IB Physics: Electromotive Force and Internal Resistance Analysis | IB物理：电池电动势与内阻分析

📚 IB Physics: Electromotive Force and Internal Resistance Analysis | IB物理：电池电动势与内阻分析

In any real battery, the chemical reactions that produce electrical energy are not perfectly efficient. Inside the battery, charge carriers encounter resistance from the electrolyte and electrodes, meaning that when current flows, some electrical energy is converted to heat before the charges ever leave the terminals. Understanding this internal resistance is essential for predicting how a battery actually behaves in a circuit.

在任何真实电池中，将化学能转化为电能的过程并非完美高效。电池内部，电荷载流子在电解质和电极中会受到阻力，这意味着当电流流动时，一部分电能尚未离开端子就被转化为热。理解这种内阻对于预测电池在电路中的真实表现至关重要。

1. Understanding EMF and Terminal Voltage | 理解电动势与端电压

Electromotive force (EMF), denoted by E, is the total energy supplied by the battery per unit charge passing through it. It is measured in volts and represents the theoretical maximum potential difference the battery can provide when no current flows. The terminal voltage V is the actual potential difference measured across the battery terminals when it delivers current.

电动势（EMF）用 E 表示，是电池在单位电荷通过时所提供的总能量。它以伏特为单位，表示电池在没有电流流动时能够提供的理论最大电势差。端电压 V 则是在电池输出电流时，实际测得的两端子之间的电势差。

When no current flows, the terminal voltage equals the EMF because there is no voltage drop across the internal resistance. As soon as the battery is connected to a load, the terminal voltage falls below the EMF. The difference between E and V is exactly the energy lost inside the battery per unit charge.

当没有电流时，端电压等于电动势，因为内阻上没有电压降。一旦电池接入负载，端电压就会低于电动势。E 与 V 之差恰好是单位电荷在电池内部损失的能量。

2. Internal Resistance and the Circuit Model | 内阻与电路模型

A real battery can be modelled as an ideal EMF source in series with a small resistor, r, known as the internal resistance. This resistance is not a separate physical component but a lumped representation of the battery’s internal opposition to charge flow. The larger the current drawn, the greater the voltage drop across r.

实际电池可以建模为一个理想电动势源与一个小电阻 r 串联，这个电阻就是内阻。它并不是一个独立的物理元件，而是电池内部对电荷流动阻碍的集中表示。输出电流越大，内阻 r 上的电压降就越大。

For a circuit consisting of the battery and an external load resistance R, the total resistance in the circuit is R + r. The current is therefore given by I = E / (R + r). This model is extremely powerful: it allows us to treat a real battery exactly like an ideal source plus a known resistor.

对于包含电池和外接负载电阻 R 的电路，回路总电阻为 R + r。因此电流为 I = E / (R + r)。这个模型非常强大：它让我们能够将真实电池视为理想电源与一个已知电阻的串联组合。

3. The Relationship Between EMF, Terminal Voltage, Current and Internal Resistance | 电动势、端电压、电流和内阻的关系

From the circuit model, the terminal voltage V is the EMF minus the voltage drop across the internal resistance. This gives the fundamental equation:

根据电路模型，端电压 V 等于电动势减去内阻上的电压降。由此得到基本方程：

V = E − I r

This linear relationship is the key to understanding battery behaviour. When the current I is zero, V equals E. As the current increases, the terminal voltage decreases linearly with a gradient of −r. The greater the internal resistance, the more sharply the voltage drops when current is drawn.

这个线性关系是理解电池行为的关键。当电流 I 为零时，V 等于 E。随着电流增大，端电压以 −r 为斜率线性下降。内阻越大，输出电流时电压下降得越显著。

For example, a battery with E = 9.0 V and r = 0.5 Ω, delivering 2.0 A, has a terminal voltage of V = 9.0 − (2.0)(0.5) = 8.0 V. The missing 1.0 V represents energy converted to heat inside the battery itself.

例如，一个 E = 9.0 V、r = 0.5 Ω 的电池，输出 2.0 A 电流时，端电压为 V = 9.0 − (2.0)(0.5) = 8.0 V。损失的 1.0 V 代表电池内部转化为热量的能量。

4. Measuring EMF and Internal Resistance | 测量电动势和内阻

To determine E and r experimentally, you can measure the terminal voltage V and the current I for several different load resistances. Using the equation V = E − I r, a graph of V against I gives a straight line with a y-intercept equal to E and a gradient equal to −r.

为了实验测定 E 和 r，可以针对不同的负载电阻测量端电压 V 和电流 I。利用方程 V = E − I r，画出 V 对 I 的图像，得到一条直线，其纵截距等于 E，斜率等于 −r。

A typical setup uses a variable resistor as the load, an ammeter in series, and a voltmeter connected directly across the battery terminals. By adjusting the variable resistor, you record pairs of (V, I) values. Ensure the switch is closed only while taking readings to avoid polarisation and temperature changes in the battery.

典型实验装置使用滑动变阻器作为负载，电流表串联，电压表直接并联在电池两端。通过调节变阻器，记录一系列 (V, I) 数据。注意只在读数时闭合开关，以避免电池极化和温度变化影响结果。

Alternatively, you can measure the open-circuit voltage with a high-resistance voltmeter to obtain E, then use a single known load to find r. However, the graphical method is preferred because it averages out random errors and reveals any systematic drift.

另一种方法是先用高内阻电压表测量断路电压得到 E，再接入一个已知负载求 r。但图像法更受青睐，因为它能平均随机误差并揭示系统漂移。

5. Graphical Analysis: V vs I | V-I 图像分析

When plotting V on the vertical axis and I on the horizontal axis, the equation V = E − I r produces a straight line of the form y = mx + c. The y-intercept is E, the x-intercept is E/r, and the absolute value of the gradient is r. This graph is a standard IB Physics analysis task.

以 V 为纵轴、I 为横轴作图时，方程 V = E − I r 给出形如 y = mx + c 的直线。纵截距为 E，横截距为 E/r，斜率的绝对值为 r。这张图是 IB 物理标准的分析任务。

Note that the x-intercept corresponds to the short-circuit current, I_s = E/r, which is the maximum current the battery could supply if the external resistance were zero. In practice, this condition is rarely achieved and can damage the battery, but the intercept can still be extrapolated from the graph.

注意横截距对应短路电流 I_s = E/r，即外电阻为零时电池能提供的最大电流。实际中这种状态很少达到，且可能损坏电池，但可以从图像外推得到。

Quantity	From V–I graph
EMF E	y-intercept
Internal resistance r	−gradient
Short-circuit current I_s	x-intercept

When drawing the best-fit line, ignore outliers and remember that the theoretical line should be straight. A curved graph indicates that r changes with current, often due to heating or electrolyte polarization.

画最佳拟合线时忽略异常点，理论线应为直线。若图像弯曲，说明 r 随电流变化，通常由发热或电解质极化引起。

6. Short Circuit and Maximum Current | 短路与最大电流

If the terminals of a battery are connected directly by a wire of negligible resistance, the external resistance approaches zero. The current is then limited only by the internal resistance, giving I_max = E/r. This is called the short-circuit current.

如果电池两端直接由电阻可忽略的导线连接，外电阻趋近于零。此时电流仅受内阻限制，即 I_max = E/r。这被称为短路电流。

During a short circuit, the terminal voltage drops to nearly zero because nearly all of the EMF is used to drive current through the internal resistance. The power dissipated inside the battery is P = I²r, which can cause rapid overheating, venting, or even rupture in real batteries.

短路时端电压几乎降至零，因为几乎所有电动势都用于在内阻上驱动电流。电池内部耗散功率为 P = I²r，可能导致电池快速过热、泄气甚至破裂。

In practical terms, a fresh AA alkaline battery with E ≈ 1.5 V and r ≈ 0.15 Ω could theoretically supply 10 A, but typical short-circuit currents are lower due to additional contact resistance. Always use a current-limiting resistor when testing batteries.

实际中，一节新的 AA 碱性电池 E ≈ 1.5 V、r ≈ 0.15 Ω，理论上可提供 10 A，但由于接触电阻，实际短路电流更低。测试电池时务必使用限流电阻。

7. Power Dissipation and Efficiency | 功率损耗与效率

The total power produced by the battery is P_total = E I. Part of this power is delivered to the external load, P_out = V I, and the remainder is dissipated inside the battery as heat, P_loss = I²r. The sum of these two equals the total power: E I = V I + I²r.

电池产生的总功率为 P_total = E I。其中一部分输出给外部负载，P_out = V I，其余部分在电池内部以热量形式耗散，P_loss = I²r。两者之和等于总功率：E I = V I + I²r。

The efficiency of the battery is the ratio of useful output power to total power:

电池的效率是有用输出功率与总功率之比：

η = V / E = R / (R + r)

When the load resistance R is very large compared to r, the terminal voltage approaches E and the efficiency approaches 100%. Conversely, when R is small, much of the energy is wasted inside the battery, and efficiency is low.

当负载电阻 R 远大于 r 时，端电压接近 E，效率接近 100%。反之，当 R 很小时，大量能量在电池内部被浪费，效率很低。

8. Matching External Resistance for Maximum Power Transfer | 外阻匹配与最大功率传输

For a given battery, the power delivered to the external load is P_out = I²R, where I = E / (R + r). As R varies, P_out has a maximum value. Some IB exam questions ask you to derive or state that maximum power is transferred when the load resistance equals the internal resistance.

对于给定电池，输出到外部负载的功率为 P_out = I²R，其中 I = E / (R + r)。当 R 变化时，P_out 存在最大值。一些 IB 考题要求推导或说明：当负载电阻等于内阻时，传输功率最大。

The condition R = r gives P_max = E² / (4r). Under this condition, the terminal voltage is exactly half of the EMF, and the efficiency is 50%. This result is called the maximum power transfer theorem.

满足 R = r 时，P_max = E² / (4r)。此时端电压恰好等于电动势的一半，效率为 50%。这一结论称为最大功率传输定理。

It is important to distinguish between maximum power transfer and maximum efficiency. In many real applications, such as power distribution, we want high efficiency rather than maximum power. Deliberately matching the load to r is rarely desirable for battery-powered devices.

务必区分最大功率传输与最大效率。在许多实际应用中，比如电力分配，我们希望高效率而非最大功率。对电池供电的设备而言，刻意让负载匹配内阻很少是理想的。

9. Effects of Internal Resistance on Real Batteries | 内阻对真实电池的影响

Internal resistance is not constant. It depends on temperature, state of charge, and the chemical composition of the battery. As a battery discharges, its internal resistance often increases because the concentration of active reactants decreases and reaction products accumulate on the electrodes.

内阻并不是常数。它取决于温度、电荷状态和电池的化学成分。随着电池放电，内阻通常会增大，因为活性反应物浓度降低，反应产物在电极上积累。

At low temperatures, chemical reactions slow down and internal resistance increases significantly. This is why a car battery may struggle to start an engine on a cold morning. Conversely, high temperatures reduce internal resistance but can accelerate unwanted side reactions and reduce battery lifetime.

低温下化学反应变慢，内阻显著增大。这就是为什么汽车电池在寒冷早晨可能难以启动发动机。相反，高温会降低内阻，但会加速有害副反应并缩短电池寿命。

For rechargeable batteries, internal resistance is also influenced by the charge-discharge cycle history. Overcharging or deep discharging can damage the electrode structure, permanently raising r. Regulators and battery management systems monitor r as a health indicator.

对于充电电池，内阻还受到充放电循环历史的影响。过充或深放电会损伤电极结构，使 r 永久增大。电源管理芯片常将 r 作为健康状态的监测指标。

10. Experimental Techniques and Sources of Error | 实验技巧与误差来源

In the IB laboratory, measuring E and r requires careful attention to reduce systematic errors. The voltmeter should have a very high resistance so that the current it draws is negligible compared with the circuit current. The ammeter should have a very low resistance to avoid introducing additional series resistance.

在 IB 实验室中，测量 E 和 r 需要仔细减小系统误差。电压表内阻应非常高，使其分走的电流相对回路电流可忽略。电流表内阻应非常低，以避免引入额外的串联电阻。

Contact resistance at the variable resistor and connecting wires can also distort results. Use thick connecting leads and ensure all connections are clean and tight. The battery itself may heat up when large currents flow, changing r during the experiment, so keep currents small and readings quick.

滑动变阻器和导线连接点的接触电阻也会使结果失真。使用粗导线并确保所有连接清洁、紧固。当大电流流过时电池本身会发热，使 r 在实验过程中改变，因此应保持小电流并快速读数。

Repeated readings and taking the best-fit line of V versus I help reduce random errors. Extrapolating to find E is more reliable than single measurements, provided the linear relationship holds. If the graph shows curvature, you should note that r is not constant and limit your conclusions to the range of currents tested.

重复读数并对 V 与 I 作最佳拟合线可减少随机误差。外推求 E 比单次测量更可靠，前提是线性关系成立。若图像出现弯曲，应说明 r 并非恒定，并只在你测试的电流范围内下结论。

11. Worked Examples | 例题分析

Example 1: A battery has E = 6.0 V and r = 0.80 Ω. It is connected to a resistor of 4.2 Ω. Find (a) the circuit current, (b) the terminal voltage, (c) the power dissipated in the internal resistance.

例 1：某电池 E = 6.0 V，r = 0.80 Ω，接一个 4.2 Ω 的电阻。求 (a) 回路电流；(b) 端电压；(c) 内阻上的功率耗散。

(a) I = E / (R + r) = 6.0 / (4.2 + 0.8) = 6.0 / 5.0 = 1.2 A

(b) V = E − I r = 6.0 − (1.2)(0.80) = 6.0 − 0.96 = 5.04 V

Example 2: When a cell supplies 0.50 A, the terminal voltage is 1.45 V. When it supplies 2.0 A, the terminal voltage is 1.30 V. Find E and r.

例 2：某电池输出 0.50 A 时端电压为 1.45 V；输出 2.0 A 时端电压为 1.30 V。求 E 和 r。

1.45 = E − 0.50r, 1.30 = E − 2.0r

Subtracting the equations: 0.15 = 1.5r → r = 0.10 Ω. Then E = 1.45 + 0.50 × 0.10 = 1.50 V.

两式相减：0.15 = 1.5r → r = 0.10 Ω。于是 E = 1.45 + 0.50 × 0.10 = 1.50 V。

12. Common Misconceptions and Exam Tips | 常见误解与考试提示

A frequent error is confusing EMF with terminal voltage. EMF is the energy supplied per unit charge, while terminal voltage is the measured potential difference under load. Another misconception is that a battery always outputs a constant voltage; in reality, the terminal voltage decreases as current increases.

常见错误是混淆电动势与端电压。电动势是单位电荷获得的能量，而端电压是带负载时实测的电势差。另一误解是认为电池总是输出恒定电压；实际上端电压随电流增大而降低。

In exam questions, always draw the circuit model with r inside the battery symbol. Label E and r explicitly. When using V = E − I r, make sure the current I is the same through the external circuit and the battery. For open-circuit questions, set I = 0 so that V = E.

在考试题中，务必画出带内阻 r 的电池电路模型，明确标出 E 和 r。使用 V = E − I r 时，确保电流 I 同时流过外部电路和电池内部。对于断路问题，令 I = 0，则 V = E。

Finally, be careful with units: resistance in ohms, current in amperes, voltage in volts. Check whether the question asks for terminal voltage, EMF, or power, as these quantities are related but not interchangeable. Reading the question carefully is the first step to full marks.

最后，注意单位：电阻用欧姆，电流用安培，电压用伏特。仔细看清题目问的是端电压、电动势还是功率，这些量相关但不可互换。认真审题是拿满分的第一步。

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IB Physics: Electromotive Force and Internal Resistance Analysis | IB物理：电池电动势与内阻分析