📚 Magnetic Fields: Forces, Flux and Induction | 磁场：力、磁通量与电磁感应
A magnetic field is a fundamental concept in AQA International A-level Physics. It explains how moving charges and current-carrying conductors interact, how charged particles follow circular paths, and how changing magnetic flux induces an emf. This unit brings together force, motion, energy transfer and the operation of generators and transformers.
磁场是 AQA International A-level 物理中的一个基本概念。它解释了运动电荷和载流导体如何相互作用、带电粒子为何做圆周运动，以及变化的磁通量如何感应出电动势。本单元将力、运动、能量转移以及发电机和变压器的工作原理联系在一起。
In the International A-level specification, Magnetic Fields appears as a dedicated topic. Understanding the vector nature of magnetic flux density, Fleming's left-hand rule, Faraday's law and Lenz's law is essential for both numerical problems and explanation questions.
在 International A-level 考试大纲中，磁场是一个独立主题。理解磁通量密度的矢量性质、弗莱明左手定则、法拉第定律和楞次定律，对解答计算题和解释题都至关重要。
1. Magnetic Fields and Flux Density | 磁场与磁通量密度
A magnetic field is a region around a permanent magnet, an electromagnet or a current-carrying conductor in which a magnetic force can be detected. The direction of the field at any point is defined as the direction in which a north pole would experience a force.
磁场是永磁体、电磁铁或载流导体周围能够检测到磁力的区域。某点磁场方向定义为该点北极受力的方向。
the flux density that produces a force of 1 newton on 1 metre of wire carrying 1 ampere perpendicular to the field.
磁通量密度 B 是量度磁场强弱的物理量。它通过垂直置于磁场中的导体所受的力来定义，关系式为 F = BIL sin θ。1 特斯拉（1 T）表示：当 1 米长、通有 1 安培且垂直于磁场的导线受到 1 牛顿力时的磁通量密度。
B = F / (IL sin θ)
Field lines are used to represent magnetic fields visually. They point from north to south outside a magnet and form closed loops. In a uniform magnetic field, the field lines are parallel and equally spaced, so B is constant in both magnitude and direction.
磁感线用于直观表示磁场。磁感线在磁体外部从 N 极指向 S 极并形成闭合回路。在匀强磁场中，磁感线平行且等距，因此 B 的大小和方向均不变。
2. Force on a Current-Carrying Conductor | 载流导体所受的力
ven by F = BIL sin θ, where θ is the angle between the current I and the magnetic field B.
放在磁场中的载流导体会受到力的作用，除非电流方向与磁场平行。力的大小由 F = BIL sin θ 给出，其中 θ 是电流 I 与磁场 B 之间的夹角。
F = BIL sin θ
The direction of the force is determined by Fleming's left-hand rule. Point the first finger in the direction of the magnetic field, the second finger in the direction of conventional current, and the thumb then gives the direction of the force on the conductor.
力的方向由弗莱明左手定则判断。将食指指向磁场方向，中指指向常规电流方向，拇指所指即为导体所受力的方向。
in electric motors, where opposing forces on two sides of a coil create a turning effect.
当 θ = 0° 或 180° 时，电流与磁场平行，力为零。当电流与磁场垂直（θ = 90°）时，力最大，为 F = BIL。这一原理应用于电动机中，线圈两边受到的相反力产生转动效果。
3. Force on a Moving Charge | 运动电荷所受的力
the charge and v is the speed of the particle. The angle θ is measured between the velocity and the magnetic field.
带电粒子在磁场中运动会受到磁力，大小为 F = BQv sin θ，其中 Q 为电荷量，v 为粒子速度。夹角 θ 是速度方向与磁场方向之间的夹角。
F = BQv sin θ
For a positive charge, the direction follows Fleming's left-hand rule with conventional current in the direction of motion. For a negative charge, the force direction is reversed. The force is zero when the particle moves parallel to the field.
对于正电荷，可使用弗莱明左手定则，电流方向取运动方向来判断受力方向。对于负电荷，受力方向相反。当粒子运动方向与磁场平行时，磁力为零。
Because the magnetic force is always perpendicular to the velocity, it does no work on the particle. It cannot change the kinetic energy or speed of the particle, only the direction of motion. This is why charged particles can be deflected in circular arcs inside magnetic fields.
由于磁力始终垂直于速度，它不对粒子做功。它不能改变粒子的动能或速率，只能改变运动方向。这就是带电粒子在磁场中会发生偏转并做圆弧运动的原因。
4. Circular Motion of Charged Particles | 带电粒子的圆周运动
equired for circular motion. Equating BQv = mv² / r gives the radius of the circular path.
当带电粒子垂直进入匀强磁场时，磁力提供圆周运动所需的向心力。令 BQv = mv² / r，可得到圆周路径的半径。
r = mv / BQ
The radius increases with momentum mv and decreases with stronger magnetic field or larger charge. The period of the circular motion is independent of speed, because a faster particle travels in a larger circle but still takes the same time for one revolution.
半径随动量 mv 增大而增大，随磁场增强或电荷增大而减小。圆周运动的周期与速度无关，因为速度较大的粒子虽然半径较大，但完成一周所需时间相同。
T = 2πm / BQ
ting voltage matches the cyclotron frequency f = BQ / 2πm, allowing repeated acceleration.
这一特性应用于回旋加速器和质谱仪中。在回旋加速器中，带电粒子在缝隙中被电场加速，同时被垂直磁场偏转。交变电压的频率与回旋频率 f = BQ / 2πm 匹配，从而实现反复加速。
5. Magnetic Flux and Flux Linkage | 磁通量与磁链
nd the normal to the area. The unit of magnetic flux is the weber (Wb), and 1 Wb = 1 T m².
通过面积 A 的磁通量 Φ 定义为 Φ = BA cos θ，其中 θ 是磁感线与面积法线之间的夹角。磁通量的单位是韦伯（Wb），1 Wb = 1 T m²。
Φ = BA cos θ
m flux linkage occurs when the field is perpendicular to the plane of the coil, at θ = 0°.
对于 N 匝线圈，磁链为 NΦ = BAN cos θ。磁链是匝数与通过每匝的磁通量的乘积。当磁场垂直于线圈平面（θ = 0°）时，磁链最大。
ro. Recognising this angular dependence is important for understanding induced emf in rotating coils and electromagnetic induction problems.
当磁场平行于线圈平面时，θ = 90°，磁链为零。理解这种角度依赖关系，对掌握旋转线圈的感应电动势以及电磁感应问题非常重要。
6. Electromagnetic Induction: Faraday's and Lenz's Laws | 电磁感应：法拉第定律与楞次定律
Electromagnetic induction occurs whenever there is a change in magnetic flux linkage. Faraday's law states that the induced emf is equal to the negative rate of change of flux linkage.
只要磁链发生变化，就会产生电磁感应。法拉第定律指出，感应电动势等于磁链变化率的负值。
ε = -N ΔΦ / Δt
Lenz's law explains the negative sign in Faraday's law. The induced current always flows in a direction that opposes the change in magnetic flux that produced it. This law is a consequence of the conservation of energy.
楞次定律解释了法拉第定律中的负号。感应电流的方向总是阻碍产生它的磁通量变化。该定律是能量守恒的必然结果。
For example, when a magnet is pushed into a coil, the coil develops a north pole at the approaching end to repel the magnet. Work must be done against this repulsion, and that mechanical work is transferred into electrical energy in the circuit.
例如，当磁铁插入线圈时，线圈靠近磁铁的一端产生同极以排斥磁铁。外力必须克服这种排斥做功，机械功由此转化为电路中的电能。
7. Induced EMF in Moving Conductors | 运动导体中的感应电动势
ux lines and induces an emf. The magnitude is given by E = BLv, often called motional emf.
长度为 L 的直导体以速度 v 垂直切割磁感线时，会产生感应电动势。其大小为 E = BLv，通常称为动生电动势。
E = BLv
component perpendicular to the field is effective. The induced emf is then E = BLv sin θ.
如果速度与磁场成 θ 角，则只有垂直于磁场的分量有效。此时感应电动势为 E = BLv sin θ。
The direction of the induced current can be found using Fleming's right-hand rule. Point the first finger in the direction of the magnetic field and the thumb in the direction of motion; the second finger then shows the direction of the conventional induced current.
感应电流的方向可用弗莱明右手定则判断。食指指向磁场方向，拇指指向导体运动方向，中指则指向常规感应电流的方向。
8. AC Generator and Rotating Coil | 交流发电机与旋转线圈
When a coil rotates uniformly in a uniform magnetic field, the flux linkage through the coil varies sinusoidally with time. This changing flux linkage produces an alternating emf.
当线圈在匀强磁场中匀速转动时，穿过线圈的磁链随时间按正弦规律变化。这种变化的磁链产生交变电动势。
NΦ = BAN cos(ωt)
flux linkage. It reaches a peak value ε₀ = BANω, where ω is the angular speed of the coil.
感应电动势是磁链变化率的负值。其峰值 ε₀ = BANω，其中 ω 是线圈的角速度。
ε = BANω sin(ωt) and ε₀ = BANω
When the coil plane is perpendicular to the field, the flux linkage is maximum but the rate of change is zero, so the induced emf is zero. The emf is greatest when the coil plane is parallel to the field, because the coil is cutting field lines at the maximum rate.
当线圈平面垂直于磁场时，磁链最大但变化率为零，因此感应电动势为零。当线圈平面平行于磁场时，切割磁感线速率最大，感应电动势最大。
9. Transformers | 变压器
A transformer changes the size of an alternating voltage using two coils wound on a soft iron core. The primary coil produces a changing magnetic flux in the core, and this changing flux links the secondary coil to induce an emf.
变压器利用绕在软铁芯上的两组线圈改变交流电压的大小。初级线圈在铁芯中产生变化的磁通，这一变化磁通与次级线圈交链，从而感应出电动势。
For an ideal transformer, the ratio of secondary voltage to primary voltage equals the ratio of turns.
对于理想变压器，次级电压与初级电压之比等于匝数之比。
Vs / Vp = Ns / Np
Assuming 100% efficiency, the input power equals the output power. This gives the current ratio Is / Ip = Np / N
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