D.4 Induction (HL only) | D.4 电磁感应（仅限HL）

📚 D.4 Induction (HL only) | D.4 电磁感应（仅限HL）

Electromagnetic induction is the generation of an electromotive force (emf) across a conductor when it experiences a changing magnetic field. Discovered by Michael Faraday, this principle underpins electric power generation, transformers, induction cooking and a vast array of modern technologies. In IB Physics topic D.4 (HL only), we explore magnetic flux, flux linkage, Faraday's law, Lenz's law, motional emf, AC generators and transformers, before extending into the HL-exclusive territory of self-inductance, LR circuits and energy storage in magnetic fields.

电磁感应是当导体处于变化的磁场中时，其两端产生电动势（emf）的现象。由法拉第发现，这一原理是发电、变压器、电磁炉以及大量现代技术的基础。在IB物理课题D.4（仅限HL）中，我们将探究磁通量、磁链、法拉第定律、楞次定律、动生电动势、交流发电机和变压器，然后进入HL专属的自感、LR电路以及磁场储能等内容。

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

rea A whose normal makes an angle θ with the field lines, the flux is Φ = BA cos θ. The SI unit of flux is the weber (Wb), where 1 Wb = 1 T m². Flux is maximum BA when the surface is perpendicular to the field (θ = 0°) and zero when the surface is parallel to the field (θ = 90°).

磁通量 Φ 是穿过某一给定面积的磁场总量的量度。对于强度为 B 的匀强磁场，若面积为 A 的平面其法线与磁感线之间的夹角为 θ，则磁通量 Φ = BA cos θ。磁通量的国际单位是韦伯（Wb），1 Wb = 1 T m²。当平面与磁场垂直（θ = 0°）时，磁通量最大，为 BA；当平面与磁场平行（θ = 90°）时，磁通量为零。

Φ = BA cos θ    1 Wb = 1 T m²

For a coil of N turns, the quantity that directly determines the induced emf is the magnetic flux linkage, defined as NΦ. If each turn encloses the same flux Φ, the total flux linkage is simply NΦ. Any change in this flux linkage – whether caused by motion, a time-varying magnetic field, or a changing area – will induce an emf.

对于 N 匝线圈，直接决定感应电动势的物理量是磁链，其定义为 NΦ。若每匝线圈包围相同的磁通量 Φ，则总磁链即为 NΦ。磁链的任何变化——无论是由运动、时变磁场还是面积变化引起——都会感应出电动势。

flux linkage = NΦ    unit: Wb (or Wb-turns)

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

Faraday' ge of magnetic flux linkage. In symbols, ε = −N ΔΦ/Δt, where the minus sign expresses Lenz' x changes quickly, when the number of turns is large, or when the field strength is large.

法拉第定律指出，感应电动势的大小等于磁链的变化率。用符号表示为 ε = −N ΔΦ/Δt，其中负号表达的就是楞次定律。因此，当磁通量变化很快、匝数很多或磁场很强时，感应电动势就很大。

ε = −N ΔΦ/Δt

The law is often written in terms of an average emf over a finite time interval, and the instantaneous emf is the gradient of an NΦ–t graph. Note carefully that a steady flux – however large – produces no emf at all; only a changing flux linkage does.

该定律通常用有限时间间隔内的平均电动势来表示，而瞬时电动势则是 NΦ–t 图像的斜率。要特别注意，恒定不变的磁通量——无论多大——都不会产生电动势；只有变化的磁链才会。

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3. Lenz's Law and Conservation of Energy | 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 that produced it. If the flux through a coil increases, the induced current creates a field opposing that increase; if the flux decreases, the induced current reinforces the original field. The induced current therefore always tries to keep the flux linkage constant.

楞次定律指出，感应电流的方向总是反抗产生它的那个变化。若穿过线圈的磁通量增加，感应电流产生的磁场方向与原来相反，阻碍其增加；若磁通量减少，感应电流则维持原来的磁场。因此感应电流总是"试图"保持磁链不变。

Lenz's law is simply conservation of energy in disguise. If the induced current helped the change, it would grow without limit and create energy from nothing. In a real generator, the opposition appears as a mechanical force that the driving engine must overcome, and the work done by that force is exactly the electrical energy produced.

楞次定律实质上是能量守恒的体现。如果感应电流有助于变化，它就会无限增大，凭空创造能量。在真实发电机中，这种"反抗"表现为原动机必须克服的机械力，克服该力所做的功恰好等于所产生的电能。

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4. Motional emf: a Rod Moving in a Field | 动生电动势：磁场中运动的导体棒

A straight conductor of length ℓ with speed v perpendicular to a uniform field B experiences a motional emf given by ε = Bℓ ential difference builds up, and this emf is independent of the resistance of the circuit.

长度为 ℓ 的直导体以速度 v 垂直于匀强磁场 B 运动时，会产生动生电动势 ε = Bℓv。导体中的自由电荷受到磁场力被推移，直到建立起稳定的电势差。该电动势与电路的电阻无关。

ε = Bℓv    (B, ℓ and v mutually perpendicular)

the rod slides on rails connected to a resistor R and completes a circuit, a current I = Bℓv/R flows. The field then exerts a braking force on the rod of magnitude F = BIℓ = B²ℓ, directed opposite to the motion, and the power dissipated in the resistor is P = Fv = B²ℓ²v²/R.

若导体棒在导轨上滑动，导轨与电阻 R 构成闭合回路，则电流 I = Bℓv/R。此时磁场对棒施加大小 F = BIℓ = B²ℓ²v/R 的制动力，方向与运动方向相反，电阻上耗散的功率为 P = Fv = B²ℓ²v²/R。

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5. The AC Generator | 交流发电机

ωt). Differentiating gives the induced emf ε = NBAω sin(ωt), so the peak emf is ε₀ = NBAω.

当 N 匝、面积为 A 的线圈在匀强磁场 B 中以恒定角速度 ω 转动时，磁链随时间变化为 NΦ = NBA cos(ωt)。对其求导得感应电动势 ε = NBAω sin(ωt)，因此峰值电动势为 ε₀ = NBAω。

ε = NBAω sin(ωt)    ε₀ = NBAω

Notice that the flux and the emf are 90° out of phase: the emf is greatest when the coil plane is parallel to the field (flux momentarily zero), and zero when the coil plane is perpendicular to the field (flux momentarily maximum). Increasing N, B, A or ω all increase the peak output.

注意磁通量与电动势相位相差 90°：当线圈平面平行于磁场时（磁通量瞬时为零）电动势最大；当线圈平面垂直于磁场时（磁通量瞬时最大）电动势为零。增大 N、B、A 或 ω 都会提高峰值输出。

With slip rings, the output alternates and gives a.c.; with a split-ring commutator the connections swap every half turn, giving a pulsating d.c. This single difference explains why the same machine can serve as an a.c. or a d.c. generator.

使用滑环时，输出方向周期性改变，得到交流电；使用分裂环换向器时，每半圈电路连接交换一次，得到脉动直流电。正是这一区别使同一台机器既可作交流发电机，也可作直流发电机。

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6. Root-Mean-Square Values and Power | 有效值与功率

rent that would dissipate the same average power in a resistor. For a sinusoidal supply, V_rms = V₀/√2 ≈ 0.707 V₀ and I_rms = I₀/√2 ≈ 0.707 I₀.

由于交流电不断变化，我们定义有效值（rms）为：在电阻中产生相同平均功率的等效直流电。对于正弦交流电，V_rms = V₀/√2 ≈ 0.707 V₀，I_rms = I₀/√2 ≈ 0.707 I₀。

V_rms = V₀/√2    I_rms = I₀/√2    P̄ = V_rms I_rms = ½V₀I₀

The average power is P̄ = V_rms I_rms ve load. Mains voltages quoted as 230 V are rms values, so the actual peak is about 325 V.

平均功率 P̄ = V_rms I_rms，对于纯电阻负载，它等于峰值功率的一半。市电标称的 230 V 就是有效值，其实际峰值为约 325 V。

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7. Transformers | 变压器

, which links the secondary and induces an emf there. For an ideal transformer, Vₛ/Vₚ = Nₛ/Nₚ.

变压器由绕在同一叠片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，该磁通量耦合到次级线圈并感应出电动势。对于理想变压器，Vₛ/Vₚ = Nₛ/Nₚ。

Vₛ/Vₚ = Nₛ/Nₚ    IₚVₚ = IₛVₛ    (ideal: 100% efficient)

Type	Turns ratio	Voltage	Current
Step-up	Nₛ > Nₚ	increases	decreases
Step-down	Nₛ < Nₚ	decreases	increases

Real transformers lose energy through resistance heating in the coils (reduced by thick wire), eddy currents in the core (reduced by lamination), hysteresis (reduced by a soft-iron core) and flux leakage. Power is transmitted at high voltage and low current so that the I²R loss in the cables is minimised.

实际变压器通过多种途径损耗能量：线圈电阻发热（用粗导线减小）、铁芯中的涡流（用叠片减小）、磁滞损耗（用软铁芯减小）以及漏磁。远距离输电采用高电压、小电流，从而把电缆中的 I²R 损耗降到最低。

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8. Self-Inductance | 自感

y (H), where 1 H = 1 Wb A⁻¹ = 1 V s A⁻¹. The back emf opposes the change in current, so an inductor resists any attempt to change the current through it.

任何通有变化电流的线圈都会在自身内部感应出电动势，这一性质称为自感 L，定义为 ε = −L ΔI/Δt。其单位是亨利（H），1 H = 1 Wb A⁻¹ = 1 V s A⁻¹。这个反电动势反抗电流的变化，因此电感器会阻碍通过它的电流发生任何改变。

ε = −L ΔI/Δt    1 H = 1 V s A⁻¹

The self-inductance of a solenoid increases with the number of turns, its cross-sectional area and the permeability of its core, and decreases with its length. This is why an iron core dramatically increases the inductance of a coil.

螺线管的自感随匝数、横截面积和磁芯磁导率的增大而增大，随长度增加而减小。这正是铁芯能显著提高线圈自感的原因。

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9. Energy Stored in a Magnetic Field | 磁场中储存的能量

ield. Integrating the power P = L I (dI/dt) over the growth of the current gives E = ½LI².

在电感器中建立电流需要克服反电动势做功，这些能量储存在磁场中。对电流增长过程积分功率 P = L I (dI/dt) 可得 E = ½LI²。

E = ½LI²

When the current is switched off, this stored energy returns to the circuit, often appearing as a spark across the switch – a familiar demonstration of self-induction and a reminder that the energy cannot simply vanish.

当电流被切断时，这些储存的能量会返回电路，常常表现为开关处的电火花——这是自感的经典演示，也提醒我们能量不会凭空消失。

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10. LR Circuits and the Time Constant | LR 电路与时间常数

it has a time constant τ = L/R, measured in seconds, and the current grows as I = I₀(1 − e^−t/τ), where I₀ = ε/R is the final steady current.

当电感器与电阻串联接上电源时，电流不会立即上升。该电路具有时间常数 τ = L/R（单位为秒），电流增长规律为 I = I₀(1 − e^−t/τ)，其中 I₀ = ε/R 是最终的稳定电流。

τ = L/R    growth: I = I₀(1 − e^−t/τ)    decay: I = I₀ e^−t/τ

After one time constant the current has reached 63% of its final value; after five time constants it is effectively steady. When the supply is removed, the current decays exponentially, falling to 37% of its initial value after one time constant.

经过一个时间常数后，电流达到最终值的 63%；经过五个时间常数后基本稳定。切断电源后，电流按指数规律衰减，一个时间常数后降至初始值的 37%。

A large inductance gives a long time constant, which is why inductors are used to smooth currents and to suppress sudden surges in circuits.

自感越大，时间常数越长。因此电感器常用于平滑电流、抑制电路中的突变电流。

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11. Graphs, Sign Conventions and Exam Technique | 图像、符号约定与应试技巧

Exam questions frequently ask you to relate three graphs: Φ–t, NΦ–t and ε–t. The gradient of the flux linkage graph gives the emf, and the area under an emf–time graph gives the change in flux linkage. Take care never to confuse flux with flux linkage – a common source of lost marks.

考试中常要求你联系三类图像：Φ–t、NΦ–t 与 ε–t。磁链图像的斜率给出电动势，而电动势–时间图像下的面积给出磁链的变化量。切记不要混淆磁通量与磁链——这是最常见的失分点。

When applying Lenz's law, always state the direction of the original flux, the direction of its change, the direction of the induced field (opposing the change) and finally the current direction from the right-hand grip rule. Marks are awarded for each step of this reasoning chain, so write it out explicitly.

应用楞次定律时，应依次写明：原磁场方向、磁通量的变化方向、感应磁场方向（反抗该变化），最后用右手螺旋定则得出电流方向。每一步都有对应得分点，因此要把推理链完整写出。

For numerical work, remember that ε = −N ΔΦ/Δt uses the change in flux, while ε = Bℓ m², V s and H are all linked, and dimensional consistency is an easy way to catch errors.

在数值计算中要记住：ε = −N ΔΦ/Δt 用的是磁通量的变化；ε = Bℓv 适用于单根直导体；ε = NBAω 适用于旋转线圈。仔细检查单位：Wb、T m²、V s 与 H 之间彼此关联，用量纲一致性检查是发现错误的好方法。

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12. Key Equations Summary | 关键公式总结

The table below collects every equation you must be able to recall and apply for topic D.4 at HL.

下表汇总了 HL 课题 D.4 中你必须能够记忆并应用的公式。

Quantity	Equation	Unit
Magnetic flux	Φ = BA cos θ	Wb
Flux linkage	NΦ	Wb
Faraday's law	ε = −N ΔΦ/Δt	V
Motional emf	ε = Bℓv	V
Rotating coil	ε₀ = NBAω	V
rms values	Vrms = V₀/√2	V
Transformer	Vₛ/Vₚ = Nₛ/Nₚ	–
Self-inductance	ε = −L ΔI/Δt	H
Magnetic energy	E = ½LI²	J
LR time constant	τ = L/R	s

Master these relationships, practise the sign reasoning behind Lenz's law, and D.4 becomes one of the most rewarding – and most mark-rich – topics in the whole IB Physics course.

掌握这些关系式，并熟练运用楞次定律的符号推理，D.4 就会成为整个 IB 物理课程中最有收获、也最容易得分的课题之一。

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