Using Induction: Eddy Currents, Generators and Transformers | 电磁感应应用：涡流、发电机与变压器

📚 Using Induction: Eddy Currents, Generators and Transformers | 电磁感应应用：涡流、发电机与变压器

Electromagnetic induction is not only a laboratory phenomenon: it is the basis of many everyday devices. Eddy currents, AC generators and transformers all rely on changing magnetic flux to produce emfs, forces and energy transfer. Understanding these applications is central to the CIE A Level Physics syllabus on electromagnetic induction.

电磁感应不仅是实验室现象：它是许多日常设备的基础。涡流、交流发电机和变压器都依靠变化的磁通量来产生电动势、力以及实现能量传递。理解这些应用是 CIE A Level 物理电磁感应部分的核心内容。

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1. Recap: Faraday’s Law and Lenz’s Law | 回顾：法拉第定律与楞次定律

When the magnetic flux through a circuit changes, an emf is induced. Faraday’s law gives the magnitude of the induced emf: it is proportional to the rate of change of magnetic flux linkage. Lenz’s law gives the direction: the induced current always opposes the change that produces it.

当穿过电路的磁通量发生变化时，会感应出电动势。法拉第定律给出感应电动势的大小：它与磁链变化率成正比。楞次定律给出方向：感应电流总是阻碍产生它的变化。

For a coil of N turns, the flux linkage is NΦ, so the induced emf is written as:

对于 N 匝线圈，磁链为 NΦ，因此感应电动势可写为：

ε = -N ΔΦ / Δt

The negative sign is the mathematical statement of Lenz’s law. It reminds us that the induced emf drives a current whose magnetic effect opposes the original flux change.

负号是楞次定律的数学表达。它提醒我们，感应电动势驱动的电流所产生的磁效应会阻碍原来的磁通量变化。

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2. What are Eddy Currents? | 什么是涡流？

Eddy currents are circulating currents induced inside a solid conductor when it experiences a changing magnetic flux. They flow in closed loops within the metal, like swirling ‘eddies’ in water.

涡流是在整块导体内感应出的环形电流，当导体处于变化的磁通量中时产生。它们在金属内部以闭合回路流动，就像水中的漩涡。

Because the resistance of a solid metal block is very low, eddy currents can be large. They produce significant heating through I²R losses and also create magnetic forces that can oppose motion.

由于整块金属的电阻非常小，涡流可能很大。它们通过 I²R 损耗产生明显的热效应，并产生阻碍运动的磁力。

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3. Magnetic Braking and Damping | 磁制动与阻尼

A metal plate moving through a magnetic field has eddy currents induced in it. By Lenz’s law, these currents flow in directions that oppose the relative motion, creating a braking force.

金属板在磁场中运动时，内部会感应出涡流。根据楞次定律，这些电流的流动方向会阻碍相对运动，从而产生制动力。

This effect is used in eddy-current brakes for trains and roller coasters. It is also used for electromagnetic damping in moving-coil meters, where a metal former or disc reduces needle oscillation and allows the pointer to settle quickly.

这种效应用于火车和过山车的涡流制动。它也用于动圈式仪表中的电磁阻尼，金属框架或圆盘可以减小指针摆动，使指针迅速稳定。

The braking force increases with speed, giving smooth non-contact braking without friction surfaces or mechanical wear.

制动力随速度增大而增大，提供平滑的非接触制动，没有摩擦表面或机械磨损。

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4. Undesirable Eddy Currents and Laminated Cores | 有害涡流与叠片铁芯

In transformers, motors and generators, eddy currents in iron cores waste energy as heat. To reduce them, the core is made of thin iron sheets called laminations, insulated from each other by varnish or oxide layers.

在变压器、电动机和发电机中，铁芯中的涡流会以热的形式浪费能量。为减少涡流，铁芯由薄铁片叠成，片间通过漆层或氧化层绝缘。

The laminations are cut parallel to the magnetic field direction. This increases the resistance to circulating currents and confines the eddy currents to thin layers, greatly reducing the total loss.

叠片平行于磁场方向切割。这增大了环流电阻，将涡流限制在薄层内，从而大大减少总损耗。

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5. The AC Generator: Construction | 交流发电机：基本结构

A simple AC generator consists of a coil of wire rotating in a uniform magnetic field. Slip rings and carbon brushes connect the rotating coil to an external circuit without twisting the wires.

简单交流发电机由在均匀磁场中旋转的线圈组成。滑环和碳刷将旋转线圈连接到外电路，而不会使导线缠绕。

As the coil rotates, the magnetic flux linkage through it changes periodically. An external mechanical force rotates the coil, so the generator converts mechanical energy into electrical energy.

线圈旋转时，穿过它的磁链周期性变化。外部机械力使线圈旋转，因此发电机将机械能转化为电能。

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6. Induced EMF in a Rotating Coil | 旋转线圈中的感应电动势

Consider a coil of N turns and area A rotating with constant angular speed ω in a uniform magnetic field B. If t = 0 is taken when the plane of the coil is perpendicular to the field, the flux linkage at time t is:

考虑 N 匝、面积为 A 的线圈在均匀磁场 B 中以恒定角速度 ω 旋转。若取 t = 0 时线圈平面与磁场垂直，则 t 时刻的磁链为：

NΦ = BAN cos(ωt)

Differentiating the flux linkage with respect to time gives the induced emf:

对磁链关于时间求导可得感应电动势：

ε = BANω sin(ωt)

The peak emf is therefore ε₀ = BANω. The output is alternating: it is zero when the coil is perpendicular to the field and a maximum when the coil is parallel to the field.

因此峰值电动势为 ε₀ = BANω。输出是交变的：线圈垂直于磁场时为零，平行于磁场时最大。

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7. Generator Output and RMS Values | 发电机输出与方均根值

The alternating emf from a generator is usually described by its root-mean-square value. For a sinusoidal emf, the rms value is:

发电机产生的交变电动势通常用方均根值描述。对于正弦电动势，方均根值为：

ε_rms = ε₀ / √2

Similarly, for alternating current, I_rms = I₀ / √2. The rms value is the equivalent steady DC value that would deliver the same average power to a resistor.

同样，对于交变电流，I_rms = I₀ / √2。方均根值是对电阻产生相同平均功率的等效稳定直流值。

When comparing AC supplies or calculating average power, rms values must be used. Average power in a resistor is P = I_rms² R = I_rms V_rms.

在比较交流电源或计算平均功率时，必须使用方均根值。电阻中的平均功率为 P = I_rms² R = I_rms V_rms。

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8. Transformer Principles: Mutual Induction | 变压器原理：互感

A transformer consists of two coils wound on a common soft-iron core. An alternating current in the primary coil produces a changing magnetic flux in the core, which links the secondary coil and induces an alternating emf in it.

变压器由绕在同一软铁芯上的两个线圈组成。初级线圈中的交变电流在铁芯中产生变化的磁通量，该磁通量穿过次级线圈并在其中感应出交变电动势。

The core concentrates the magnetic flux so that nearly all the flux produced by the primary passes through the secondary. The changing flux shared by both coils is an example of mutual induction.

铁芯集中磁通量，使初级线圈产生的几乎所有磁通都通过次级线圈。两个线圈共享的变化磁通就是互感的一个例子。

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9. Ideal Transformer Equation | 理想变压器方程

For an ideal transformer with no energy losses, the ratio of secondary voltage to primary voltage equals the ratio of the number of turns:

对于无能量损耗的理想变压器，次级电压与初级电压之比等于匝数之比：

Vₛ / Vₚ = Nₛ / Nₚ

If the secondary coil has more turns than the primary, the output voltage is increased and the device is a step-up transformer. If the secondary has fewer turns, it is a step-down transformer.

如果次级线圈匝数多于初级线圈，输出电压升高，该设备为升压变压器。如果次级匝数较少，则为降压变压器。

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10. Current and Power in an Ideal Transformer | 理想变压器中的电流与功率

An ideal transformer is 100% efficient, so the input electrical power equals the output electrical power:

理想变压器效率为 100%，因此输入电功率等于输出电功率：

IₚVₚ = IₛVₛ

Combining this with the turns ratio gives the current ratio:

将其与匝数比结合可得电流比：

Iₛ / Iₚ = Nₚ / Nₛ

Thus a step-up transformer increases voltage but reduces current. This is the key reason high voltages are used in power transmission: for a given power, a higher voltage means a lower current, so the I²R heating loss in transmission cables is greatly reduced.

因此升压变压器提高电压但降低电流。这是输电中使用高电压的关键原因：在功率一定时，电压越高电流越小，因此输电电缆中的 I²R 热损耗大大减少。

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11. Energy Losses in Real Transformers | 实际变压器中的能量损耗

Real transformers are not 100% efficient. The main energy losses include copper losses, eddy-current losses, hysteresis losses and magnetic flux leakage.

实际变压器效率并非 100%。主要能量损耗包括铜损、涡流损耗、磁滞损耗和磁通泄漏。

Copper losses are I²R heating in the primary and secondary windings. They are reduced by using low-resistance copper wire of large cross-sectional area.

铜损是初级和次级绕组中的 I²R 发热。通过使用大截面积的低电阻铜线可以减少铜损。

Eddy-current losses are reduced by using a laminated core. Hysteresis losses occur because the core is repeatedly magnetised and demagnetised; they are reduced by using soft magnetic materials such as soft iron.

涡流损耗通过使用叠片铁芯来减少。磁滞损耗因铁芯反复磁化和去磁而产生；可通过使用软铁等软磁材料来减少。

Flux leakage is reduced by designing the core so that it forms a nearly closed loop, ensuring almost all the primary flux links the secondary coil. Transformer efficiency is calculated as:

通过将铁芯设计成近似闭合回路，使几乎所有初级磁通都通过次级线圈，可以减少漏磁。变压器效率按下式计算：

Efficiency = (useful output power / total input power) × 100%

Large modern transformers can exceed 98% efficiency, but the small percentage loss still matters when very large powers are involved.

大型现代变压器效率可超过 98%，但当涉及非常大的功率时，很小的百分比损耗仍然很重要。

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12. Power Transmission and Transformer Applications | 电力传输与变压器应用

In a typical power grid, a power station generates electricity at around 25 kV. A step-up transformer raises the voltage to several hundred kilovolts for long-distance transmission.

在典型电网中，发电厂约以 25 kV 发电。升压变压器将电压升高至数百千伏，用于远距离输电。

At the consumer end, local step-down transformers reduce the voltage to safer levels for homes and industry. This combination of step-up and step-down transformers makes large-scale electricity distribution efficient and practical.

在用户端，本地降压变压器将电压降至家庭和工业使用的安全水平。升压和降压变压器的组合使大规模电力分配既高效又实用。

High-voltage transmission reduces the current for the same transmitted power, so the power lost as I²R in the cables is much smaller. This is the main reason transformers are essential in national electricity grids.

高压输电在相同传输功率下降低电流，因此电缆中 I²R 损耗大大减少。这是变压器在国家电网中必不可少的主要原因。

Published by TutorHao | Physics Revision Series | aleveler.com

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