Electromotive Force, Internal Resistance and Circuit Analysis | 电动势、内电阻与电路分析

📚 Electromotive Force, Internal Resistance and Circuit Analysis | 电动势、内电阻与电路分析

In IGCSE Science, understanding how real batteries behave is essential. A perfect battery is an ideal voltage source, but every real battery has internal resistance. This article explains electromotive force (e.m.f.), terminal potential difference, and how to analyse circuits using these ideas.

在 IGCSE 科学中，理解真实电池的表现至关重要。理想电池是一个完美的电压源，但每个真实电池都有内电阻。本文解释电动势（e.m.f.）、端电压，以及如何用这些概念分析电路。

1. Ideal Cells and Real Batteries | 理想电池与真实电池

An ideal cell has no internal resistance. It always provides the same voltage, regardless of how much current flows through it. This ideal behaviour is used in many simple circuit calculations.

理想电池没有内电阻。无论通过多少电流，它始终提供相同的电压。这种理想行为用于许多简单电路计算中。

In reality, every cell or battery is made of materials that resist the flow of charge. This is called internal resistance, shown by the symbol r. The internal resistance is inside the battery, not a separate resistor in the circuit.

在现实中，每个电池或蓄电池由阻碍电荷流动的材料制成。这称为内电阻，用符号 r 表示。内电阻在电池内部，而不是电路中的单独电阻。

An ideal cell has r = 0.
理想电池：r = 0。
A real cell behaves as an ideal e.m.f. source in series with a small internal resistor r.
真实电池可视为一个理想电动势源串联一个小内电阻 r。

2. Electromotive Force (e.m.f.) | 电动势

The electromotive force (e.m.f.) of a source is the total energy supplied to each coulomb of charge as it passes through the source. The unit of e.m.f. is the volt, where 1 V = 1 J/C.

电源的电动势是每库仑电荷通过电源时被提供的总能量。电动势的单位是伏特，其中 1 V = 1 J/C。

Do not confuse e.m.f. with potential difference. E.m.f. is energy transferred per unit charge from chemical (or other) energy into electrical energy inside the source. Potential difference is energy transferred per unit charge from electrical energy into other forms in a component.

不要混淆电动势与电位差。电动势是电源内部每单位电荷从化学能（或其他能量）转化为电能的能量。电位差是元件中每单位电荷从电能转化为其他形式的能量。

ε = E / Q

Here ε is the e.m.f., E is the energy transformed, and Q is the charge.

这里 ε 是电动势，E 是转化的能量，Q 是电荷。

E.m.f. is measured across a source when no current is drawn.
电动势是在没有电流输出时跨电源测量的。
E.m.f. is often denoted by the Greek letter ε.
电动势常用希腊字母 ε 表示。

3. Terminal Potential Difference | 端电压

The terminal potential difference (terminal p.d.) is the voltage actually measured across the terminals of a cell when it is supplying current. It is always less than the e.m.f. when current flows, because the internal resistance r uses some of the voltage.

端电压是电池在输出电流时，实际测量到的电池两极之间的电压。当电流流过时，它总是小于电动势，因为内电阻 r 消耗了一部分电压。

V = ε − Ir

Where V is the terminal p.d., I is the current, and r is the internal resistance.

其中 V 是端电压，I 是电流，r 是内电阻。

If the external circuit has resistance R, then the terminal p.d. is also equal to the voltage across the external resistor: V = IR.

如果外部电路有电阻 R，那么端电压也等于外部电阻上的电压：V = IR。

4. The Full Circuit Equation | 全电路方程

For a complete circuit containing a source of e.m.f. ε, internal resistance r, and an external resistor R, the total resistance in the circuit is R + r. The current is therefore given by:

对于包含电动势 ε、内电阻 r 和外部电阻 R 的完整电路，电路中的总电阻为 R + r。因此电流由下式给出：

ε = I(R + r)

This can also be written as ε = IR + Ir = V + Ir.

这也可以写成 ε = IR + Ir = V + Ir。

If R is very large (circuit open), then I = 0 and V = ε.
如果 R 非常大（断路），则 I = 0，V = ε。
If R is very small (short circuit), the current is maximum: I = ε / r.
如果 R 非常小（短路），电流最大：I = ε / r。

5. Investigating Internal Resistance | 探究内电阻

To measure the e.m.f. and internal resistance of a cell, you can use a circuit with a variable resistor (rheostat), an ammeter, and a voltmeter connected across the cell.

要测量电池的电动势和内电阻，可以使用一个含变阻器、电流表和跨接电池两端的电压表的电路。

Start with a high resistance on the rheostat to keep the current small.
首先将变阻器阻值调大，使电流较小。
Record pairs of readings: terminal p.d. V and current I.
记录成对的读数：端电压 V 和电流 I。
Change the rheostat several times to get multiple readings.
多次改变变阻器以获得多组读数。

Plot a graph of V on the y-axis against I on the x-axis. The graph will be a straight line with a negative gradient.

以 V 为纵轴、I 为横轴作图，得到的是一条负斜率的直线。

V = −rI + ε

From the graph, the intercept on the y-axis gives the e.m.f., and the negative gradient gives the internal resistance r.

从图中，y 轴截距给出电动势，负斜率给出内电阻 r。

6. Graphical Analysis of V–I | V–I 图像的图像分析

The equation V = −rI + ε is in the form y = mx + c. Here y is V, x is I, the gradient m = −r, and the y-intercept c = ε.

方程 V = −rI + ε 符合 y = mx + c 的形式。这里 y 是 V，x 是 I，斜率 m = −r，y 轴截距 c = ε。

The vertical intercept (at I = 0) gives the e.m.f. ε.
纵截距（当 I = 0）给出电动势 ε。
The slope is negative and equals −r.
斜率为负，等于 −r。
The horizontal intercept (where V = 0) gives the short-circuit current ε / r.
横截距（当 V = 0）给出短路电流 ε / r。

Quantity	How to find from graph
e.m.f. ε	y-intercept
internal resistance r	negative of the gradient
short-circuit current	x-intercept

7. Circuit Symbols and Components | 电路符号与元件

To analyse circuits correctly, you must know the standard symbols for components. A cell is drawn as a long line and a short thin line. A battery is two or more cells. A resistor is a rectangle (or a zigzag in some conventions).

要正确分析电路，必须知道元件的标准符号。电池画为一长一短两条线。电池组是两个或多个电池。电阻用矩形表示（有些惯例用锯齿线）。

Voltmeter is always connected in parallel with the component.
电压表总是与被测元件并联。
Ammeter is always connected in series with the component.
电流表总是与被测元件串联。
An ideal voltmeter has infinite resistance; an ideal ammeter has zero resistance.
理想电压表有无限大电阻；理想电流表有零电阻。

8. Series and Parallel Circuits | 串联与并联电路

In a series circuit, the same current flows through every component. The total resistance is the sum of all resistances:

在串联电路中，通过每个元件的电流相同。总电阻等于所有电阻之和：

R_total = R₁ + R₂ + R₃ + …

In a parallel circuit, the voltage across each branch is the same. The total resistance is given by:

在并联电路中，每个支路两端的电压相同。总电阻由下式给出：

1/R_total = 1/R₁ + 1/R₂ + …

For two resistors in parallel, a simpler formula is:

对于两个并联电阻，可使用更简单的公式：

R_total = (R₁ × R₂) / (R₁ + R₂)

9. Kirchhoff’s Laws at IGCSE Level | IGCSE 层次的基尔霍夫定律

IGCSE physics does not require the full use of Kirchhoff’s laws, but two simple ideas are extremely useful for circuit problems.

IGCSE 物理不要求全面使用基尔霍夫定律，但两个简单的概念对电路问题非常有用。

Kirchhoff’s first law: at any junction, the total current entering equals the total current leaving. This is a statement of conservation of charge.
基尔霍夫第一定律：在任何节点，进入的总电流等于离开的总电流。这是电荷守恒的表述。
Kirchhoff’s second law: around any closed loop, the sum of the e.m.f.s equals the sum of the potential differences. This is a statement of conservation of energy.
基尔霍夫第二定律：在任何闭合回路中，电动势之和等于电位差之和。这是能量守恒的表述。

For a single loop containing one cell and several resistors, the second law gives ε = IR₁ + IR₂ + … + Ir.

对于包含一个电池和几个电阻的单回路，第二定律给出 ε = IR₁ + IR₂ + … + Ir。

10. Power and Energy in Circuits | 电路中的功率与能量

Electrical power is the rate at which energy is transferred. The power dissipated in a resistor depends on both the current through it and the voltage across it.

电功率是能量转移的速率。电阻中消耗的功率取决于通过它的电流和它两端的电压。

P = VI

Using Ohm’s law, V = IR, this can be written as P = I²R or P = V²/R.

利用欧姆定律 V = IR，可以写成 P = I²R 或 P = V²/R。

In a battery, power is wasted in the internal resistance as heat, given by P = I²r. The useful power delivered to the external circuit is P = VI.

在电池中，内电阻上会以热的形式浪费功率，表达式为 P = I²r。传递到外部电路的有用功率是 P = VI。

11. Worked Example | 例题解析

A battery has an e.m.f. of 6.0 V and an internal resistance of 0.50 Ω. It is connected to a 2.5 Ω external resistor. Calculate (a) the current in the circuit, (b) the terminal p.d., (c) the power dissipated in the internal resistance.

一个电池的电动势为 6.0 V，内电阻为 0.50 Ω。它连接到一个 2.5 Ω 的外部电阻。计算（a）电路中的电流，（b）端电压，（c）内电阻上消耗的功率。

(a) Total resistance R_total = R + r = 2.5 + 0.50 = 3.0 Ω.

（a）总电阻 R_total = R + r = 2.5 + 0.50 = 3.0 Ω。

I = ε / (R + r) = 6.0 / 3.0 = 2.0 A

(b) Terminal p.d. V = IR = 2.0 × 2.5 = 5.0 V. Check: V = ε − Ir = 6.0 − (2.0 × 0.50) = 5.0 V.

（b）端电压 V = IR = 2.0 × 2.5 = 5.0 V。验证：V = ε − Ir = 6.0 − (2.0 × 0.50) = 5.0 V。

（c）内电阻上浪费的功率 P = I²r = (2.0)² × 0.50 = 2.0 W。

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

Many students forget that the internal resistance is in series with the external circuit. Always add r to R when calculating the current.

许多学生忘记内电阻与外部电路串联。计算电流时，务必把 r 加到 R 上。

Do not confuse e.m.f. with terminal voltage. Terminal voltage is V = ε − Ir.
不要混淆电动势与端电压。端电压是 V = ε − Ir。
When the battery is open circuit, the voltmeter reading equals the e.m.f., because I = 0.
当电池断路时，电压表读数等于电动势，因为 I = 0。
On a V–I graph, the slope is negative. The magnitude of the slope is r.
在 V–I 图像中，斜率为负。斜率的绝对值是 r。
Always state units: volts for e.m.f. and p.d., ohms for resistance, amps for current.
始终注明单位：电动势和电位差用伏特，电阻用欧姆，电流用安培。

13. Summary | 小结

Real batteries have internal resistance, which causes the terminal p.d. to be less than the e.m.f. when current flows. The key equations are V = ε − Ir and ε = I(R + r). A V–I graph is a straight line with y-intercept ε and gradient −r.

真实电池有内电阻，导致电流输出时端电压小于电动势。关键方程是 V = ε − Ir 和 ε = I(R + r)。V–I 图像是一条直线，y 轴截距为 ε，斜率为 −r。

Understanding these ideas helps you solve circuit problems involving real sources, measure internal resistance experimentally, and avoid common errors in IGCSE Science exams.

理解这些概念有助于你解决涉及真实电源的电路问题，通过实验测量内电阻，并避免 IGCSE 科学考试中的常见错误。

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Electromotive Force, Internal Resistance and Circuit Analysis | 电动势、内电阻与电路分析