📚 Relationship Between EMF and Internal Resistance of a Power Supply | 电源内阻与电动势的关系
In A-Level Physics, every real battery is modelled as a perfect source of electromotive force (EMF) in series with an internal resistance. Understanding how EMF and internal resistance interact is essential for analysing circuits, predicting terminal voltage and calculating power delivery.
在 A-Level 物理中，任何真实电池都可以建模为一个理想的电动势源与一个内阻串联。理解电动势与内阻如何相互作用，对于分析电路、预测端电压以及计算功率输出至关重要。
1. What is EMF? | 什么是电动势？
Electromotive force (EMF), symbol E, is the total energy supplied by the battery per unit charge passing through it. It is measured in volts (V). EMF represents the energy conversion from chemical (or other) forms to electrical energy.
电动势（符号 E）是电池为通过它的每单位电荷所提供的总能量，单位为伏特（V）。电动势表示从化学能（或其他形式的能量）到电能的转换能力。
E = W / Q
where W is the energy supplied and Q is the charge moved.
其中 W 是提供的能量，Q 是移动的电荷量。
2. What is Internal Resistance? | 什么是内阻？
Internal resistance, symbol r, is the resistance offered by the material of the battery itself to the flow of current. It is measured in ohms (Ω). Even a new battery has a small internal resistance; an old or damaged battery may have a much larger internal resistance.
内阻（符号 r）是电池本身材料对电流流动所呈现的电阻，单位为欧姆（Ω）。即使是新电池也有很小的内阻；旧电池或损坏电池的内阻可能大得多。
Internal resistance is not a separate physical component; it is a distributed property of the electrodes and electrolyte. In circuit diagrams, it is drawn as a resistor in series with an ideal EMF source.
内阻并不是一个独立的物理元件，而是电极和电解液的分布属性。在电路图中，它被画成一个与理想电动势源串联的电阻。
3. Relationship Between EMF, Terminal Voltage and Internal Resistance | 电动势、端电压与内阻的关系
When current flows through a battery, the internal resistance causes a voltage drop equal to Ir. The terminal voltage V across the battery terminals is therefore less than the EMF.
当电流流过电池时，内阻会产生一个等于 Ir 的电压降。因此，电池两端的端电压 V 小于电动势。
V = E − Ir
This is the master equation that links EMF, internal resistance, current and terminal voltage.
这是联系电动势、内阻、电流和端电压的基本方程。
Rearranging gives E = V + Ir, showing that the EMF is equal to the sum of the terminal voltage and the voltage lost inside the cell.
变形后可得 E = V + Ir，表明电动势等于端电压与电池内部损耗电压之和。
4. Open-Circuit and Short-Circuit Conditions | 开路与短路情况
When no current flows (open circuit), I = 0, so the terminal voltage equals the EMF. Thus a voltmeter with very high resistance connected across a battery measures approximately E.
当没有电流流动（开路）时，I = 0，因此端电压等于电动势。因此，用高内阻电压表并联在电池两端，测得的电压近似为 E。
When the terminals are connected by a wire of negligible resistance (short circuit), V ≈ 0, so the current is maximum:
当两端被一根电阻可忽略的导线连接（短路）时，V ≈ 0，因此电流达到最大值：
Ishort = E / r
A short circuit can be dangerous because the current is limited only by the internal resistance, which is usually very small.
短路可能非常危险，因为电流仅受内阻限制，而内阻通常很小。
5. Terminal Voltage vs Current Graph | 端电压与电流的关系图像
Rearranging the equation V = E − Ir gives the straight-line form V = −r I + E. If V is plotted against I, the gradient is −r and the intercept on the V-axis is E.
将方程 V = E − Ir 变形为直线形式 V = −r I + E。若以 V 为纵轴、I 为横轴作图，则斜率为 −r，与 V 轴的截距为 E。
| Quantity | From graph |
| EMF E | Intercept on V-axis |
| Internal resistance r | Negative gradient = −r |
| Short-circuit current | Intercept on I-axis (E/r) |
This graph is commonly used in the laboratory to determine both E and r by varying the external load resistance.
该图像常在实验中通过改变外部负载电阻来确定 E 和 r。
6. Power Delivered to the External Circuit | 输出到外电路的功率
The power dissipated in the external load resistor R is Pout = I²R. Using I = E / (R + r), the output power becomes:
外部负载电阻 R 上消耗的功率为 Pout = I²R。由 I = E / (R + r)，输出功率可写为：
Pout = ( E² R ) / ( R + r )²
This power is zero when R = 0 (short circuit) and approaches zero as R becomes very large, because the current tends to zero.
当 R = 0（短路）时该功率为零；当 R 很大使得电流趋于零时，该功率也趋近于零。
7. Maximum Power Transfer Theorem | 最大功率传输定理
The maximum power is delivered to the external load when the load resistance equals the internal resistance of the source, i.e. R = r.
当外部负载电阻等于电源内阻时，即 R = r，外电路获得的功率最大。
Pmax = E² / (4r)
At this point, the efficiency is only 50%, because half of the total power is dissipated in the internal resistance.
此时效率仅为 50%，因为总功率的一半消耗在内阻上。
In power transmission systems, we want high efficiency, so R is made much larger than r. In electronic matching circuits, maximum power transfer may be more important than efficiency.
在电力传输系统中，我们希望高效率，因此使 R 远大于 r。而在电子匹配电路中，最大功率传输可能比效率更重要。
8. Efficiency of the Cell | 电池的效率
The efficiency of a cell is the ratio of the useful power delivered to the load to the total power produced by the EMF.
电池的效率是输送到负载的有用功率与电动势产生的总功率之比。
η = V / E = R / (R + r)
As R increases, efficiency improves. When R = r, efficiency is 0.5 (50%). When R >> r, efficiency approaches 1.
随着 R 增大，效率提高。当 R = r 时，效率为 0.5（50%）。当 R >> r 时，效率趋近于 1。
9. Why Internal Resistance Increases with Age | 为什么内阻会随使用而增大
As a battery discharges or ages, chemical reactions inside the cell become less efficient. The electrodes may corrode, the electrolyte concentration changes, and reaction by-products accumulate. These effects increase the internal resistance.
随着电池放电或老化，电池内部的化学反应效率降低。电极可能腐蚀，电解液浓度改变，反应副产物积累。这些效应都会使内阻增大。
Therefore a battery that still shows a reasonable EMF when measured open-circuit may fail to deliver a large current, because the high internal resistance limits the current and causes a large voltage drop inside the cell.
因此，一个开路测量时电动势仍正常的电池，可能无法提供大电流，因为高内阻限制了电流，并在电池内部产生很大的电压降。
10. Experimental Determination of E and r | 实验测定 E 和 r
A simple experiment uses a variable resistor (rheostat) connected across a cell. For each resistance value, record the terminal voltage V and current I. Then plot V against I.
一个简单的实验是用一个可变电阻（变阻器）并联在电池两端。对每个电阻值，记录端电压 V 和电流 I，然后绘制 V–I 图像。
Alternatively, use a fixed resistor and a voltmeter; change the load resistance by adding known resistors in series or parallel. Use a high-resistance voltmeter to measure V accurately.
另一种方法是使用固定电阻和电压表；通过串联或并联已知电阻来改变负载电阻。使用高内阻电压表准确测量 V。
From the best-fit line, the y-intercept gives E and the negative gradient gives r. Errors can be reduced by using a digital voltmeter and taking multiple readings.
根据最佳拟合直线，y 轴截距给出 E，负斜率给出 r。使用数字电压表并多次读数可以减少误差。
11. Common Mistakes and Exam Tips | 常见错误与考试提示
- Do not confuse EMF with terminal voltage: EMF is always greater than terminal voltage when current flows.
- 不要混淆电动势与端电压：当有电流流动时，电动势总是大于端电压。
- When drawing the V–I graph, remember the gradient is negative, not positive.
- 绘制 V–I 图像时，记住斜率是负的，不是正的。
- In calculations, always use the internal resistance r when calculating total circuit resistance: R_total = R + r.
- 计算时，计算总电路电阻一定要加上内阻：Rtotal = R + r。
- For maximum power transfer, R = r, not R as large as possible.
- 最大功率传输条件是 R = r，而不是 R 越大越好。
- When asked for the ‘lost volts’, calculate Ir, not E − V separately if V is given.
- 当题目问“损失的电压”时，计算 Ir；如果已知 V，则 E − V 等于 Ir。
12. Worked Example | 例题解析
A battery has EMF 9.0 V and internal resistance 1.5 Ω. It is connected to a 4.5 Ω resistor. Calculate (a) the current, (b) the terminal voltage, (c) the power dissipated in the internal resistance.
某电池电动势为 9.0 V，内阻为 1.5 Ω，连接一个 4.5 Ω 的电阻。计算：(a) 电流；(b) 端电压；(c) 内阻消耗的功率。
(a) Total resistance = 4.5 + 1.5 = 6.0 Ω. Current I = E / Rtotal = 9.0 / 6.0 = 1.5 A.
(a) 总电阻 = 4.5 + 1.5 = 6.0 Ω。电流 I = E / Rtotal = 9.0 / 6.0 = 1.5 A。
(b) Terminal voltage V = IR = 1.5 × 4.5 = 6.75 V. Or V = E − Ir = 9.0 − (1.5 × 1.5) = 6.75 V.
(b) 端电压 V = IR = 1.5 × 4.5 = 6.75 V。或 V = E − Ir = 9.0 − (1.5 × 1.5) = 6.75 V。
(c) Power inside battery P = I²r = (1.5)² × 1.5 = 3.375 W ≈ 3.4 W.
(c) 电池内部功率 P = I²r = (1.5)² × 1.5 = 3.375 W ≈ 3.4 W。
Published by TutorHao | Physics Revision Series | aleveler.com
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