空间中有效传播，并可用尺寸合理的天线发射。

In digital communication, modulation takes the form of shifting between discrete states — such as amplitude-shift keying (ASK), frequency-shift keying (FSK), and phase-shift keying (PSK).

在数字通信中，调制表现为在离散状态之间切换——如幅移键控（ASK）、频移键控（FSK）和相移键控（PSK）。

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12. Multiplexing | 多路复用

Multiplexing is a technique that allows multiple signals to share a single communication channel simultaneously, maximising bandwidth utilisation. This is essential for cost-effective communication networks.

多路复用是一种允许多个信号同时共享单一通信信道的技术，可最大限度地提高带宽利用率。这对经济高效的通信网络至关重要。

The two main types relevant to A-Level are:

与A-Level相关的两种主要类型是：

• Frequency-division multiplexing (FDM): each signal is assigned a different carrier frequency band, and all are transmitted simultaneously. Used in radio and television broadcasting.
• Time-division multiplexing (TDM): each signal occupies the entire bandwidth but is transmitted in rapid succession in time slots. Used in digital telephone systems.

• 频分复用（FDM）：每个信号分配不同的载波频段，同时传输。用于广播和电视。
• 时分复用（TDM）：每个信号占用整个带宽，但在时间片上轮流快速传输。用于数字电话系统。

Optical fibres use a third form called wavelength-division multiplexing (WDM), where different wavelengths of light, each carrying an independent data stream, are combined into one fibre — multiplying the fibre’s capacity enormously.

光纤使用第三种形式，称为波分复用（WDM），将承载独立数据流的不同波长的光组合进一根光纤中——极大地提高了光纤的容量。

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Key Exam Tips | 考试要点提示

For CIE A-Level Physics Paper 4, focus on mastering the following:

对于CIE A-Level物理Paper 4，重点掌握以下内容：

• Know how to calculate channel capacity using C = B log₂(1 + S/N) and be able to discuss its implications.
• Understand the difference between analogue and digital signals, including advantages of digital.
• Know how to calculate attenuation in dB and use power ratios correctly.
• Be able to explain the principle of total internal reflection in optical fibres and its role in communication.
• Understand the relationship between bandwidth, data rate, and the Nyquist criterion.

• 会用 C = B log₂(1 + S/N) 计算信道容量，并能讨论其含义。
• 理解模拟信号与数字信号的区别，包括数字信号的优点。
• 会以dB为单位计算衰减，并正确使用功率比。
• 能够解释光纤中全内反射的原理及其在通信中的作用。
• 理解带宽、数据速率和奈奎斯特定理之间的关系。

Common exam question types include calculating signal-to-noise ratio improvements, deriving received power after attenuation, explaining why optical fibre is superior to copper for high-speed data, and describing the characteristics of geostationary satellites.

常见考题类型包括：计算信噪比的改善、推导衰减后的接收功率、解释为什么光纤在高速度数据传输中优于铜缆，以及描述地球同步卫星的特性。

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📚 Analogue vs Digital Signals in A-Level Physics | A-Level 物理：模拟信号与数字信号的区别

In A-Level Physics (CIE), the distinction between analogue and digital signals is a fundamental topic in the ‘Communication’ section of the syllabus. Understanding how information is encoded, transmitted, and decoded underpins nearly every modern technology, from radio broadcasting to fibre-optic internet. This article provides a thorough, exam-focused comparison of analogue and digital signals, including waveform characteristics, sampling principles, advantages of digital transmission, and common CIE exam questions.

在 A-Level 物理（CIE 考试局）中，模拟信号与数字信号的区别是“通信”章节的基础考点。理解信息如何被编码、传输和解码，是现代几乎所有技术（从无线电广播到光纤互联网）的核心前提。本文提供全面、紧扣考点的比较分析，涵盖波形特征、采样原理、数字传输的优势以及常见 CIE 考题类型。

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1. Definitions | 定义

An analogue signal is a continuously varying quantity, such as voltage or current, whose instantaneous value is directly proportional to the physical quantity being measured. It can take an infinite number of values within a given range. Examples include the electrical signal from a microphone, a thermocouple reading, or the traditional analogue television broadcast.

模拟信号是一种连续变化的物理量（如电压或电流），其瞬时值与被测物理量成正比。在给定范围内，它可以取无限多个数值。典型例子包括麦克风产生的电信号、热电偶读数以及传统模拟电视广播。

A digital signal, by contrast, is a discrete signal that can only take a limited number of defined values — in practice, usually two: HIGH (logic 1) and LOW (logic 0). These values are represented by voltage pulses, light pulses in optical fibre, or other binary states. Most digital signals in electronics are binary, meaning each pulse carries one ‘bit’ of information.

数字信号则是一种离散信号，只能取有限个定义值——在实际中通常只有两个：高电平（逻辑 1）和低电平（逻辑 0）。这些值通过电压脉冲、光纤中的光脉冲或其他二进制状态来表示。电子学中的大多数数字信号是二进制的，意味着每个脉冲携带一个“比特”的信息。

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2. Waveform Characteristics | 波形特征

The graphical representation of an analogue signal is a smooth, continuous curve. Its amplitude, frequency, and phase can vary continuously in response to the information being transmitted. A sound wave converted to a voltage by a microphone is a perfect example — the voltage waveform mirrors the air pressure variations exactly.

模拟信号的图像是一条平滑、连续的曲线。其振幅、频率和相位都会随所传输的信息连续变化。由麦克风将声波转换而成的电压波形就是一个典型例子——电压波形精确地镜像了空气压强的变化。

A digital signal, when displayed on an oscilloscope or in a timing diagram, appears as a square wave. The voltage alternates between two discrete levels — for example, 0 V representing binary ‘0’ and +5 V representing binary ‘1’. The transitions between these levels are steep and occur at specific clock instants.

数字信号在示波器上或时序图中显示为方波。电压在两个离散电平之间交替——例如，0 V 代表二进制“0”，+5 V 代表二进制“1”。电平之间的切换非常陡峭，且发生在特定的时钟时刻。

Feature	Analogue	Digital
Waveform shape	Continuous, smooth curve	Discrete square wave
Number of values	Infinite	Finite (usually 2)
Voltage levels	Any value in a range, e.g. 0–5 V	Only HIGH or LOW, e.g. 0 V and 5 V

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3. Quantisation and Sampling | 量化与采样

To transmit an analogue signal digitally, the analogue signal must first be converted using an analogue-to-digital converter (ADC). This process involves two key stages: sampling and quantisation. Sampling measures the amplitude of the analogue signal at regular time intervals. The number of samples taken per second is called the sampling frequency.

为了以数字方式传输模拟信号，必须首先使用模数转换器（ADC）对模拟信号进行转换。这个过程包含两个关键步骤：采样和量化。采样以固定的时间间隔测量模拟信号的振幅，每秒采集的样本数称为采样频率。

Quantisation is the process of mapping the sampled amplitudes to a finite set of discrete levels. The number of quantisation levels is typically expressed as a power of two, e.g. 256 levels (8-bit), 1024 levels (10-bit), etc. The quantisation error (or resolution) is the difference between the actual analogue value and its nearest quantised level. A larger number of bits provides smaller quantisation error but requires more bandwidth to transmit.

量化是将采样得到的振幅映射到有限个离散电平的过程。量化电平数通常用 2 的幂次表示，例如 256 级（8 比特）、1024 级（10 比特）等。量化误差（或分辨率）是实际模拟值与最接近的量化电平之间的差值。比特数越多，量化误差越小，但传输所需的带宽也越大。

Quantisation error = Actual value − Quantised value

Furthermore, the sampling theorem (Nyquist theorem) states that the sampling frequency must be at least twice the highest frequency component of the analogue signal in order to reconstruct it faithfully. That is:

此外，采样定理（奈奎斯特定理）表明：为了无失真地重建模拟信号，采样频率必须至少是该信号中最高频率分量的两倍，即：

f_sampling ≥ 2 × f_max

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4. Advantages of Digital Signals | 数字信号的优点

Digital signals have several significant advantages over analogue signals, which is why modern communication systems are almost exclusively digital:

数字信号相比模拟信号具有若干显著优势，这也是现代通信系统几乎全部数字化的原因：

• Immunity to noise: Because digital signals only need to be detected as HIGH or LOW, moderate noise or attenuation does not alter the information content. A regenerator (repeater) can reconstruct a perfect clean signal.
• 抗噪能力强：由于数字信号只需识别高、低两种电平，适度的噪声或衰减不会改变信息内容。中继器（再生器）可以重建出完美干净的信号。
• Error detection and correction: Digital data can include parity bits, checksums, and other error-correcting codes, enabling receivers to detect and even correct transmission errors.
• 可检错与纠错：数字数据可以包含奇偶校验位、校验和以及其他纠错编码，使接收端能够检测甚至纠正传输错误。
• Storage and processing: Digital data can be stored reliably in memory, compressed, encrypted, and processed by computers without degradation over time.
• 便于存储与处理：数字数据可以可靠地存储在存储器中，可压缩、加密并由计算机处理，且不会随时间劣化。
• Long-distance transmission: Over long distances, analogue signals accumulate noise that cannot be removed, whereas digital signals can be regenerated at intervals with no loss of quality.
• 远距离传输：在长距离传输中，模拟信号会不断累积噪声且无法去除，而数字信号可以每隔一段距离再生，不会损失质量。

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5. Disadvantages of Digital Signals | 数字信号的缺点

Despite the many benefits, digital signals are not without drawbacks, and these are frequently tested in CIE exams:

尽管数字信号优势众多，但也并非没有缺点，以下内容在 CIE 考试中经常出现：

• Increased bandwidth requirement: Representing a continuous analogue wave in digital form requires many bits per second, occupying a wider bandwidth than the original analogue signal.
• 带宽需求增大：将连续的模拟波形以数字形式表达需要每秒传输大量比特，占用的带宽比原始模拟信号更宽。
• Quantisation error: The digitisation process introduces an inherent loss of information — the recovered signal is only an approximation of the original.
• 量化误差：数字化过程会带来固有的信息丢失——恢复出来的信号只是原信号的近似值。
• Complexity: Digital systems require ADC, DAC, encoding, and decoding circuits, making the hardware more complex and expensive than simple analogue circuits.
• 系统复杂：数字系统需要 ADC、DAC、编码和解码电路，硬件比简单的模拟电路更复杂、更昂贵。

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6. Analogue-to-Digital Conversion (ADC) and DAC | 模数转换（ADC）与数模转换（DAC）

In an exam, you may be asked to describe the process of converting an analogue signal to a digital signal. The ADC operates in three steps:

在考试中，你可能会被要求描述模拟信号转换为数字信号的过程。ADC 的工作分为三个步骤：

(1) Sampling: The analogue signal is sampled at regular intervals of time, called the sampling period. Each sample holds the instantaneous voltage.

(1) 采样：以固定的时间间隔（称为采样周期）对模拟信号进行采样，每个样本保存瞬时电压值。

(2) Quantisation: Each sampled voltage is rounded to the nearest permitted discrete level. The number of levels is often 2ⁿ, where n is the number of bits.

(2) 量化：每个采样电压被四舍五入到最接近的允许离散电平。电平数通常为 2ⁿ，其中 n 是比特数。

(3) Encoding: Each quantised level is assigned a unique binary code, which is then transmitted as a sequence of HIGH/LOW pulses.

(3) 编码：每个量化电平被分配一个唯一的二进制码，然后以高/低脉冲序列的形式传输。

The reverse process, digital-to-analogue conversion (DAC), converts the binary signal back into a smooth analogue voltage, for example to drive a loudspeaker. The DAC produces a ‘staircase’ waveform, which is then passed through a low-pass filter to smooth it into the original-like continuous signal.

逆过程——数模转换（DAC）——将二进制信号重新转换为平滑的模拟电压，例如用于驱动扬声器。DAC 产生“阶梯状”波形，随后通过低通滤波器平滑处理，使其成为近似原始信号的连续波形。

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7. Analogue vs Digital in Communication Systems | 通信系统中的模拟与数字对比

In CIE A-Level Physics, this comparison is often presented in table form in both the syllabus and past papers. You should be prepared to reproduce a comparison table with the following key points:

在 CIE A-Level 物理中，这种对比在考纲和真题中常以表格形式出现。你应该能够写出包含以下关键点的对比表格：

Property	Analogue	Digital
Signal type	Continuous	Discrete (binary)
Degradation with noise	Noise accumulates irreversibly	Noise can be removed by regeneration
Bandwidth needed	Narrower	Wider
Error checking	Not possible	Possible via parity/checksum
Hardware complexity	Simple	Complex (requires ADC/DAC, encoders)
Example	Analogue radio (AM/FM), older telephones	Fibre-optic internet, mobile phone data, CDs

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8. Attenuation and Regeneration | 衰减与再生

Attenuation refers to the loss of signal power as it travels along a transmission medium. It is measured in decibels (dB) using the formula:

衰减是指信号在传输介质中传播时的功率损失，用分贝（dB）表示，公式如下：

Loss (dB) = 10 log₁₀(P₁ / P₂)

where P₁ is the input power and P₂ is the output power. For analogue signals, attenuation reduces the amplitude of the signal, and any noise picked up along the way remains and is amplified along with the signal. This is an irreversible loss.

其中 P₁ 是输入功率，P₂ 是输出功率。对于模拟信号，衰减会降低信号幅度，而沿途引入的噪声将伴随信号一起被放大，这种损失是不可逆的。

For digital signals, however, regeneration is possible. A digital repeater reads the incoming binary signal, detects whether each bit is HIGH or LOW, and transmits a fresh, full-amplitude signal. Noise that was added to the digital pulse does not need to be preserved — only the logical value matters. Thus, digital signals can be transmitted over arbitrarily long distances without cumulative degradation.

然而对于数字信号，再生是可行的。数字中继器读取输入的二进制的信号，判断每个比特是高还是低，然后重新发射一个满幅度的新信号。叠加在数字脉冲上的噪声无需保留——重要的是逻辑值。因此，数字信号可以在任意长的距离上传输而不会产生累积劣化。

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9. Bandwidth and Bit Rate | 带宽与比特率

The bandwidth of a communication channel is the range of frequencies it can carry, measured in hertz (Hz). The bit rate is the number of bits transmitted per second, measured in bits per second (bps). For a digital signal, a longer bit stream requires a larger bandwidth. Quantisation levels relate directly to bit rate as follows:

通信信道的带宽是它能够承载的频率范围，单位为赫兹（Hz）。比特率是每秒传输的比特数，单位为比特每秒（bps）。对于数字信号，比特流越长，所需带宽越大。量化电平与比特率的关系如下：

Bit rate = (number of samples per second) × (number of bits per sample)

For example, if an audio signal is sampled at 40,000 samples per second, with each sample encoded as 8 bits, the bit rate would be 40,000 × 8 = 320,000 bps = 320 kbps. This is a common calculation in CIE paper 4 questions.

例如，如果一个音频信号以每秒 40,000 次采样，每个样本编码为 8 比特，那么比特率就是 40,000 × 8 = 320,000 bps = 320 kbps。这是 CIE 第四卷常见的一道计算题。

Note that the bandwidth required to transmit this digital signal is related to the bit rate — a higher bit rate needs a greater bandwidth. This is why digital communication systems require channels with wide bandwidths, such as optical fibres carrying GHz-range signals.

需要注意，传输该数字信号所需的带宽与比特率相关——比特率越高，所需带宽越大。这就是为什么数字通信系统需要大带宽信道，例如承载吉赫兹量级信号的光纤。

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10. Typical CIE Exam Questions | 常见 CIE 考题类型

Here are the question styles most frequently encountered in CIE A-Level Physics past papers, along with the reasoning you should demonstrate in your answers:

以下是 CIE A-Level 物理真题中最常见的题型，以及你在作答时应该展示的推理过程：

• (a) Compare the waveforms of analogue and digital signals: Draw a smooth continuous sine wave for analogue and a square wave for digital; label the two voltage levels of the digital waveform.
• (a) 比较模拟与数字信号波形：为模拟信号画平滑连续正弦波，为数字信号画方波，并标出数字波形的两个电压电平。
• (b) Explain why digital signals are less affected by noise: State that a digital receiver only needs to distinguish between HIGH and LOW; small disturbances do not affect the logical state.
• (b) 解释为什么数字信号受噪声影响较小：指出数字接收器只需区分高低电平，小幅扰动不会影响逻辑状态。
• (c) Calculate the bit rate or sampling rate from given data: Use the formula: bit rate = sampling rate × bits per sample. Alternatively, determine the minimum sampling frequency using the Nyquist criterion.
• (c) 根据给定数据计算比特率或采样率：使用公式：比特率 = 采样率 × 每样本比特数；或者用奈奎斯特定理确定最小采样频率。
• (d) Describe the steps of an ADC: Mention sampling at regular intervals, quantising to finite levels, and encoding to binary.
• (d) 描述 ADC 的工作步骤：提及定时采样、有限电平量化以及二进制编码。
• (e) Discuss the advantages of digital transmission over analogue: Mention noise immunity, error detection/correction, regeneration, and faithful long-distance transmission.
• (e) 讨论数字传输相比模拟传输的优势：提及抗噪能力、检错纠错、信号再生和长距离保真传输。

A common exam misconception is to write that digital signals have ‘no noise’. This is incorrect — digital signals still experience noise, but the noise does not affect the interpretation of the data if it does not exceed the threshold levels. Be precise in your wording: digital signals are less affected by noise, not noise-free.

一个常见的考场误区是写“数字信号没有噪声”。这是错误的——数字信号同样会受噪声影响，但只要噪声不越过阈值电平，就不会影响数据解读。措辞要准确：数字信号是受噪声影响更小，而非完全无噪声。

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11. Unifying Summary | 总结对比

To consolidate your revision, keep the following principle in mind: analogue signals are continuous and exact but fragile; digital signals are discrete and approximate but robust. This single sentence captures the essence of why the world has switched from analogue to digital communication.

为了巩固复习，请牢记这一原则：模拟信号连续而精确，但易受干扰；数字信号离散而近似，却十分稳健。这一句话概括了为什么世界已经从模拟通信全面转向数字通信。

In terms of exam technique, whenever you are asked to ‘explain the advantage of digital signals’, always mention three things: (1) noise immunity and regeneration, (2) error checking capability, and (3) storage/processing compatibility. Whenever you are asked to ‘suggest why analogue signals are still used’, mention that they require simpler circuits and less bandwidth, and that some physical quantities are naturally analogue.

在答题技巧方面，每当被要求“解释数字信号的优点”时，务必提到三点：(1) 抗噪与再生能力，(2) 检错能力，(3) 存储与处理兼容性。每当被要求“说明为什么模拟信号仍在使用”时，应提到模拟电路更简单、占用带宽更少，以及某些物理量天然就是模拟量。

Finally, practise drawing both types of waveforms with correct labels: ‘amplitude’ on the y-axis, ‘time’ on the x-axis, and for digital signals, label the HIGH and LOW levels explicitly with their respective voltage values. Accurate diagrams are often awarded marks independently of the written explanation.

最后，请练习绘制这两种波形图并正确标注：纵轴为“振幅”，横轴为“时间”；对数字信号要明确标出高、低电平及其电压值。准确的图形常常可以独立于文字描述获得分数。

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📚 A-Level Physics: Resonance and Its Conditions | A-Level 物理：共振现象及其产生条件

Resonance is one of the most elegant and practical ideas in A-Level Physics. It explains why a small repeated push can make a heavy swing rise higher and higher, why some bridges vibrate dangerously in the wind, and why a note from a singer can shatter a glass. In this revision article, we will define resonance precisely, examine the conditions required for it to occur, explore the resonance curve and damping, and look at real-life applications and dangers.

共振是 A-Level 物理中既优美又实用的概念之一。它可以解释为什么持续的小推力能让沉重的秋千越荡越高、为什么有些桥梁会在风中危险地振动、为什么歌手的一个音符可以震碎玻璃。在这篇复习文章中，我们将精确定义共振、分析其产生条件、研究共振曲线和阻尼，并考察现实中的应用与危害。

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1. What Is Resonance? | 什么是共振？

Resonance is the phenomenon in which a system oscillates with a much larger amplitude when the frequency of an applied periodic force is equal to the natural frequency of the system. At resonance, the driving force transfers energy to the system in the most efficient way, so the amplitude reaches a maximum value.

共振是当外加周期力的频率等于系统固有频率时，系统的振幅显著增大的现象。在共振时，驱动力以最高效率向系统传递能量，因此振幅达到最大值。

For example, a child on a swing has a fixed natural frequency. If a parent pushes the swing at exactly this frequency, each push adds energy at the right moment and the swing goes higher and higher. If the pushes come at the wrong time, the swing barely moves.

例如，荡秋千的孩子有一个固定的固有频率。如果家长恰好以这个频率推秋千，每一次推力都在正确的时刻输入能量，秋千就会越荡越高。如果推的时机不对，秋千几乎不会升高。

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2. Key Terms: Free Vibration and Natural Frequency | 关键术语：自由振动与固有频率

A free vibration occurs when a system oscillates on its own after being displaced from equilibrium, with no external periodic force acting on it. The frequency of this vibration is called the natural frequency, written as f₀. It depends on the physical properties of the system, such as mass, stiffness, or length.

自由振动是指系统偏离平衡位置后，在没有任何外部周期力作用的情况下自行振动。这种振动的频率称为固有频率，记作 f₀。它取决于系统本身的物理性质，例如质量、劲度系数或长度。

For a simple pendulum, the period is related to its length:

对单摆而言，其周期与摆长有关：

T = 2π√(l/g)

For a mass on a spring, the period is related to the mass m and the spring constant k:

对于弹簧振子，其周期与质量 m 和劲度系数 k 有关：

T = 2π√(m/k)

In both cases, the natural frequency is f₀ = 1/T. You should be able to state these formulas and explain what happens when l, m or k changes.

在这两种情况下，固有频率都是 f₀ = 1/T。你应该能够写出这些公式，并解释当 l、m 或 k 改变时会发生什么。

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3. Forced Vibration and Driving Frequency | 受迫振动与驱动力频率

A forced vibration happens when an external periodic force continuously drives an oscillator. The system is then made to vibrate at the driving frequency, also called the forcing frequency or applied frequency, not necessarily at its own natural frequency.

受迫振动是指外部周期力持续驱动振子时所发生的振动。此时系统被迫以驱动频率振动，而非一定以其固有频率振动。驱动频率也叫受迫频率或施加频率。

After a short initial period, the system settles into a steady state in which it oscillates at the driving frequency. The size of the steady amplitude depends strongly on how close the driving frequency is to the natural frequency.

经过短暂的起始阶段后，系统会进入稳态，并以驱动频率振动。稳态振幅的大小强烈依赖于驱动频率与固有频率的接近程度。

If the driving frequency is very different from f₀, the response is small. If the driving frequency is close to f₀, even a small force can produce a very large amplitude. This is the key idea behind resonance.

如果驱动频率与 f₀ 相差很远，系统的响应很小；如果驱动频率接近 f₀，即使很小的力也能产生很大的振幅。这就是共振背后的核心思想。

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4. Conditions for Resonance | 共振的产生条件

Four conditions are usually required for resonance to be observed clearly:

要清楚地观察到共振，通常需要满足以下四个条件：

• 1. The system must have a natural frequency f₀.

  系统必须具有固有频率 f₀。

• 2. An external periodic driving force must act on the system.

  必须有一个外部周期力作用在系统上。

• 3. The driving frequency must be equal to the natural frequency.

  驱动频率必须等于固有频率。

• 4. The damping must be small enough for the amplitude to grow large.

  阻尼必须足够小，振幅才能增大到明显程度。

In examination answers, the standard definition is usually enough: resonance occurs when the driving frequency is equal to the natural frequency, causing maximum amplitude and maximum energy transfer.

在考试作答中，通常写出标准定义即可：当驱动频率等于固有频率时发生共振，此时振幅最大，能量传递效率最高。

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5. The Resonance Curve | 共振曲线

The relationship between amplitude and driving frequency is shown by a resonance curve. The horizontal axis represents the driving frequency, while the vertical axis represents the steady amplitude of the forced oscillator.

振幅与驱动频率之间的关系可以用共振曲线表示。横轴代表驱动频率，纵轴代表受迫振子的稳态振幅。

As the driving frequency approaches f₀, the amplitude rises sharply. At f = f₀, the amplitude is at its maximum. With very little damping, the peak is very tall and narrow. With greater damping, the peak is shorter and broader.

当驱动频率接近 f₀ 时，振幅急剧增大。当 f = f₀ 时，振幅达到最大值。阻尼很小时，峰值又高又窄；阻尼较大时，峰值较低且较宽。

In CIE questions, you may be asked to sketch this curve for two different amounts of damping. Remember: the undamped or lightly damped curve has a much higher and sharper peak around f₀.

在 CIE 考试中，你可能会被要求画出两种不同阻尼情况下的共振曲线。请记住：无阻尼或轻阻尼曲线的峰值在 f₀ 附近更高、更尖锐。

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6. Energy Transfer at Resonance | 共振时的能量传递

At resonance, the driving force does maximum positive work on the oscillator. This happens because the force is always in phase with the velocity of the oscillator, so each push acts in the direction of motion and adds energy.

在共振时，驱动力对振子做最大的正功。这是因为驱动力始终与振子的速度同相，每一次推动都沿着运动方向，从而不断输入能量。

If there were no damping, the amplitude would keep increasing without limit. In the real world, damping is always present. The amplitude stops growing when the energy supplied per cycle equals the energy lost per cycle due to damping.

如果没有阻尼，振幅将无限增大。但在现实世界中，阻尼总是存在的。当每个周期输入的能量等于每个周期因阻尼而损失的能量时，振幅就不再增大了。

Therefore, at resonance the amplitude is large and constant because of an energy balance. This idea is often examined in explanation-based questions.

因此，共振时的振幅大而稳定，是因为能量收支达到平衡。这一思路常出现在解释类考题中。

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7. Phase Relationship at Resonance | 共振时的相位关系

The phase relationship between the driving force and the displacement is important in describing resonance.

驱动力与位移之间的相位关系对于描述共振非常重要。

At very low driving frequency, the displacement is nearly in phase with the driving force. At very high driving frequency, the displacement is almost exactly out of phase with the driving force, lagging by about 180°.

当驱动频率很低时，位移与驱动力几乎同相。当驱动频率很高时，位移与驱动力几乎反相，位移大约滞后 180°。

At resonance, the phase difference between the driving force and displacement is 90°. This means the driving force is in phase with velocity, which is the condition for maximum energy input.

在共振时，驱动力与位移之间的相位差为 90°。这意味着驱动力与速度同相，这正是能量输入最大的条件。

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8. Effect of Damping on Resonance | 阻尼对共振的影响

Damping removes energy from an oscillating system. It can be caused by air resistance, friction, or internal forces in a material.

阻尼会使振动系统损失能量。空气阻力、摩擦或材料内部作用力都可能引起阻尼。

When damping increases, the maximum amplitude of the resonance curve decreases. The curve becomes flatter and broader, so resonance becomes less sharp. With very heavy damping, resonance may barely be noticeable.

当阻尼增大时，共振曲线的最大振幅减小。曲线变得更平坦、更宽，因此共振变得不那么尖锐。如果阻尼很重，共振可能几乎观察不到。

In addition, the frequency at which maximum amplitude occurs shifts slightly downward compared with the undamped natural frequency. In many A-Level questions, you only need to state that increased damping reduces the amplitude and makes the peak broader.

此外，最大振幅对应的频率会相对于无阻尼固有频率略微下移。在许多 A-Level 考题中，你只需说明增大阻尼会降低振幅并使峰值变宽。

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9. Applications of Resonance | 共振的应用

Resonance is used deliberately in many devices. Learning a few examples helps you answer application questions quickly.

共振在许多设备中被有意利用。掌握几个典型例子有助于你快速回答应用类问题。

• Radio tuning: in a radio receiver, the electrical circuit has a natural frequency. You change the capacitance until the circuit resonates with the signal from a particular station, making that station much stronger than others.

  收音机调谐：收音机接收电路具有固有频率。你改变电容大小，直到电路与某个电台的信号发生共振，使该电台的信号比其他台强得多。

• Microwave ovens: water molecules in food absorb microwaves strongly because the frequency of the microwaves is close to the natural rotational frequency of the water molecules. This raises the temperature of the food.

  微波炉：食物中的水分子强烈吸收微波，因为微波频率接近水分子的自然转动频率。这会使食物的温度升高。

• Musical instruments: the air column in a flute or the string in a violin vibrates at its natural frequency. When another part of the instrument provides a matching driving frequency, the sound is amplified by resonance.

  乐器：长笛中的空气柱或小提琴中的琴弦以固有频率振动。当乐器的其他部分提供相匹配的驱动频率时，声音会通过共振而增强。

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10. Dangers of Resonance and Engineering Prevention | 共振的危害与工程防护

Resonance can also damage structures. Engineers try to avoid resonance whenever possible.

共振也可能破坏结构。工程人员会尽可能避免共振。

One classic example is soldiers marching across a bridge. If their marching frequency matches the bridge’s natural frequency, the bridge can vibrate violently. To prevent this, soldiers are ordered to break step when crossing a bridge.

一个典型例子是士兵列队过桥。如果他们的步频与桥梁的固有频率相同，桥梁可能会剧烈振动。为了防止这种情况，士兵们过桥时会被命令打乱步伐。

Suspension bridges can also be set into oscillation by wind. The Tacoma Narrows Bridge collapse in 1940 is a famous case of large-amplitude oscillation caused by wind and resonance-like effects. Modern bridges are designed with a natural frequency far away from likely driving frequencies, and some use tuned mass dampers to absorb unwanted vibration.

吊桥也可能因风而产生强烈振荡。1940 年塔科马海峡大桥坍塌就是风致大幅振荡的著名案例。现代桥梁在设计时会让固有频率远离可能的驱动频率，有些还会使用调谐质量阻尼器来吸收有害振动。

In general, engineers can reduce the risk of resonance by changing the natural frequency, increasing damping, or avoiding periodic forces at dangerous frequencies.

一般来说，工程人员可以通过改变固有频率、增大阻尼或避免在危险频率下施加周期力来降低共振风险。

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11. Examination Tips | 考试要点

When answering CIE resonance questions, the most common mistake is to state only that amplitudes become large, without mentioning the equality of the two frequencies. A complete answer should say:

在回答 CIE 共振类题目时，最常见的错误是只提到振幅变大，而没有说明两个频率相等。完整的回答应该这样写：

“Resonance occurs when the driving frequency equals the natural frequency, causing maximum amplitude and maximum energy transfer.”

Remember the symbol conventions used in exam papers: f₀ is the natural frequency, while the driving frequency may be written as fᵈ or simply stated in words. Always label your axes clearly when sketching a resonance curve: frequency on the x-axis and amplitude on the y-axis.

请注意考试试卷中的符号约定：f₀ 是固有频率，驱动频率可能写作 fᵈ，或直接用文字说明。在画共振曲线时，一定要标清坐标轴：横轴为频率，纵轴为振幅。

Finally, be ready to link resonance to energy. When examiners ask why the amplitude becomes large, the best answer is that the driving force transfers energy into the system at the correct frequency, and the energy gained per cycle balances the energy lost due to damping.

最后，要准备好将共振与能量联系起来。当考官问为什么振幅会变大时，最好的答案是：驱动力以正确的频率向系统输入能量，每个周期获得的能量与阻尼损失的能量达到平衡。

Published by TutorHao | Physics Revision Series | aleveler.com

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