Faraday’s Law of Electromagnetic Induction & Induced EMF | 法拉第电磁感应定律与感应电动势

📚 Faraday’s Law of Electromagnetic Induction & Induced EMF | 法拉第电磁感应定律与感应电动势

Faraday’s law of electromagnetic induction is one of the most fundamental principles in physics, explaining how a changing magnetic field can produce an electric current. It forms the basis for generators, transformers, and countless modern technologies. In this revision guide, we will break down the law, its mathematical form, Lenz’s law, and the key problem-solving techniques you need for exam success.

法拉第电磁感应定律是物理学中最基本的原则之一，它解释了变化的磁场如何产生电流。它是发电机、变压器以及无数现代技术的基础。在本复习指南中，我们将详细拆解这一定律、其数学形式、楞次定律以及考试中必备的解题技巧。

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1. What Is Electromagnetic Induction? | 什么是电磁感应？

Electromagnetic induction is the production of an electromotive force (EMF) across an electrical conductor when it experiences a change in magnetic flux. The key idea is that a changing magnetic field creates an electric field, which pushes charges around a circuit.

电磁感应是指当导体经历磁通量变化时，在导体两端产生电动势（EMF）的现象。核心思想是变化的磁场会产生电场，从而推动电荷在电路中运动。

There are two classic ways to induce an EMF:

产生电动势有两种经典方式：

• Move a conductor through a stationary magnetic field, e.g., moving a wire upward between magnet poles.

  让导体在静止磁场中运动，例如在磁极之间向上移动导线。

• Hold the conductor stationary and change the magnetic field around it, e.g., pushing a bar magnet into a coil.

  保持导体静止，而改变其周围的磁场，例如将条形磁铁插入线圈。

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2. Magnetic Flux Φ | 磁通量 Φ

Before we can state Faraday’s law, we must define magnetic flux. Magnetic flux Φ through an area A in a uniform magnetic field B is given by:

在陈述法拉第定律之前，我们必须定义磁通量。均匀磁场 B 中穿过面积 A 的磁通量 Φ 为：

Φ = B × A × cos θ

where θ is the angle between the magnetic field direction and the normal (perpendicular) to the area. The SI unit of magnetic flux is the weber (Wb), where 1 Wb = 1 T·m².

其中 θ 是磁场方向与面积法线（垂直方向）之间的夹角。磁通量的国际单位是韦伯（Wb），1 Wb = 1 T·m²。

Notice that if the magnetic field is parallel to the surface (θ = 90°), cos θ = 0 and the flux is zero. Maximum flux occurs when the field is perpendicular to the surface (θ = 0°).

注意，如果磁场平行于表面（θ = 90°），则 cos θ = 0，磁通量为零。当磁场垂直于表面（θ = 0°）时，磁通量最大。

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3. Faraday’s Law of Induction | 法拉第感应定律

Faraday’s law states that the induced EMF in a circuit is equal to the negative rate of change of magnetic flux linkage through the circuit. For a coil with N turns, the flux linkage is NΦ, and the law is written as:

法拉第定律指出，回路中感应电动势的大小等于穿过回路的磁通链变化率的负值。对于有 N 匝的线圈，磁通链为 NΦ，该定律写作：

EMF = −N × (ΔΦ / Δt)

where ΔΦ/Δt is the rate of change of magnetic flux in webers per second (Wb/s). The negative sign is a reminder of Lenz’s law, which tells us the direction of the induced EMF.

其中 ΔΦ/Δt 是磁通量的变化率，单位为韦伯每秒（Wb/s）。负号提醒我们楞次定律的存在，它指示了感应电动势的方向。

If the flux is not changing uniformly, we use calculus:

如果磁通量不是均匀变化，我们使用微积分形式：

EMF = −N × (dΦ / dt)

This is the instantaneous induced EMF, where dΦ/dt represents the instantaneous rate of change of flux.

这是瞬时感应电动势，其中 dΦ/dt 表示磁通量的瞬时变化率。

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4. Lenz’s Law: The Direction of Induced EMF | 楞次定律：感应电动势的方向

Lenz’s law states that the direction of the induced current is such that it opposes the change that produced it. This is a consequence of the conservation of energy: the induced current creates its own magnetic field that tries to maintain the original flux.

楞次定律指出，感应电流的方向总是阻碍引起感应电流的磁通量变化。这是能量守恒的结果：感应电流会产生自己的磁场，试图维持原有的磁通量。

For example, when a north pole of a magnet moves toward a coil, the coil acts like a magnet with its north pole facing the incoming magnet, repelling it. When the magnet moves away, the coil’s polarity reverses to attract the magnet.

例如，当磁铁的 N 极移向线圈时，线圈表现得像一个 N 极朝向磁铁的磁体，排斥它。当磁铁移开时，线圈的极性反转以吸引磁铁。

Lenz’s law can be used to determine the direction of induced current:

楞次定律可用于确定感应电流的方向：

• Determine the direction of the external magnetic field.

  确定外部磁场的方向。

• Determine whether the flux through the coil is increasing or decreasing.

  确定穿过线圈的磁通量是增加还是减少。

• The induced magnetic field will oppose this change (opposite direction if flux increases, same direction if flux decreases).

  感应磁场将阻碍这一变化（磁通量增加时方向相反，磁通量减少时方向相同）。

• Use the right-hand grip rule to find the current direction that produces this induced magnetic field.

  使用右手螺旋定则找出产生该感应磁场的电流方向。

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5. Motional EMF | 动生电动势

Motional EMF arises when a conductor moves through a magnetic field. For a straight conductor of length L moving with velocity v perpendicular to a uniform magnetic field B, the induced EMF is:

当导体在磁场中运动时会产生动生电动势。对于长度为 L 的直导体，以速度 v 垂直于均匀磁场 B 运动时，感应电动势为：

EMF = B × L × v

This formula assumes that B, L, and v are mutually perpendicular. If the angle between the velocity and the field is φ, then the effective component is v sin φ, giving:

该公式假设 B、L 和 v 两两垂直。如果速度与磁场之间的夹角为 φ，则有效分量为 v sin φ，即：

EMF = B × L × v × sin φ

Motional EMF can also be understood in terms of the magnetic force on charge carriers. The magnetic force qvB separates charges, creating an electric field that balances the magnetic force when equilibrium is reached.

动生电动势也可以用载流子所受的磁力来理解。磁力 qvB 使电荷分离，产生电场，当达到平衡时电场力与磁力平衡。

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6. Magnetic Flux Linkage | 磁通链

For a coil of N turns, the total flux linkage is NΦ. Flux linkage is sometimes denoted by the symbol Λ or simply written as NΦ. Its SI unit is the weber-turn (Wb-turns).

对于 N 匝线圈，总磁通链为 NΦ。磁通链有时用符号 Λ 表示，或直接写作 NΦ。其国际单位是韦伯匝（Wb-turns）。

For an area A with its plane perpendicular to a uniform field B, and a coil of N turns, the flux linkage is simply:

对于平面垂直于均匀磁场 B 的面积 A，且有 N 匝的线圈，磁通链简化为：

NΦ = N × B × A

In exam problems, you are often asked to calculate the change in flux linkage when a coil rotates in a magnetic field or when a magnet moves relative to a coil. Remember that flux linkage changes can be due to a change in B, a change in A, a change in θ, or any combination of these.

在考试题目中，你经常需要计算线圈在磁场中转动或磁铁相对线圈移动时磁通链的变化。请记住，磁通链的变化可能由 B 的变化、A 的变化、θ 的变化或这些因素的任意组合引起。

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7. Faraday’s Law with a Rotating Coil | 转动线圈中的法拉第定律

In a generator, a coil of area A and N turns rotates at constant angular velocity ω in a uniform magnetic field B. The angle between the normal to the coil and the field is θ = ωt (if starting from perpendicular to the field). The flux linkage at time t is:

在发电机中，面积为 A、匝数为 N 的线圈在均匀磁场 B 中以恒定角速度 ω 转动。线圈法线与磁场之间的角度为 θ = ωt（若从垂直于磁场开始）。t 时刻的磁通链为：

NΦ(t) = N × B × A × cos(ωt)

Using Faraday’s law, the induced EMF is the negative derivative of flux linkage with respect to time:

利用法拉第定律，感应电动势是磁通链对时间的负导数：

EMF(t) = N × B × A × ω × sin(ωt)

This is a sinusoidal alternating EMF. The maximum EMF is:

这是正弦交流电动势。最大电动势为：

EMF₀ = N × B × A × ω

The graph of EMF vs time is a sine wave, with the EMF zero when flux linkage is maximum and EMF maximum when flux linkage is zero. This phase relationship is a common exam trap.

电动势随时间变化的图像是正弦波，当磁通链最大时电动势为零，当磁通链为零时电动势最大。这种相位关系是常见的考试陷阱。

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8. Eddy Currents | 涡电流

Eddy currents are loops of electric current induced within conductors by a changing magnetic field. They circulate in closed paths within the material, like whirlpools in water. Eddy currents produce their own magnetic fields and often cause unwanted heating and energy loss in transformers and other electrical machines.

涡电流是变化磁场在导体内感应的电流环流。它们在材料内部沿闭合路径流动，就像水中的漩涡。涡电流会产生自己的磁场，通常在变压器和其他电气设备中导致不必要的发热和能量损失。

To reduce eddy currents, transformer cores are laminated: they are made of thin sheets of iron, each insulated from the next. This reduces the size of the loops and thus the magnitude of the eddy currents.

为了减少涡电流，变压器铁芯采用叠片结构：由彼此绝缘的薄铁片制成。这减小了电流环的大小，从而降低了涡电流的大小。

However, eddy currents are useful in some applications, such as electromagnetic braking in trains, metal detectors, and induction cooktops.

然而，涡电流在某些应用中是有用的，例如列车的电磁制动、金属探测器和电磁炉。

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9. Average vs Instantaneous EMF | 平均电动势与瞬时电动势

Faraday’s law can be applied in two ways. If you need the average EMF over a time interval Δt, use:

法拉第定律有两种应用方式。如果你需要时间间隔 Δt 内的平均电动势，使用：

EMF_avg = −N × (ΔΦ / Δt)

If you need the instantaneous EMF at a specific moment, use the derivative:

如果你需要某一特定时刻的瞬时电动势，使用导数形式：

EMF_inst = −N × (dΦ / dt)

In solving problems, always identify whether the flux is changing uniformly. If the flux-time graph is a straight line, the rate of change is constant and equal to the slope. If it is curved, the instantaneous rate is the slope of the tangent at that point.

在解题时，务必判断磁通量是否均匀变化。如果 Φ-t 图像是直线，变化率为常数且等于斜率。如果是曲线，瞬时变化率是该点切线的斜率。

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10. Worked Example: Bar Magnet and Coil | 例题：条形磁铁与线圈

A coil has 200 turns and a cross-sectional area of 4.0 × 10⁻³ m². A bar magnet is pushed into the coil, increasing the magnetic field perpendicular to the coil from 0 T to 0.15 T in 0.20 s. Calculate the average induced EMF.

一个线圈有 200 匝，横截面积为 4.0 × 10⁻³ m²。将条形磁铁插入线圈，使垂直于线圈的磁场在 0.20 s 内从 0 T 增加到 0.15 T。计算平均感应电动势。

Step 1: Calculate the change in flux.

步骤 1：计算磁通量变化量。

ΔΦ = ΔB × A = 0.15 × 4.0 × 10⁻³ = 6.0 × 10⁻⁴ Wb

Step 2: Apply Faraday’s law.

步骤 2：应用法拉第定律。

EMF = −N × (ΔΦ / Δt) = −200 × (6.0 × 10⁻⁴ / 0.20) = −0.60 V

The magnitude of the average induced EMF is 0.60 V. The negative sign indicates that the induced current opposes the increase in flux (Lenz’s law).

平均感应电动势的大小为 0.60 V。负号表示感应电流阻碍磁通量的增加（楞次定律）。

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11. Comparison of Key Equations | 关键公式对比表

Quantity | 物理量	Equation | 公式	When to Use | 使用时机
Magnetic flux | 磁通量	Φ = B × A × cos θ	Flux through a surface | 穿过某个面的磁通量
Flux linkage | 磁通链	NΦ = N × B × A × cos θ	Coil of N turns | N 匝线圈
Average induced EMF | 平均感应电动势	EMF = −N × (ΔΦ / Δt)	Flux change over a time interval | 一段时间内的磁通变化
Instantaneous induced EMF | 瞬时感应电动势	EMF = −N × (dΦ / dt)	Flux changing non-uniformly | 磁通量非均匀变化
Motional EMF | 动生电动势	EMF = B × L × v × sin φ	Conductor moving through a field | 导体在磁场中运动

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12. Common Pitfalls and Exam Tips | 常见错误与考试技巧

Many students lose marks on electromagnetic induction questions due to a few common mistakes:

许多学生在电磁感应题目中因为一些常见错误而丢分：

• Forgetting the cos θ factor when the field is not perpendicular to the area.

  当磁场不垂直于面积时，忘记乘 cos θ 因子。

• Confusing flux with flux linkage: always multiply by N for coils.

  混淆磁通量与磁通链：对于线圈务必乘以匝数 N。

• Ignoring Lenz’s law when describing direction: the induced EMF always opposes the change.

  描述方向时忽略楞次定律：感应电动势总是阻碍变化。

• Using the wrong sign convention: define a positive direction first, then apply the negative sign consistently.

  使用错误的符号约定：先定义正方向，然后一致地应用负号。

• Using area in cm² without converting to m². Always convert to SI units first.

  使用 cm² 面积而未转换为 m²。务必先转换为国际单位。

Finally, always sketch a flux-time graph when solving multi-step problems. This helps you see whether flux is increasing or decreasing, which directly determines the direction of the induced current.

最后，在解决多步问题时，务必画出 Φ-t 图像草图。这有助于你判断磁通量是增加还是减少，从而直接确定感应电流的方向。

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Published by TutorHao | Physics Revision Series | aleveler.com

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