📚 Electromagnetic Induction (HL) | 电磁感应（HL）
Electromagnetic induction is a cornerstone of the IB Physics Higher Level syllabus, seamlessly linking the concepts of magnetism, electricity and energy conservation. From Faraday's fundamental flux rule to the self-inductance that governs RL circuits, this topic demands both qualitative insight and quantitative rigour. In this article, we systematically cover magnetic flux, Lenz's law, motional emf, AC generators, transformers, self-induction, energy storage in inductors and eddy currents, all tailored to the Cambridge IB HL examination style.
电磁感应是 IB 物理高级课程的核心支柱，将磁场、电路与能量守恒的思想完美贯通。从法拉第的磁通量变化法则到控制 RL 电路的自感现象，该专题需要定性的物理直觉与严格的定量分析并重。本文系统讲解磁通量与磁链、楞次定律、动生电动势、交流发电机、变压器、自感、电感储能及涡流，紧扣剑桥 IB HL 考纲要点，以中英对照的方式帮助读者精准掌握每一个关键概念。
1. Magnetic Flux and Flux Linkage | 磁通量和磁链
Magnetic flux Φ (phi) quantifies the net magnetic field passing perpendicularly through a surface. For a uniform magnetic field B and a plane area A, the flux is Φ = BA cos θ, where θ is the angle between the field direction and the normal to the area. The SI unit is the weber (Wb), equivalent to T·m².
磁通量 Φ 描述磁场垂直穿过某一曲面的净量。在匀强磁场 B 中，通过面积为 A 的平面的磁通量为 Φ = BA cos θ，其中 θ 是磁场方向与面积法线之间的夹角。其国际单位是韦伯（Wb），等同于 T·m²。
When a coil of N turns is considered, the relevant quantity is the magnetic flux linkage, defined as NΦ. A changing flux linkage is the sole driver of induced electromotive force (emf) in a circuit; if the flux is steady, no emf is induced regardless of how strong the magnetic field is.
对于含有 N 匝的线圈，我们关心的物理量是磁链 NΦ。磁链的变化是电路中产生感应电动势的唯一驱动力；如果磁通量保持不变，无论磁场有多强都不会产生感应电动势。
2. Faraday's Law of Induction | 法拉第电磁感应定律
Faraday's law states that the magnitude of the induced emf in a coil is equal to the rate of change of magnetic flux linkage. Mathematically, the instantaneous emf is ε = –N (dΦ/dt). In many IB problems, we use the average form εavg = –N (ΔΦ/Δt) when the flux change is uniform.
法拉第电磁感应定律指出，线圈中感应电动势的大小等于磁链的变化率。瞬时电动势的数学表达式为 ε = –N (dΦ/dt)。在大量 IB 考题中，若磁通量均匀变化，我们采用平均电动势形式 εavg = –N (ΔΦ/Δt)。
The negative sign comes from Lenz's law and indicates the direction of the induced emf. The magnitude of emf can be increased by using more turns, a stronger magnetic field, a larger coil area, or a faster rate of change of flux. Faraday's law is the operational principle behind every generator and transformer on the planet.
公式中的负号来源于楞次定律，表明感应电动势的方向。增大匝数、增强磁场、增大线圈面积或加快磁通变化率都能提高电动势的幅度。法拉第定律是世间所有发电机和变压器工作的根本原理。
3. Lenz's Law and Energy Conservation | 楞次定律与能量守恒
Lenz's law states that the direction of the induced current is such that it opposes the change in magnetic flux that produced it. This opposition is a direct consequence of the conservation of energy. If the induced current instead reinforced the flux change, a small movement would cause an ever-increasing current, creating energy out of nothing.
楞次定律指出，感应电流的方向总是反抗产生它的磁通量变化。这种反抗是能量守恒定律的直接结果。假如感应电流反而助长磁通变化，那么一个微小的动作就能产生源源不断增大的电流，等同于无中生有创造能量，违反了物理根基。
For example, when the north pole of a bar magnet is pushed into a solenoid, the induced current makes the solenoid's near face a north pole, repelling the magnet. Work must be done against this repulsion, and that mechanical work is transformed into electrical energy.
例如，把条形磁铁的 N 极推入螺线管时，感应电流使螺线管靠近磁铁的一端形成 N 极，产生排斥力。推进磁铁必须克服排斥力做功，这部分机械功就转化为电能。
4. Motional EMF | 动生电动势
When a straight conductor of length L moves with velocity v perpendicular to a uniform magnetic field B, free electrons experience a magnetic force F = qvB, driving them to one end. The resulting charge separation establishes an electric field, and equilibrium gives the motional emf: ε = B L v.
当长度为 L 的直导体以速度 v 垂直于匀强磁场 B 运动时，自由电子受到磁力 F = qvB 而向一端聚集。电荷分离形成的电场力与磁力平衡，产生的动生电动势为 ε = B L v。
If the velocity is not perpendicular to the field, we use the component v⊥ = v sin θ, giving ε = B L v sin θ. This motional emf is entirely consistent with Faraday's law: the moving conductor sweeps out an area per unit time, changing the flux through a hypothetical circuit.
若速度不与磁场垂直，则采用垂直分量 v⊥ = v sin θ，得到 ε = B L v sin θ。这种动生电动势与法拉第定律完全相容：运动的导体在单位时间内扫过一定面积，改变了假想回路内的磁通量。
5. The AC Generator | 交流发电机
An alternating current (AC) generator consists of a coil of N turns, each of area A, rotating with angular speed ω in a uniform magnetic field B. The instantaneous induced emf is sinusoidal: ε = B A N ω sin (ωt). The peak emf is ε₀ = B A N ω.
交流发电机由 N 匝面积为 A 的线圈在匀强磁场 B 中以角速度 ω 旋转构成。瞬时感应电动势按正弦规律变化：ε = B A N ω sin (ωt)，其峰值为 ε₀ = B A N ω。
The rotation period T = 2π/ω, so the frequency of the AC output is f = ω/(2π). Slip rings and brushes maintain electrical contact while allowing continuous rotation. The sinusoidal nature of the emf makes AC power particularly convenient for transformers.
旋转周期为 T = 2π/ω，因此交流电的频率为 f = ω/(2π)。滑环和电刷在保持电接触的同时允许线圈连续旋转。电动势的正弦特性使得交流电能极其方便地配合变压器使用。
6. The Ideal Transformer | 理想变压器
A transformer uses electromagnetic induction to step an AC voltage up or down. For an ideal transformer with no energy losses, the ratio of secondary to primary voltage equals the turn ratio: Vs / Vp = Ns / Np. Since power in equals power out, Ip Vp = Is Vs, the current ratio is inverse: Is / Ip = Np / Ns.
变压器利用电磁感应升高或降低交流电压。对于无能量损耗的理想变压器，次级与初级电压之比等于匝数比：Vs / Vp = Ns / Np。由于输入功率与输出功率相等，即 Ip Vp = Is Vs，电流比与匝数比成反比：Is / Ip = Np / Ns。
Real transformers suffer losses from eddy currents in the iron core, hysteresis in the magnetic material, and resistive heating in the windings. Laminated cores and soft magnetic materials are used to minimise these losses. Transformers only operate with alternating current—a steady DC would produce zero change in flux and burn out the windings.
实际变压器存在铁芯涡流损耗、磁性材料的磁滞损耗和绕组电阻发热。采用叠片铁芯和软磁材料可以大幅降低这些损耗。变压器只能使用交流电——恒定直流电不会产生磁通变化，反而会烧毁线圈。
7. Self-Induction and Inductance | 自感与电感
Self-induction is the phenomenon where a changing current in a coil induces an emf in the same coil that opposes the current change. The induced self-emf is ε = –L (ΔI/Δt), where L is the self-inductance (or simply inductance), measured in henries (H). A device designed to possess significant inductance is called an inductor.
自感现象是指线圈中变化的电流在自身产生感应电动势，并且反抗电流的变化。自感电动势为 ε = –L (ΔI/Δt)，其中 L 为自感系数（电感），单位是亨利（H）。具有显著电感量的器件称为电感器。
The inductance of a long solenoid depends on its geometry and core material: L = μ₀ μᵣ N² A / l, where μ₀ is the permeability of free space, μᵣ the relative permeability of the core, N the number of turns, A the cross-sectional area, and l the length. An inductor resists rapid current changes, making it a crucial component in filters and energy-storage circuits.
长直螺线管的电感取决于其几何结构和磁芯材料：L = μ₀ μᵣ N² A / l，其中 μ₀ 为真空磁导率，μᵣ 为磁芯的相对磁导率，N 为匝数，A 为截面积，l 为长度。电感器抵抗电流的快速变化，因此在滤波器与储能电路中扮演关键角色。
8. RL Circuits and the Time Constant | RL 电路与时间常数
When an inductor is combined with a resistor in a DC circuit, the current does not change instantaneously. Upon connecting to a battery, the current grows according to I = I₀ (1 – e–t/τ), where the steady-state current I₀ = V/R and the time constant is τ = L/R. When the circuit is disconnected, the current decays as I = I₀ e–t/τ.
当电感与电阻组成直流回路时，电流不会瞬间突变。接通电源后，电流按 I = I₀ (1 – e–t/τ) 的规律增长，其中稳态电流 I₀ = V/R，时间常数 τ = L/R。断开电源时，电流则按 I = I₀ e–t/τ 衰减。
The time constant τ gives the time for the current to reach about 63% of its final value during growth, or to fall to 37% during decay. A large inductance or small resistance yields a long time constant, meaning the circuit responds slowly to changes. This exponential behaviour mirrors that of RC circuits, with energy storage shifting from electric fields to magnetic fields.
时间常数 τ 标志着电流增长到终值的约 63% 或衰减到初值的 37% 所需的时间。电感越大、电阻越小，时间常数越长，电路对变化的响应越慢。这一指数规律与 RC 电路互为镜像，只不过储能形式从电场变成了磁场。
9. Energy Stored in an Inductor | 电感中储存的能量
An inductor stores energy in its magnetic field. By integrating the power P = ε I = L I (dI/dt) over time, the stored energy is found to be E = ½ L I². This formula is analogous to the kinetic energy of a moving mass and the energy stored in a capacitor (½ C V²).
电感器将能量储存在其磁场中。对功率 P = ε I = L I (dI/dt) 积分可得储存的能量为 E = ½ L I²。这一表达式与运动物体的动能以及电容器储能 ½ C V² 高度类似。
When the current through an inductor is interrupted, the collapse of the magnetic field releases this stored energy, often causing a high-voltage spike. This principle is used in boost converters and automotive ignition coils. Moreover, the magnetic energy density in a vacuum is u = B²/(2μ₀), revealing that energy resides in the field itself.
当流过电感的电流被突然切断时，磁场崩塌释放出储存的能量，往往引发高压尖峰。升压变换器和汽车点火线圈正是利用了这一原理。此外，真空中的磁场能量密度为 u = B²/(2
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