Lenz's Law | 楞次定律

📚 Lenz’s Law | 楞次定律

Lenz’s law is a fundamental rule in electromagnetism that determines the direction of an induced current. It is often tested together with Faraday’s law of electromagnetic induction in CIE A-Level Physics, and it explains why induced effects always resist the change that creates them.

楞次定律是电磁学中决定感应电流方向的基本规律。在 CIE A-Level 物理中，它常与法拉第电磁感应定律一起考查，并且解释了为什么感应效果总是反抗产生它的变化。

1. Electromagnetic Induction and Magnetic Flux | 电磁感应与磁通量

When a conductor experiences a changing magnetic flux, an electromotive force (emf) is induced. Magnetic flux is defined as Φ = BA cos θ, where B is the magnetic flux density, A is the area of the loop, and θ is the angle between the magnetic field and the normal to the area.

当导体经历的磁通量发生变化时，就会产生感应电动势。磁通量定义为 Φ = BA cos θ，其中 B 是磁通密度，A 是回路面积，θ 是磁场方向与面积法线之间的夹角。

Magnetic flux is measured in webers (Wb). A change in flux can be caused by changing the field strength B, changing the area A, or changing the orientation angle θ.

磁通量的单位是韦伯（Wb）。磁通量变化可以由磁感应强度 B 变化、面积 A 变化或取向角度 θ 变化引起。

In CIE exam questions, it is important to distinguish between magnetic flux and magnetic flux linkage. Flux linkage is the product NΦ, where N is the number of turns of a coil.

在 CIE 考试题中，区分磁通量和磁链非常重要。磁链是 NΦ 的乘积，其中 N 是线圈的匝数。

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 is directly proportional to the rate of change of magnetic flux linkage.

法拉第定律指出，感应电动势的大小与磁链变化率成正比。

emf = -N ΔΦ / Δt

Here N is the number of turns, ΔΦ is the change in magnetic flux through one turn, and Δt is the time interval over which the change occurs. The negative sign is the contribution of Lenz’s law and shows the direction of the induced emf.

其中 N 是线圈匝数，ΔΦ 是单匝线圈的磁通量变化量，Δt 是发生变化的时间间隔。负号正是楞次定律的贡献，它表示感应电动势的方向。

Faraday’s law gives the size of the induced emf, while Lenz’s law gives its direction. Together they form the complete law of electromagnetic induction.

法拉第定律给出感应电动势的大小，而楞次定律给出其方向。两者共同构成完整的电磁感应定律。

3. Statement of Lenz’s Law | 楞次定律的表述

Lenz’s law states that the direction of an induced current is such that it opposes the change in magnetic flux that produces it.

楞次定律指出：感应电流的方向总是使它产生的效果反抗引起感应电流的磁通量变化。

This opposition can appear as an induced magnetic field, an induced mechanical force, or an induced back emf. The key word is change, not the flux itself.

这种反抗可以表现为感应磁场、感应机械力或反电动势。关键词是变化，而不是磁通量本身。

Lenz’s law is a direct consequence of the conservation of energy. If the induced current helped the change instead of opposing it, energy would be created from nothing, which is impossible.

楞次定律是能量守恒定律的直接结果。如果感应电流帮助变化而不是反抗变化，能量就会无中生有，这是不可能的。

4. Determining the Direction of Induced Current | 判断感应电流的方向

To find the direction of the induced current, first identify the direction of the external magnetic field and whether the magnetic flux through the loop is increasing or decreasing.

要判断感应电流的方向，首先确定外磁场方向以及穿过回路的磁通量是增大还是减小。

If the flux through the loop is increasing, the induced magnetic field must point opposite to the external field so as to reduce the increase. If the flux is decreasing, the induced magnetic field must point in the same direction as the external field so as to reduce the decrease.

如果穿过回路的磁通量增大，感应磁场方向必须与外磁场相反，以削弱这种增大。如果磁通量减小，感应磁场方向必须与外磁场相同，以削弱这种减小。

After finding the required direction of the induced magnetic field, use the right-hand grip rule to determine the direction of the induced current that would create that field.

在确定所需感应磁场方向后，用右手螺旋定则判断能够产生该磁场的感应电流方向。

Change in external flux	Induced magnetic field	Effect
Increasing	Opposite to external field	Opposes increase
Decreasing	Same direction as external field	Opposes decrease

In CIE structured questions, always show these three steps: state the flux change, state the direction of the induced field required by Lenz’s law, and then give the induced current direction using the right-hand rule.

在 CIE 结构化问题中，一定要展示这三个步骤：说明磁通量变化，根据楞次定律说明所需感应磁场的方向，然后使用右手定则给出感应电流方向。

5. Worked Example: Magnet and Solenoid | 例题：磁铁与螺线管

Consider a bar magnet moving north pole first towards a solenoid. The magnetic flux through the solenoid increases in the direction of the magnet’s field, say to the right.

考虑一根条形磁铁以 N 极朝前靠近螺线管。通过螺线管的磁通量沿磁铁磁场方向增大，例如向右增大。

By Lenz’s law, the solenoid must produce a magnetic field to the left to oppose the increasing flux to the right. Using the right-hand grip rule, the induced current must circulate in a direction that makes the left end of the solenoid a north pole.

根据楞次定律，螺线管必须产生向左的磁场来反抗向右磁通量的增大。根据右手螺旋定则，感应电流必须沿使螺线管左端成为 N 极的方向流动。

The left end of the solenoid therefore repels the approaching north pole of the magnet. This repulsion means external work must be done to push the magnet closer, and that work is transferred to electrical energy in the circuit.

因此螺线管左端排斥靠近的磁铁 N 极。这种排斥意味着必须做外功才能把磁铁推近，这部分功转化为电路中的电能。

If the magnet is then pulled away, the flux to the right decreases. The solenoid now produces a field to the right to oppose the decrease, and the induced current reverses direction.

如果随后把磁铁拉远，向右的磁通量减小。螺线管此时产生向右的磁场来反抗减小，感应电流反向。

6. Energy Conservation and Lenz’s Law | 能量守恒与楞次定律

Lenz’s law can be understood as nature’s way of preventing a runaway gain in energy. The induced current always creates effects that resist the original change, so external work is required to maintain that change.

楞次定律可以理解为自然避免能量无限增加的一种方式。感应电流总是产生反抗原变化的效果，因此要保持变化就必须做外功。

When a magnet falls through a copper tube, eddy currents are induced in the tube. These currents create a magnetic field that opposes the falling magnet, so the magnet falls more slowly than under gravity alone.

当磁铁穿过铜管下落时，铜管中感应出涡流。这些涡流产生阻碍磁铁下落的磁场，因此磁铁下落比仅在重力作用下更慢。

Gravitational potential energy is converted into electrical energy in the eddy currents and then into thermal energy in the copper tube. This is a direct demonstration of energy conservation through Lenz’s law.

重力势能转化为涡流中的电能，再转化为铜管中的热能。这是楞次定律体现能量守恒的直接演示。

7. Eddy Currents and Magnetic Braking | 涡流与磁制动

Eddy currents are circulating currents induced inside a bulk conductor when it is exposed to a changing magnetic flux or when it moves through a non-uniform magnetic field.

涡流是块状导体处于变化磁通量中或在非均匀磁场中运动时，在导体内部感应出的环状电流。

According to Lenz’s law, eddy currents produce magnetic fields that oppose the relative motion or the flux change. This gives rise to magnetic braking and damping in moving metal parts.

根据楞次定律，涡流产生反抗相对运动或磁通量变化的磁场。这导致运动金属部件中出现磁制动和阻尼作用。

In CIE questions, eddy currents are often used to explain why a metal pendulum slows down between the poles of a magnet, or why a copper disc rotating under a magnet experiences a braking force.

在 CIE 考题中，涡流常用于解释金属摆在磁极之间为什么会减速，或者磁铁下方的铜盘旋转时为什么会受到制动力。

Eddy currents also cause unwanted energy loss in transformer cores. To reduce them, the iron core is laminated with thin sheets separated by insulating layers, which increases the resistance to circulating currents.

涡流还会在变压器铁芯中造成不必要的能量损耗。为了减少涡流，铁芯由薄片叠成，片间有绝缘层，从而增大环流的电阻。

8. Applications: Electric Generators and Motors | 应用：发电机与电动机

In an electric generator, a coil is rotated in a magnetic field. The changing flux through the coil induces an emf. Lenz’s law tells us that the induced current produces a torque opposing the rotation of the coil.

在发电机中，线圈在磁场中旋转。通过线圈的磁通量变化感应出电动势。楞次定律告诉我们，感应电流会产生反抗线圈旋转的力矩。

This opposing torque explains why a generator becomes harder to turn when it is connected to a load and current flows. The mechanical work done against the opposing torque is converted into electrical energy.

这个反抗力矩解释了为什么发电机接上负载有电流流过时会变得更难转动。克服反抗力矩所做的机械功转化为电能。

In an electric motor, a current-carrying coil rotates in a magnetic field. The changing flux through the rotating coil induces a back emf, which by Lenz’s law opposes the applied voltage.

在电动机中，载流线圈在磁场中旋转。旋转线圈中变化的磁通量感应出反电动势，根据楞次定律，反电动势反抗外加电压。

This back emf limits the current through the motor at normal operating speed. If the motor is stalled, no back emf is generated, and the current may become dangerously large because there is no induced opposition.

该反电动势在正常转速下限制通过电动机的电流。如果电动机堵转，就不会产生反电动势，电流可能大到危险的程度，因为没有感应电动势的反抗。

9. Transformers and Mutual Induction | 变压器与互感

A transformer works by mutual induction. An alternating current in the primary coil creates a changing magnetic flux in the iron core, which links the secondary coil and induces an emf in it.

变压器通过互感工作。初级线圈中的交变电流在铁芯中产生变化的磁通量，该磁通量耦合到次级线圈并在其中感应出电动势。

Lenz’s law implies that the primary current experiences a back emf due to the changing core flux. This back emf limits the primary current when the secondary circuit is open, so an ideal transformer draws very little current on no load.

楞次定律意味着初级电流会因铁芯磁通量变化而受到反电动势作用。当次级电路开路时，这个反电动势限制初级电流，因此理想变压器空载时电流很小。

When the secondary circuit is closed, the secondary current produces its own flux that opposes the core flux. By Lenz’s law, the primary current then increases to supply an opposing flux and maintain the core flux almost constant.

当次级电路闭合时，次级电流产生自己的磁通量，反抗铁芯磁通量。根据楞次定律，初级电流随之增大，产生相反的磁通量以维持铁芯磁通量几乎不变。

This mutual opposition ensures that power is transferred from the primary to the secondary circuit while the core flux remains nearly constant in an ideal transformer.

这种相互反抗确保功率从初级传递到次级，同时理想变压器中的铁芯磁通量几乎保持不变。

10. Common Misconceptions and Exam Tips | 常见误区与应试技巧

A common misconception is that Lenz’s law means the induced field is always opposite to the external magnetic field. In fact, it only opposes the change in flux, not the flux itself.

一个常见误区是认为楞次定律意味着感应磁场总是与外磁场相反。实际上，它只反抗磁通量的变化，而不是反抗磁通量本身。

When the external flux is decreasing, the induced field is in the same direction as the external field to try to keep the flux from falling. This often surprises students.

当外部磁通量减小时，感应磁场方向与外磁场相同，以尽量阻止磁通量减少。这一点常常让学生感到意外。

Exam questions often ask for the direction of induced current when a magnet moves towards or away from a coil, or when a conducting loop enters or leaves a magnetic field. Always state the change in flux, the opposition required by Lenz’s law, and then the current direction from the right-hand rule.

考题经常要求判断磁铁靠近或远离线圈、导体回路进入或离开磁场时的感应电流方向。答题时一定要陈述磁通量变化、楞次定律要求的反抗方向，再用右手定则确定电流方向。

Remember that the negative sign in Faraday’s law is not just a mathematical detail; it represents Lenz’s law and energy conservation. In calculations, you may omit the negative sign when only the magnitude is needed, but you must include it when explaining direction.

记住法拉第定律中的负号不只是数学细节，它代表了楞次定律和能量守恒。在计算中，如果只需要大小，可以省略负号；但在解释方向时，必须使用负号。

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Lenz’s Law | 楞次定律