📚 Faraday's Law of Electromagnetic Induction | 法拉第电磁感应定律
Electromagnetic induction is the process by which a changing magnetic flux produces an electromotive force (emf) in a conductor. Faraday's law of electromagnetic induction gives a quantitative description of this effect and is central to generators, transformers, induction motors, wireless chargers and many sensing devices. It is one of the most heavily examined topics in A-Level Physics, because it links magnetic fields, circuit theory, energy conservation and graph analysis in a single idea.
电磁感应是指变化的磁通量在导体中产生电动势（emf）的过程。法拉第电磁感应定律定量描述了这一效应，它是发电机、变压器、感应电动机、无线充电器以及许多传感设备的核心原理。它是 A-Level 物理中被考查最多的主题之一，因为它把磁场、电路理论、能量守恒与图像分析联系在同一个概念之中。
1. The Idea of Induction | 电磁感应的基本思想
In 1831 Michael Faraday showed that a changing magnetic field near a closed loop of wire can drive a current around the loop, even with no battery present. The current is produced by an induced emf, and its size depends on how rapidly the magnetic influence changes, not on the total strength alone.
1831 年，迈克尔·法拉第证明：闭合导线回路附近的磁场发生变化时，即使没有电池，也能在回路中驱动电流。该电流由感应电动势产生，其大小取决于磁场影响变化的快慢，而不仅仅是磁场总强度。
A steady magnetic field produces no induced emf in a stationary coil. The key is change: move the magnet, switch the electromagnet on or off, rotate the coil, stretch the loop, or alter the circuit geometry in any way, and an emf appears.
恒定磁场不会在静止线圈中产生感应电动势。关键在于变化：移动磁铁、接通或断开电磁铁、转动线圈、拉伸回路，或以任何方式改变回路几何形状，都会产生感应电动势。
The induced emf exists whether or not the circuit is complete. If the circuit is closed, the emf drives an induced current; if the circuit is open, the emf is still there across the gap, but no current flows. This distinction is frequently tested.
无论电路是否闭合，感应电动势都存在。若电路闭合，电动势会驱动感应电流；若电路断开，电动势仍然存在于断口两端，但没有电流流过。这一区别常常被考查。
- Change of magnetic field strength B | 磁感应强度 B 的变化
- Change of the area A enclosed by the loop | 回路所围面积 A 的变化
- Change of the angle θ between B and the normal to the area | B 与面积法线夹角 θ 的变化
- Combinations of the above, e.g. a rotating coil | 以上几种变化的组合，例如旋转线圈
2. Magnetic Flux and Flux Linkage | 磁通量与磁链
θ is the angle between B and the normal to the area. Flux is a scalar with units of weber (Wb).
通过平面面积 A 的磁通量 Φ 在均匀磁场 B 垂直于面积时定义为 Φ = BA。一般情况下，Φ = BA cos θ，其中 θ 是 B 与面积法线之间的夹角。磁通量是标量，单位为韦伯（Wb）。
Φ = BA cos θ
For a coil of N turns, the flux linkage is NΦ. Faraday's law is concerned with the rate of change of this flux linkage, not simply the flux through one loop. This is why the number of turns always appears in calculations involving solenoids and transformers.
对于 N 匝线圈，磁链为 NΦ。法拉第定律关注的是这个磁链的变化率，而不仅仅是单匝线圈的磁通量。这正是螺线管与变压器计算中匝数总会出现的原因。
less. It is also useful to remember that 1 Wb = 1 V s, which follows directly from Faraday's law.
磁通量的国际单位是韦伯：1 Wb = 1 T m²。磁链的单位相同，因为匝数无量纲。同样值得记住的是 1 Wb = 1 V s，这可以直接由法拉第定律推出。
| Quantity | 物理量 | Symbol | 符号 | Unit | 单位 |
| Magnetic flux | 磁通量 | Φ | Wb |
| Flux linkage | 磁链 | NΦ | Wb (Wb-turn) |
| Magnetic flux density | 磁感应强度 | B | T |
| Induced emf | 感应电动势 | ε | V |
3. Statement of Faraday's Law | 法拉第定律的表述
Faraday' cuit. Written as an equation, the magnitude is ε = N ΔΦ / Δt for an average value over a time interval, and ε = −N dΦ/dt in differential form.
法拉第定律指出：回路中感应电动势的大小与通过回路的磁链变化率成正比。写成方程，在一段时间间隔内的平均值为 ε = N ΔΦ / Δt，微分形式为 ε = −N dΦ/dt。
ε = −N ΔΦ / Δt = −Δ(NΦ) / Δt
The minus sign expresses Lenz's law, which will be discussed in the next section. For magnitude-only calculations you may drop it, but in explanations of direction you must include it. The negative sign is not a licence to write a negative magnitude.
负号表达的是楞次定律，下一节将详细讨论。只计算大小时可以省略它，但在解释方向时必须包含它。负号并不意味着可以把大小写成负数。
The law applies to any closed conducting path, including a single wire moving in a field, a coil rotating in a field and a secondary coil wound on the same iron core as a primary. In each case you must first identify what quantity is changing: B, A, θ, or N.
该定律适用于任何闭合导电路径，包括在磁场中运动的单根导线、在磁场中旋转的线圈，以及绕在同一铁芯上的次级线圈。每一种情形下，你都必须先确定变化的是哪个量：B、A、θ 还是 N。
4. Lenz's Law and the Negative Sign | 楞次定律与负号
Lenz's law states that the direction of the induced current is such as to oppose the change producing it. If the flux through a loop increases, the induced current creates its own field opposing that increase; if the flux decreases, the induced current supports the original field.
楞次定律指出：感应电流的方向总是阻碍产生它的变化。若通过回路的磁通量增加，感应电流产生的磁场阻碍这一增加；若磁通量减少，感应电流则维持原有磁场。
Lenz's law is a consequence of conservation of energy. If the induced current helped the change, it would grow without limit and produce free energy, which is impossible. The mechanical work you do pushing a magnet into a coil is converted into electrical energy in the circuit.
楞次定律是能量守恒的必然结果。如果感应电流助长这一变化，它就会无限增长并产生免费能量，这是不可能的。你把磁铁推入线圈所做的机械功，正是转化为回路中的电能。
A reliable algorithm for direction questions: (1) determine the direction of the existing flux; (2) decide whether it is increasing or decreasing; (3) choose the induced field direction that opposes the change; (4) use the right-hand grip rule to convert that field direction into a current direction.
解决方向问题的可靠步骤：（1）确定原有磁通的方向；（2）判断它在增加还是减少；（3）选择能阻碍该变化的感应磁场方向；（4）用右手螺旋定则把磁场方向转换为电流方向。
5. Average and Instantaneous emf | 平均电动势与瞬时电动势
xact answer. When the rate varies, the formula gives the average over the chosen interval.
当磁通量以恒定速率变化时，平均电动势与瞬时电动势相等，简单公式 ε = N ΔΦ / Δt 给出精确结果。当变化率不恒定时，该公式给出所选时间间隔内的平均值。
The instantaneous emf is the gradient of the flux-linkage–time graph, multiplied by −1. A steep straight line means a large constant emf; a curve means a changing emf. The maximum instantaneous emf occurs where the NΦ–t graph is steepest.
瞬时电动势等于磁链—时间图像斜率的相反数。陡直的直线表示较大且恒定的电动势；曲线表示变化的电动势。最大瞬时电动势出现在 NΦ–t 图像最陡的位置。
Because the gradient of a straight-line segment is Δ(NΦ)/Δt, you can read emf values directly from a graph without any additional formula. This is a very common exam command: "determine the maximum emf from the graph".
由于直线段的斜率就是 Δ(NΦ)/Δt，你可以直接从图像读出电动势值，无需其他公式。这是非常常见的考题指令："从图像中确定最大电动势"。
6. Worked Calculation: Changing Field | 例题精讲：磁场变化
niform field. The flux density falls uniformly from 0.40 T to 0.10 T in 0.050 s. Find the average induced emf.
考虑一个 200 匝、面积 2.5 × 10⁻³ m² 的线圈，垂直于均匀磁场放置。磁感应强度在 0.050 s 内从 0.40 T 均匀降到 0.10 T。求平均感应电动势。
ge in flux linkage Δ(NΦ) = 200 × 7.5 × 10⁻⁴ = 0.15 Wb. Step 4: emf = 0.15 / 0.050 = 3.0 V.
第一步：磁感应强度变化 ΔB = 0.40 − 0.10 = 0.30 T。第二步：单匝磁通变化 ΔΦ = ΔB × A = 0.30 × 2.5 × 10⁻³ = 7.5 × 10⁻⁴ Wb。第三步：磁链变化 Δ(NΦ) = 200 × 7.5 × 10⁻⁴ = 0.15 Wb。第四步：电动势 = 0.15 / 0.050 = 3.0 V。
ε = N ΔΦ / Δt = 200 × (7.5 × 10⁻⁴) / 0.050 = 3.0 V
A very common error is to multiply by N twice, or to forget N altogether. Write the flux for one turn first, then multiply by N once. Always check the unit of area: cm² must be converted to m² by multiplying by 10⁻⁴.
非常常见的错误是把 N 乘两次，或者完全忘记乘 N。先写出单匝磁通，再乘以一次 N。务必检查面积单位：cm² 需乘以 10⁻⁴ 换算为 m²。
7. Motional emf and the Rod on Rails | 动生电动势与导轨上的导体棒
ular to a uniform field B, the emf induced across it is ε = BLv. This follows from Faraday's law because the area swept per second is Lv, so the flux cut per second is BLv.
当长度为 L 的直导线以速率 v 垂直于均匀磁场 B 运动时，其上产生的感应电动势为 ε = BLv。这可由法拉第定律推出，因为每秒扫过的面积为 Lv，所以每秒切割的磁通量为 BLv。
ε = BLv (B, L and v mutually perpendicular)
n is F = B²L²v / R. The mechanical power input Fv equals the electrical power I²R, which is another illustration of energy conservation.
若导体棒在连接电阻 R 的无摩擦平行导轨上滑动，感应电流为 I = BLv / R，阻碍运动的磁场力为 F = B²L²v / R。输入的机械功率 Fv 等于电功率 I²R，这再次体现了能量守恒。
d v are mutually perpendicular. If the rod moves at an angle, use the perpendicular component of velocity, v sin θ.
注意几何条件：公式 ε = BLv 仅在 B、L 与 v 两两垂直时适用。若导体棒斜向运动，应使用速度的垂直分量 v sin θ。
- Speed doubled → emf doubled → current doubled → force quadrupled | 速率加倍 → 电动势加倍 → 电流加倍 → 磁场力变为四倍
- Resistance doubled → current halved → force halved | 电阻加倍 → 电流减半 → 磁场力减半
- Field reversed → current direction reversed | 磁场反向 → 电流方向反向
8. Flux Cutting and Flux Linking | 切割磁感线与磁链变化
Two descriptions of induction appear in textbooks: cutting magnetic field lines, and changing flux linkage. Both give the same numerical answers, but the flux-linkage approach is more general and is the one required by most specifications.
教材中出现两种描述电磁感应的方式：切割磁感线和磁链变化。两者给出的数值结果相同，但磁链方法更具普遍性，也是多数考纲要求使用的方法。
Flux cutting is intuitive for a moving wire, where you can picture the conductor sweeping through field lines. Flux linkage is essential for a stationary coil with a changing field, where nothing is moving at all but an emf still appears.
对于运动导线，切割磁感线很直观，你可以想象导体扫过磁感线。而对于磁场变化、本身静止的线圈，磁链方法必不可少，因为此时没有任何东西在运动，但电动势依然出现。
A useful check: any situation described as "cutting flux" can be re-analysed as "changing the area linked by flux". If the two approaches disagree, you have almost certainly made an algebra or geometry error.
一个有用的检验方法：任何"切割磁通"的情形都可以重新分析为"磁链所围面积变化"。若两种方法结果不一致，几乎可以肯定你在代数或几何上出了错。
9. Generators, Transformers and Rotating Coils | 发电机、变压器与旋转线圈
kage is NΦ = NAB cos ωt, so the induced emf is ε = NABω sin ωt, with peak value ε₀ = NABω.
在简单交流发电机中，N 匝、面积 A 的线圈在均匀磁场 B 中以角频率 ω 旋转。磁链为 NΦ = NAB cos ωt，因此感应电动势为 ε = NABω sin ωt，峰值为 ε₀ = NABω。
ε₀ = NABω, ε = ε₀ sin ωt
The emf is zero when the coil plane is perpendicular to the field, because at that instant the flux is momentarily unchanging. The emf is maximum when the coil plane is parallel to the field, because that is where the flux changes fastest.
当线圈平面垂直于磁场时，电动势为零，因为此刻磁通量瞬时不变。当线圈平面平行于磁场时，电动势最大，因为此时磁通量变化最快。
ₛ = Nₛ / Nₚ. Real transformers lose energy through eddy currents, hysteresis and resistive heating, which is why laminated cores are used.
在理想变压器中，同一次磁通变化同时穿过两个线圈，因此 εₛ / εₚ = Nₛ / Nₚ。对于无损耗的理想变压器，功率守恒，Iₚ / Iₛ = Nₛ / Nₚ。实际变压器会因涡流、磁滞和电阻发热而损耗能量，这正是使用叠片铁芯的原因。
10. Graphs of Flux, emf and Current | 磁通、电动势与电流的图像
Graph questions are extremely common. The rule to remember is that the emf graph is the negative gradient of the flux-linkage graph. Peaks, zeros and turning points of the flux graph map onto specific features of the emf graph.
图像题极为常见。需要记住的规则是：电动势图像是磁链图像的负斜率。磁链图像的峰值、零点和拐点都对应电动势图像的特定特征。
| Feature of Φ–t graph | Φ–t 图像特征 | Corresponding emf | 对应电动势 |
| Horizontal line | 水平线 | Zero | 零 |
| Straight slope | 直线斜段 | Constant emf | 恒定电动势 |
| Maximum or minimum | 极大或极小值 | Zero (for smooth curve) | 零（平滑曲线时） |
| Steepest slope | 最陡处 | Maximum emf | 最大电动势 |
| Sinusoid cos ωt | 余弦型曲线 | Sinusoid sin ωt, shifted by 90° | 正弦型曲线，相位差 90° |
If the circuit resistance is constant, the current graph has exactly the same shape as the emf graph. If the resistance changes during the process, that is no longer true, so read the question carefully.
若回路电阻恒定，电流图像与电动势图像形状完全相同。若过程中电阻发生变化，则不再成立，因此要仔细读题。
11. Experimental Investigation and Data Analysis | 实验探究与数据分析
A standard practical uses a small coil connected to a data logger, with a magnet dropped through it or pushed in and out at controlled speeds. The recorded emf–time graph shows a positive then negative pulse as the magnet enters and leaves.
一个标准实验是用连接数据记录仪的小线圈，让磁铁穿过线圈，或以受控速度推入拉出。记录到的电动势—时间图像会先出现一个正向脉冲，随后出现一个反向脉冲，对应磁铁进入与离开的过程。
The area under the emf– −d(NΦ)/dt, so integrating gives the flux-linkage change. The two pulses have equal areas if the magnet starts and ends outside the coil.
电动势—时间图像下的面积等于磁链变化量 N ΔΦ。这是因为 ε = −d(NΦ)/dt，积分即得磁链变化量。若磁铁起始和结束时都在线圈之外，两个脉冲的面积相等。
Increasing the number of turns, using a stronger magnet, or moving the magnet faster all increase the peak emf. Increasing the coil area has the same effect. Turning the coil so that its plane is parallel to the field direction reverses or reduces the effect.
增加匝数、使用更强的磁铁或加快磁铁运动速度，都会提高峰值电动势。增大线圈面积也有同样效果。转动线圈使其平面平行于磁场方向，则会反向或减弱该效应。
- Independent variable: speed of the magnet or rate of field change | 自变量：磁铁速度或磁场变化率
- Dependent variable: peak induced emf from the data logger | 因变量：数据记录仪给出的峰值感应电动势
- Control variables: number of turns, coil area, magnet strength, starting position | 控制变量：匝数、线圈面积、磁铁强度、起始位置
12. Common Misconceptions and Exam Tips | 常见误区与考试要点
A steady magnetic field never induces an emf in a stationary coil, however strong it is. Many students wrongly believe that a large B alone is enough. Only the rate of change of flux linkage matters.
恒定磁场无论多强，都不会在静止线圈中产生感应电动势。许多学生错误地认为只要 B 足够大就可以。实际上只有磁链的变化率才起作用。
Another frequent error is confusing flux with flux density. B is measured in tesla and describes the field itself; Φ is measured in weber and describes the field passing through a particular area. Always write the quantity you are calculating with its unit.
另一个常见错误是把磁通量与磁感应强度混淆。B 以特斯拉为单位，描述磁场本身；Φ 以韦伯为单位，描述穿过某一特定面积的磁场。计算时务必写出所算物理量的单位。
In explanation questions, marks are awarded for correct reasoning order: state that flux linkage changes, quote Faraday's law, apply Lenz's law for direction, and refer to energy conservation if asked. Vague answers such as "the magnet induces a current" score poorly.
在解释题中，得分点在于推理顺序正确：说明磁链发生变化、引用法拉第定律、用楞次定律判断方向，如被要求则提及能量守恒。诸如"磁铁产生电流"这类含糊答案得分很低。
| Misconception | 错误认识 | Correct physics | 正确表述 |
| A strong static field induces emf | 强恒定磁场产生电动势 | Only changing flux linkage induces emf | 只有磁链变化才产生电动势 |
| Lenz's law wastes energy | 楞次定律浪费能量 | It enforces energy conservation | 它是能量守恒的要求 |
| Flux and flux density are the same | 磁通与磁感应强度相同 | Φ = BA cos θ, different units | Φ = BA cos θ，单位不同 |
| emf exists only in closed circuits | 电动势只存在于闭合回路 | emf exists even in open circuits | 断路时电动势依然存在 |
13. Summary of Key Formulas | 核心公式小结
The following relationships cover almost every A-Level question on this topic. Learn them as a connected set rather than isolated facts: flux feeds into flux linkage, flux linkage feeds into Faraday's law, and Faraday's law plus Lenz's law gives the full physical picture.
以下关系式几乎覆盖本主题所有 A-Level 题目。要把它们作为相互联系的整体来掌握，而不是孤立的事实：磁通通向磁链，磁链通向法拉第定律，法拉第定律加上楞次定律才给出完整的物理图像。
Φ = BA cos θ
ε = −N ΔΦ / Δt
ε = BLv (motional emf)
ε₀ = NABω (rotating coil)
εₛ / εₚ = Nₛ / Nₚ (ideal transformer)
F = B²L²v / R (rod on rails)
Practise by writing down, for each new problem, exactly which of B, A, θ or N is changing. That single habit prevents the majority of mistakes in calculations and explanations on electromagnetic induction.
练习时，对每一道新题都先写清楚 B、A、θ 或 N 中究竟哪一个在变化。仅这一个习惯，就能避免电磁感应计算与解释题中的绝大多数错误。
Published by TutorHao | A-Level Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图