📚 Magnetic Fields and Electromagnetism | 磁场与电磁学
Magnetic fields and electromagnetism form one of the central pillars of CIE A-Level Physics. From the force on a current-carrying wire to electromagnetic induction and transformers, these ideas link electric currents, force and energy transfer in a coherent way.
磁场与电磁学是 CIE A-Level 物理的核心支柱之一。从通电导线在磁场中的受力，到电磁感应和变压器，这些概念将电流、力与能量传递有机地联系起来。
This revision guide works through the definition of the tesla, the motor effect, the force on a moving charge, circular motion in a magnetic field, magnetic flux, Faraday's and Lenz's laws, the a.c. generator, the transformer and the Hall effect, with worked calculations and exam tips at the end.
本篇复习指南依次梳理特斯拉的定义、电动机效应、运动电荷受力、磁场中的圆周运动、磁通量、法拉第定律与楞次定律、交流发电机、变压器以及霍尔效应，并在文末给出计算例题与应试要点。
1. Magnetic Field Concepts | 磁场基本概念
A magnetic field is a region of space in which a moving charge, a current-carrying conductor or a magnetic material experiences a force. The direction of a magnetic field at a point is the direction of the force on a north pole placed there.
磁场是空间中运动电荷、通电导线或磁性材料会受到力的区域。磁场的方向定义为置于该点的北极所受力的方向。
The strength of a magnetic field is described by the magnetic flux density B placed perpendicular to the field. Its SI unit is the tesla (T), where 1 T = 1 N A⁻¹ m⁻¹.
磁场的强弱用磁通密度 B 描述，它定义为垂直于磁场的直导线单位电流、单位长度所受的力。其国际单位是特斯拉（T），1 T = 1 N A⁻¹ m⁻¹。
B = F / (I L) and 1 T = 1 N A⁻¹ m⁻¹ = 1 kg A⁻¹ s⁻²
Field lines are used to represent magnetic fields. They point from north to south outside a magnet, and their spacing indicates field strength: closer lines mean a stronger field. Field lines never cross one another.
磁感线用于表示磁场。磁体外部的磁感线从北极指向南极，其疏密表示磁场强弱：磁感线越密，磁场越强。磁感线永不相交。
Outside a bar magnet the field is non-uniform, but between the poles of a large horseshoe magnet, or inside a long solenoid, the field is uniform. A uniform field is drawn as equally spaced parallel straight lines, and this is the standard set-up used in A-Level experiments.
条形磁铁外部的磁场并不均匀，但在大型马蹄形磁铁两极之间或长螺线管内部，磁场是均匀的。均匀磁场画成等间距的平行直线，这也是 A-Level 实验中最常用的装置。
2. Magnetic Force on a Current-Carrying Conductor | 电流在磁场中的力
A current-carrying conductor placed in a magnetic field experiences a force provided the current is not parallel to the field. The magnitude of the force is given by F = B I L sin θ, where θ is the angle between the field B and the current I.
通电导线放在磁场中时，只要电流方向与磁场不平行，就会受到力的作用。其大小由 F = B I L sin θ 给出，其中 θ 是磁场 B 与电流 I 之间的夹角。
F = B I L sin θ
The direction of this force is given by Fleming's left-hand rule: the first finger points in the field direction, the second finger in the conventional current direction, and the thumb gives the force direction. The three directions are mutually perpendicular.
该力的方向由弗莱明左手定则判断：食指指向磁场方向，中指指向电流方向，拇指指向力的方向。三者两两垂直。
When the conductor is perpendicular to the field, sin θ = 1 and the force is maximum: F = B I L. When the conductor is parallel to the field, sin θ = 0 and no force acts.
当导线与磁场垂直时，sin θ = 1，力最大：F = B I L。当导线与磁场平行时，sin θ = 0，不受力。
Note that L is the length of conductor inside the field, not the total length of the wire. If only part of the wire lies between the poles, only that part contributes to the force. Reversing either the current or the field reverses the direction of the force.
注意 L 是处于磁场内部的那段导线长度，而不是整根导线的长度。如果只有一部分导线位于两极之间，就只有这部分贡献力。把电流方向或磁场方向反过来，力的方向也随之反转。
The current balance is the classic experiment for measuring B: a known current is passed through a wire of measured length placed perpendicular to the field, and the force is measured by the change in reading on a top-pan balance. Since F = B I L, the gradient of a graph of F against I gives B L.
电流天平是测量 B 的经典实验：让已知电流通过一段长度可测、垂直于磁场的导线，通过电子天平读数的变化测出力。由 F = B I L 可知，F 对 I 作图的斜率等于 B L。
3. Force on a Moving Charge | 运动电荷在磁场中的力
A single charge q moving with speed v in a magnetic field B experiences a force of magnitude F = B q v sin θ, where θ is the angle between v and B. For a negative charge, the force direction is opposite to that predicted for a positive charge.
单个电荷 q 以速度 v 在磁场 B 中运动时，会受到大小为 F = B q v sin θ 的力，其中 θ 是 v 与 B 之间的夹角。对于负电荷，受力方向与按正电荷判断的方向相反。
F = B q v sin θ
This force is always perpendicular to both the velocity and the field, so it does no work on the charge. The speed, and therefore the kinetic energy, stays constant; only the direction of motion changes.
该力始终同时垂直于速度与磁场，因此对电荷不做功。电荷的速率因而动能保持不变，改变的只是运动方向。
If the charge moves parallel to the field, sin θ continues in a straight line at constant speed. If it moves perpendicular to the field, the force is maximum and the path becomes a circle.
若电荷沿磁场方向运动，sin θ = 0，不受力，电荷以恒定速率沿直线运动。若电荷垂直磁场运动，力最大，轨迹变成圆。
The two force expressions are consistent: a current is a flow of charge, so with I = n A v q the expression F = B I L can be rewritten as F = B q v multiplied by the number of charge carriers in the length L.
两个受力表达式是一致的：电流就是电荷的流动，利用 I = n A v q，可以把 F = B I L 改写成 F = B q v 乘以长度 L 内载流子的数目。
Remember that q is the magnitude of the charge. Protons, electrons and ions all obey the same equation; only the direction differs because of the sign of the charge.
切记 q 取电荷量的大小。质子、电子和离子都遵循同一公式，只是因电荷正负不同，受力方向不同。
4. Magnetic Fields Produced by Currents | 电流产生的磁场
A current produces a magnetic field around it. For a long straight wire the field lines are concentric circles in a plane perpendicular to the wire, and the flux density at distance r from the wire is B = μ₀ I / (2 π r), where μ₀ = 4 π × 10⁻⁷ T m A⁻¹.
电流会在其周围产生磁场。对长直导线，磁感线是垂直于导线平面内的一系列同心圆，距导线 r 处的磁通密度为 B = μ₀ I / (2 π r)，其中 μ₀ = 4 π × 10⁻⁷ T m A⁻¹。
B = μ₀ I / (2 π r) B = μ₀ n I B = μ₀ N I / (2 r)
The direction is found with the right-hand grip rule: point the thumb of the right hand along the conventional current, and the curled fingers show the direction of the field lines.
方向用右手螺旋定则判断：右手拇指指向常规电流方向，弯曲的四指指示磁感线的环绕方向。
For a solenoid of n turns per metre carrying current I, the field inside is uniform and given by B = μ₀ n I. The field is much stronger than that of a single straight wire because the contributions of all the turns add together.
对于每米 n 匝、通有电流 I 的螺线管，内部磁场均匀，由 B = μ₀ n I 给出。由于各匝线圈的贡献叠加，其磁场比单根直导线强得多。
At the centre of a flat circular coil of N turns and radius r, the flux density is B = μ₀ N I / (2 r). Note that doubling the number of turns doubles B, while doubling the radius halves it.
对半径为 r、匝数为 N 的平面圆形线圈，其中心处的磁通密度为 B = μ₀ N I / (2 r)。匝数加倍则 B 加倍，半径加倍则 B 减半。
5. Charged Particles in Circular Motion | 带电粒子的圆周运动
When a charged particle enters a uniform field perpendicular to its velocity, the magnetic force acts as a centripetal force. Equating the two gives B q v = m v² / r, so the radius of the circular path is r = m v / (B q).
当带电粒子垂直进入匀强磁场时，磁场力充当向心力。令两者相等得 B q v = m v² / r，因此圆周轨道的半径为 r = m v / (B q)。
r = m v / (B q) T = 2 π m / (B q)
The period of the circular motion is T = 2 π m / (B q), which is independent of the speed. This surprising result is the basis of the cyclotron, in which particles of increasing speed still take the same time for each revolution.
圆周运动的周期为 T = 2 π m / (B q)，与速率无关。这一令人意外的结论正是回旋加速器的基础：即使粒子速率不断增大，每转一圈所用的时间仍然相同。
A faster particle travels a larger circle, a more massive particle also travels a larger circle, and a stronger field or a larger charge bends the path more tightly.
速率越大，圆的半径越大；质量越大，半径也越大；磁场越强或电荷量越大，轨迹弯曲得越紧。
In a mass spectrometer, ions of the same charge but different masses are accelerated through the same potential difference and enter a uniform magnetic field. They follow circles of different radii, so the ions separate and can be detected at different positions.
在质谱仪中，电荷量相同但质量不同的离子经相同电势差加速后射入匀强磁场，它们沿半径不同的圆周运动，因而被分离开来，可在不同位置被检测到。
If the particle moves along a helical path, its velocity can be resolved into a component parallel to the field, which is unchanged, and a component perpendicular to the field, which produces the circular motion. The two combine to give a helix.
若粒子沿螺旋线运动，可把速度分解为平行于磁场的分量（保持不变）和垂直于磁场的分量（产生圆周运动），两者合成即为螺旋线。
6. Magnetic Flux and Flux Linkage | 磁通量与磁链
Magnetic flux Φ is a measure of the total magnetic field passing through a given area. For a uniform field B perpendicular to a flat area A, Φ = B A. The SI unit of magnetic flux is the weber (Wb), where 1 Wb = 1 T m².
磁通量 Φ 是穿过给定面积的总磁场的量度。对垂直于平面面积 A 的匀强磁场 B，Φ = B A。磁通量的国际单位是韦伯（Wb），1 Wb = 1 T m²。
If the field makes an angle θ th the normal to the area, only the component perpendicular to the surface counts, and Φ = B A cos θ. When the field is parallel to the surface, cos θ = 0 and the flux is zero.
若磁场与面积法线成 θ 角，只有垂直于表面的分量才有贡献，故 Φ = B A cos θ。当磁场平行于表面时，cos θ = 0，磁通量为零。
Φ = B A cos θ 1 Wb = 1 T m²
For a coil of N turns, the magnetic flux linkage is NΦ, measured in webers (or weber-turns). The flux linkage, not the flux alone, determines the induced e.m.f.
对匝数为 N 的线圈，磁链为 NΦ，单位为韦伯（或韦伯·匝）。决定感应电动势的是磁链，而不仅仅是磁通量。
Because B = Φ / A, the flux density B is also called the magnetic flux density: it is the flux per unit area perpendicular to the field.
由于 B = Φ / A，磁通密度 B 也被称为磁感应强度：它是单位垂直面积上的磁通量。
7. Faraday's Law of Electromagnetic Induction | 法拉第电磁感应定律
Faraday' is directly proportional to the rate of change of magnetic flux linkage. In symbols, ε = −N ΔΦ / Δ
法拉第定律指出：感应电动势的大小与磁链的变化率成正比。用符号表示为 ε = −N ΔΦ / Δt，微分形式为 ε = −N dΦ/dt。
