Magnetic Fields | 磁场

📚 Magnetic Fields | 磁场

Magnetic fields are one of the most richly tested areas in AQA International A Level Physics. From the force on a current-carrying wire to the complex trajectories of charged particles in particle accelerators, this topic connects electromagnetic theory to real-world applications. This revision guide covers every core concept, equation, and exam technique you need for your topic test.

磁场是 AQA 国际 A Level 物理中考查最丰富的领域之一。从载流导线所受的力到带电粒子在粒子加速器中的复杂轨迹，这一主题将电磁理论与现实应用紧密相连。本复习指南涵盖了你需要在章节测试中掌握的每一个核心概念、公式和考试技巧。

1. Origin and Representation of Magnetic Fields | 磁场的起源与表示

Magnetic fields are regions of space where magnetic forces can be detected. They are produced by either permanent magnets or moving electric charges (currents). The field is a vector quantity, represented at every point by a magnetic flux density vector B.

磁场是能够检测到磁力作用的空间区域。它由永磁体或运动的电荷（电流）产生。磁场是矢量，在每一点上由磁通密度矢量 B 表示。

You must be able to draw and interpret magnetic field lines. By convention, field lines run from the north pole to the south pole outside a magnet, and from south to north inside it. Key properties include: lines never cross, their density indicates field strength, and the tangent at any point gives the direction of the force on a north pole placed there.

你必须能够绘制和解释磁感线。按惯例，磁感线在磁体外部从北极指向南极，在内部从南极指向北极。关键特性包括：磁感线永不相交；其密度表示场强大小；任意一点的切线方向表示放在该处的北极所受力的方向。

For current-carrying wires, use the right-hand grip rule: grasp the wire with your right hand, thumb pointing in the direction of conventional current; your curled fingers show the circular direction of the field lines. For solenoids, the same rule applies – the field inside a long solenoid is uniform and parallel to the axis, given by B = μ₀nI for an ideal solenoid in a vacuum.

对于载流导线，使用右手螺旋定则：用右手握住导线，拇指指向传统电流方向；弯曲的手指表示磁感线的环绕方向。对于螺线管，应用同样的规则——长直螺线管内部的磁场是均匀的，且平行于轴线，在真空中理想螺线管的磁场为 B = μ₀nI。

B = μ₀nI（其中 n 为单位长度匝数，μ₀ = 4π × 10⁻⁷ T m A⁻¹）

2. Magnetic Flux Density and the Tesla | 磁通密度与特斯拉

Magnetic flux density B is defined as the force per unit length per unit current on a current-carrying conductor placed perpendicular to the magnetic field. The SI unit of B is the tesla (T). One tesla is defined as the magnetic flux density that produces a force of one newton per metre of wire carrying one ampere of current, perpendicular to the field.

磁通密度 B 定义为：垂直于磁场放置的载流导线上，单位长度、单位电流所受到的力。B 的国际单位是特斯拉（T）。一特斯拉定义为：与磁场垂直的导线，每米长度通过一安培电流时产生一牛顿力的磁通密度。

In equation form, when the wire is perpendicular to the field: F = BIL. Rearranging gives B = F/(IL), which yields the unit:

用方程表示，当导线与磁场垂直时：F = BIL。变形得 B = F/(IL)，由此得到单位：

1 T = 1 N A⁻¹ m⁻¹ = 1 kg s⁻² A⁻¹

Typical values you should memorise for comparison: the Earth’s magnetic field is roughly 5 × 10⁻⁵ T, a small bar magnet produces about 10⁻² T, and an electromagnet can reach 1–2 T. This sense of scale helps in estimating answers and checking calculations.

你应该记住以下典型值以便比较：地球磁场约为 5 × 10⁻⁵ T，小型条形磁铁产生约 10⁻² T，电磁铁可达 1–2 T。这种量级感有助于估算答案和检查计算结果。

Source	Typical B (T)	来源	典型 B 值 (T)
Earth’s surface	~5 × 10⁻⁵	地球表面	~5 × 10⁻⁵
Bar magnet	~10⁻²	条形磁铁	~10⁻²
Laboratory electromagnet	1–2	实验室电磁铁	1–2

3. Force on a Current-Carrying Conductor | 载流导体的受力

When a current-carrying conductor is placed in a magnetic field, each moving charge within the wire experiences a magnetic force. These microscopic forces add together to produce a macroscopic force on the wire. The magnitude of this force depends on four factors: magnetic flux density B, current I, length of conductor L in the field, and the angle θ between the wire and the field direction.

当载流导体置于磁场中时，导线内每个运动电荷都会受到磁力作用。这些微观力叠加在一起，对导线产生宏观力。该力的大小取决于四个因素：磁通密度 B、电流 I、处于磁场中的导体长度 L，以及导线与磁场方向之间的夹角 θ。

F = BIL sin θ

When the wire is perpendicular to the field (θ = 90°), sin θ = 1, giving maximum force F = BIL. When the wire is parallel to the field (θ = 0°), sin θ = 0, and the force is zero. This is why a wire experiences no magnetic force when aligned with the field lines. The direction of the force is found using Fleming’s left-hand rule (motor rule).

当导线与磁场垂直（θ = 90°）时，sin θ = 1，力最大，F = BIL。当导线与磁场平行（θ = 0°）时，sin θ = 0，力为零。这就是导线与磁感线同向时不受磁力的原因。力的方向用弗莱明左手定则（电动机定则）判断。

Fleming’s left-hand rule: hold your left hand so that the first finger (index) points in the direction of the magnetic field, the second finger (middle) points in the direction of conventional current, and the thumb then points in the direction of the force (motion). It is crucial to remember that the first finger is the Field, the second finger is the Current, and the thuMb is Motion – the mnemonic ‘FBI’ helps for the thumb, first, and second fingers respectively.

弗莱明左手定则：伸出左手，食指指向磁场方向，中指指向传统电流方向，大拇指即为力的方向（运动方向）。记住：食指示 F（场）、中指示 B（电流）、大拇指示 Motion（运动）——可用 “FBI” 助记。

In an exam, you must clearly state the direction of the force in words. For example, if a wire carries current horizontally from left to right in a uniform vertical magnetic field pointing downward, the force acts horizontally towards the observer. Practice by sketching the three mutually perpendicular vectors in 3D perspective.

在考试中，你必须用语言清楚地描述力的方向。例如，一根导线从左向右水平通以电流，置于竖直向下方向的均匀磁场中，则力的方向水平指向观察者。练习在三维透视图中画出三个互相垂直的矢量。

4. Force on a Moving Charge | 运动电荷的受力

A single charged particle moving through a magnetic field experiences a force given by the Lorentz force equation. Since current is the rate of flow of charge, and the drift velocity of charges is v, we can derive the force on a single charge q moving with velocity v perpendicular to a field B.

单个带电粒子在磁场中运动会受到洛伦兹力作用。由于电流是电荷流动的速率，电荷漂移速度为 v，我们可以推导出单个电荷 q 以速度 v 垂直于磁场 B 运动时所受的力。

F = BQv sin θ

Here, Q is the charge of the particle, v is its speed, and θ is the angle between the velocity vector and the magnetic field. For a proton, Q = +1.6 × 10⁻¹⁹ C; for an electron, Q = −1.6 × 10⁻¹⁹ C. The direction of the force on a negative charge is opposite to that on a positive charge moving in the same direction – a critical detail in problem solving.

其中 Q 是粒子的电荷量，v 是速率，θ 是速度矢量与磁场之间的夹角。对于质子，Q = +1.6 × 10⁻¹⁹ C；对于电子，Q = −1.6 × 10⁻¹⁹ C。负电荷所受力的方向与同方向运动的正电荷相反——这是解题中的关键细节。

The magnetic force is always perpendicular to both the velocity and the magnetic field. Because of this, magnetic forces do no work on the particle – they change the direction of motion but never the speed. The kinetic energy of a charged particle in a purely magnetic field remains constant.

磁力始终垂直于速度和磁场。因此，磁力不对粒子做功——它只改变运动方向，从不改变速率。在纯磁场中，带电粒子的动能保持不变。

This is a common exam point: students are asked to explain why a magnetic field cannot increase the kinetic energy of a charged particle. The answer is that the force is always perpendicular to the displacement, so the work done, W = Fs cos θ, is zero (since cos 90° = 0).

这是一个常见的考点：要求学生解释为什么磁场不能增加带电粒子的动能。答案是：力始终垂直于位移，因此做功 W = Fs cos θ 为零（因为 cos 90° = 0）。

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

When a charged particle moves perpendicular to a uniform magnetic field, the magnetic force acts as a centripetal force, causing the particle to follow a circular path. This is one of the most important applications of the Lorentz force in the AQA specification.

当带电粒子垂直于匀强磁场运动时，磁力充当向心力，使粒子沿圆周轨迹运动。这是 AQA 考纲中洛伦兹力最重要的应用之一。

Equating the magnetic force to the centripetal force:

将磁力与向心力相等：

BQv = (mv²)/r ⇒ r = (mv)/(BQ)

The radius r is directly proportional to the momentum mv and inversely proportional to the magnetic flux density B and charge Q. For a given particle in a fixed field, the radius is constant, and the particle moves with uniform circular motion. The time period T of the orbit is:

半径 r 与动量 mv 成正比，与磁通密度 B 和电荷 Q 成反比。对于固定场中的给定粒子，半径恒定，粒子做匀速圆周运动。轨道周期 T 为：

T = (2πm)/(BQ)

Notice that the time period is independent of the speed or radius of the particle – it depends only on mass, charge, and magnetic flux density. Faster particles orbit in larger circles but take exactly the same time per revolution. This principle is used in cyclotrons for particle acceleration.

请注意，周期与粒子速度或半径无关——它只取决于质量、电荷和磁通密度。较快的粒子在更大的圆轨道上运动，但每圈所需时间完全相同。这一原理用于回旋加速器中的粒子加速。

Exam tip: When solving circular motion problems, always start by writing the equality F_magnetic = F_centripetal. Be careful to use the correct mass – if the question gives you an alpha particle, use m = 6.64 × 10⁻²⁷ kg and Q = +2e = 3.2 × 10⁻¹⁹ C.

考试技巧：解圆周运动问题时，始终从 F_磁 = F_向心入手。注意使用正确的质量——如果题目给出 α 粒子，则用 m = 6.64 × 10⁻²⁷ kg，Q = +2e = 3.2 × 10⁻¹⁹ C。

6. Magnetic Flux and Flux Linkage | 磁通量与磁链

Magnetic flux Φ is a measure of the total number of field lines passing through a given area. It is defined as the product of magnetic flux density B and the component of the area perpendicular to the field. The SI unit of magnetic flux is the weber (Wb).

磁通量 Φ 是穿过给定面积的磁感线总数量的度量。它定义为磁通密度 B 与垂直于场的面积分量之积。磁通量的 SI 单位是韦伯（Wb）。

Φ = BA cos θ

Here, A is the area of the surface and θ is the angle between the magnetic field direction and the normal (perpendicular) to the surface. When the field is perpendicular to the surface (θ = 0°), Φ = BA, the maximum flux. When the field is parallel to the surface (θ = 90°), Φ = 0 because no field lines pass through the area.

其中 A 是表面面积，θ 是磁场方向与表面法线（垂线）之间的夹角。当场垂直于表面时（θ = 0°），Φ = BA，磁通量最大。当场平行于表面时（θ = 90°），Φ = 0，因为没有磁感线穿过该面积。

Magnetic flux linkage is defined as the product of the number of turns N of a coil and the magnetic flux Φ passing through each turn. Its unit is the weber-turn (Wb).

磁链定义为线圈匝数 N 与穿过每匝的磁通量 Φ 之积。其单位是韦伯-匝（Wb）。

Flux linkage = NΦ = BAN cos θ

If a coil of cross-sectional area A has N turns and rotates in a uniform magnetic field, the flux linkage varies sinusoidally with the rotation angle. This varying flux linkage is the basis for electromagnetic induction in generators, which we explore next.

如果截面积为 A 的线圈有 N 匝，并在均匀磁场中旋转，磁链随旋转角按正弦规律变化。这种变化的磁链是发电机中电磁感应的基础，我们接下来将探讨。

7. Faraday’s Law of Electromagnetic Induction | 法拉第电磁感应定律

Faraday’s law states that the induced electromotive force (e.m.f.) in a circuit is equal to the negative rate of change of magnetic flux linkage through the circuit. This is the fundamental equation of electromagnetic induction.

法拉第定律指出：电路中感应电动势的大小等于穿过电路的磁链随时间的变化率的负值。这是电磁感应的基本方程。

E = −N(dΦ/dt)

The negative sign represents Lenz’s law, which we will discuss in the next section. For a conductor of length L moving at speed v perpendicular to a magnetic field B, the induced e.m.f. simplifies to:

负号代表楞次定律，我们将在下一节讨论。对于以速度 v 垂直于磁场 B 运动的长度为 L 的导体，感应电动势简化为：

E = BLv

This motional e.m.f. arises because the free electrons in the moving conductor experience a magnetic force F = BQv, causing them to accumulate at one end of the conductor, creating a potential difference. This equation is valid when the velocity is perpendicular to both B and the length of the conductor.

这种动生电动势的产生是因为运动导体中的自由电子受到磁力 F = BQv 的作用，导致电子在导体一端积聚，形成电势差。该方程在速度垂直于 B 和导体长度时成立。

A key exam application: a metal rod rolling along two parallel rails in a magnetic field. The induced current flows through the circuit formed by the rails and rod. The direction of the induced current can be predicted, and the magnitude of the e.m.f. is simply E = BLv as long as the rod moves perpendicular to the field.

一个关键考试应用：一根金属棒在磁场中沿两条平行导轨滚动。感应电流流过由导轨和金属棒构成的回路。感应电流的方向可以预判，只要金属棒垂直于磁场运动，感应电动势的大小就是 E = BLv。

8. Lenz’s Law and Conservation of Energy | 楞次定律与能量守恒

Lenz’s law states that the direction of an induced current is such that it opposes the change producing it. In other words, the induced magnetic field acts to resist the change in magnetic flux. The negative sign in Faraday’s law is the mathematical expression of this principle.

楞次定律指出：感应电流的方向总是阻碍引起它的磁通量变化。换言之，感应磁场的作用是抵抗磁通量的变化。法拉第定律中的负号就是这个原理的数学表达。

Lenz’s law is a direct consequence of the conservation of energy. If the induced current aided the change producing it, energy would be created from nothing, violating conservation. Instead, when a magnet is pushed into a coil, work must be done against the opposing magnetic force; this work is converted into electrical energy.

楞次定律是能量守恒的直接结果。如果感应电流助长产生它的变化，能量就会凭空产生，违反守恒定律。相反，当磁铁被推入线圈时，必须克服排斥力做功；这些功转化为电能。

An exam favourite: explain why a magnet dropped through a long copper tube falls slowly. As the magnet falls, the changing flux induces currents in the tube. By Lenz’s law, these currents create a magnetic field opposing the motion of the magnet, reducing its acceleration. Eventually, the magnet reaches a terminal velocity. Energy is conserved because the loss in gravitational potential energy is dissipated as heat in the tube due to the induced currents.

一个考试常见题：解释为什么磁铁从长铜管中落下时速度很慢。磁铁下落时，变化的磁通量在管中感应出电流。根据楞次定律，这些电流产生与磁铁运动相反的磁场，减小其加速度。最终磁铁达到终速。能量守恒因为重力势能的损失以热量的形式在管中因感应电流而耗散。

To determine the direction of induced current for a coil, follow these steps: (1) determine the direction of the external magnetic field through the coil; (2) decide whether the flux is increasing or decreasing; (3) the induced field must oppose this change – i.e., opposite direction if flux is increasing, same direction if flux is decreasing; (4) use the right-hand grip rule to find the current direction that produces this induced field.

确定线圈感应电流方向的步骤：(1) 确定穿过线圈的外磁场方向；(2) 判断磁通量是增加还是减少；(3) 感应磁场必须阻碍这种变化——即磁通增加时方向相反，磁通减少时方向相同；(4) 用右手螺旋定则找出产生该感应磁场的电流方向。

9. Applications: The Electric Motor | 应用：电动机

The electric motor is a direct application of the force on a current-carrying conductor. A rectangular coil of wire is placed in a radial magnetic field produced by permanent magnets. When current flows through the coil, the two vertical sides experience forces in opposite directions, creating a couple that rotates the coil.

电动机是载流导体受力的直接应用。一个矩形线圈置于永磁体产生的径向磁场中。当电流通过线圈时，两条竖直边受到方向相反的力，形成使线圈转动的力偶矩。

A radial magnetic field ensures that the plane of the coil is always parallel to the field direction, so the force on each side remains constant at its maximum value throughout the rotation. The torque is given by:

径向磁场确保线圈平面始终平行于磁场方向，因此每边所受的力在转动过程中始终保持在最大值。力矩为：

T = BANI

where N is the number of turns on the coil and A is the area of the coil. For a motor to produce continuous rotation, a split-ring commutator reverses the direction of the current every half-turn. This ensures the torque always acts in the same rotational direction.

其中 N 是线圈匝数，A 是线圈面积。为了使电动机持续转动，换向器（半环）每半圈反转一次电流方向。这确保力矩始终作用在相同的转动方向上。

Exam questions often ask about the role of the commutator, the advantage of a radial field, or how the motor’s speed can be increased (increase B, I, or number of turns). Understand that the motor converts electrical energy into mechanical energy, with energy losses occurring as heat due to resistance and friction.

考试题目常问换向器的作用、径向磁场的优势，或如何提高电动机转速（增大 B、I 或匝数）。要理解电动机将电能转化为机械能，能量损失以熱和摩擦形式损耗。

10. Applications: Generators and Transformers | 应用：发电机与变压器

A generator is the reverse of a motor – it converts mechanical energy into electrical energy using electromagnetic induction. A coil rotates in a magnetic field, causing the flux linkage to vary sinusoidally. By Faraday’s law, this produces a sinusoidal alternating e.m.f.

发电机是电动机的反向应用——它利用电磁感应将机械能转化为电能。线圈在磁场中旋转，使磁链呈正弦变化。根据法拉第定律，这产生正弦交流电动势。

For a coil of N turns and area A rotating at angular speed ω in a magnetic field B, the flux linkage is NΦ = BAN cos(ωt), and the induced e.m.f. is:

对于 N 匝、面积为 A 的线圈，在磁场 B 中以角速度 ω 旋转，磁链为 NΦ = BAN cos(ωt)，感应电动势为：

E = BANω sin(ωt)

The maximum e.m.f. is E₀ = BANω. The e.m.f. is zero when the coil is perpendicular to the field (maximum flux, zero rate of change) and maximum when the coil is parallel to the field (zero flux, maximum rate of change).

最大电动势为 E₀ = BANω。当线圈垂直于磁场时（磁通量最大、变化率为零），电动势为零；当线圈平行于磁场时（磁通量为零、变化率最大），电动势最大。

A transformer consists of two coils wound on a soft iron core. An alternating current in the primary coil produces a changing magnetic flux in the core, which links with the secondary coil. Faraday’s law then induces an e.m.f. in the secondary coil. For an ideal transformer:

变压器由绕在软铁芯上的两个线圈组成。初级线圈中的交流电在铁芯中产生变化的磁通量，与次级线圈耦合。根据法拉第定律，在次级线圈中感应出电动势。对于理想变压器：

Vₛ/Vₚ = Nₛ/Nₚ and VₚIₚ = VₛIₛ

Step-up transformers increase voltage and decrease current; step-down transformers do the reverse. The national grid uses step-up transformers for transmission to reduce I²R power losses in the cables. Real transformers have power losses due to eddy currents in the core (reduced by laminating the core), hysteresis, and resistance of the windings.

升压变压器升高电压、降低电流；降压变压器则相反。国家电网使用升压变压器进行输电，以减少电缆中的 I²R 功率损耗。实际变压器存在损耗：铁芯中的涡流（通过叠片减少）、磁滞损耗和绕组电阻损耗。

11. Comparing Magnetic and Electric Fields | 磁场与电场的比较

A common examination question asks you to compare and contrast electric and magnetic fields. Both are vector fields that exert forces on charges, but their behaviours differ significantly. The table below summarises the key differences.

一个常见的考试题目要求你比较电场和磁场。两者都是对电荷施力的矢量场，但行为有显著差异。下表总结了关键区别。

Property	Electric Field	Magnetic Field	特性	电场	磁场
Source	Stationary charges	Moving charges / magnets	来源	静止电荷	运动电荷 / 磁体
Force on charge	F = QE, always parallel to E	F = BQv sin θ, perpendicular to v and B	对电荷的力	F = QE，始终平行于 E	F = BQv sin θ，垂直于 v 和 B
Work done	Can do work, changes KE	Does no work, KE constant	做功	可以做功，改变动能	不做功，动能不变
Field lines	Start at +, end at −	Continuous closed loops	场线	从 + 到 −	连续的闭合环

A stationary charge experiences a force in an electric field but not in a magnetic field. A moving charge experiences forces in both fields, but the magnetic force depends on velocity and is always perpendicular to it. You should be comfortable explaining these differences with reference to the equations.

静止电荷在电场中受力，但在磁场中不受力。运动电荷在两种场中都受力，但磁力取决于速度且始终垂直于速度。你应该能够结合方程熟练解释这些差异。

12. Exam Techniques and Common Mistakes | 考试技巧与常见错误

Finally, let us review the most common pitfalls and the techniques to avoid them. The marker’s reports consistently highlight the same errors, so mastering these will give you a significant advantage.

最后，让我们回顾最常见的错误和避免它们的技巧。阅卷报告反复强调同样的错误，因此掌握这些将为你带来显著优势。

Mistake 1: Confusing Fleming’s rules. Use the left-hand rule for motors (force on a current in a field) and the right-hand rule for generators (induced current direction). Write ‘FBI’ on your hand or use the position of the fingers as a physical reminder during the exam.

错误一：混淆弗莱明定则。左手定则用于电动机（磁场中电流受力），右手定则用于发电机（感应电流方向）。考试时可在手上写 “FBI” 或用手指位置作物理提示。

Mistake 2: Incorrect angle in F = BIL sin θ. The angle θ is always measured between the wire and the magnetic field, not between the force and anything else. If the wire is perpendicular to the field, use 90°; if the question gives the angle to the normal, convert it first.

错误二：F = BIL sin θ 中角度取错。角 θ 始终是导线与磁场之间的夹角，而非力与其他方向的夹角。若导线垂直于磁场则为 90°；若题目给出的是与法线的夹角，先转换。

Mistake 3: Direction for negative charges. When using Fleming’s left-hand rule for electrons, remember that conventional current flows opposite to electron motion. Point your second finger in the direction of conventional current (opposite to electron velocity), not the direction the electron is moving.

错误三：负电荷方向判断错误。对电子使用弗莱明左手定则时，记住传统电流方向与电子运动方向相反。中指指向传统电流方向（与电子速度相反），而非电子运动方向。

Mistake 4: Forgetting units and conversions. Always convert cm to m, g to kg,

Published by TutorHao | Physics Revision Series | aleveler.com

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Magnetic Fields | 磁场