📚 IB Physics: Electromotive Force and Internal Resistance | IB物理：电池电动势与内阻
In IB Physics, the electromotive force (EMF) of a battery and its internal resistance are fundamental concepts that bridge theoretical circuit analysis and real-world energy sources. Understanding these ideas is essential for solving problems involving terminal voltage, power transfer, and efficiency of electrical devices.
在IB物理中，电池的电动势和内阻是连接理论电路分析与现实能源核心的基础概念。理解这些内容对于解决涉及端电压、功率传输和电气设备效率的问题至关重要。
1. What is Electromotive Force (EMF)? | 什么是电动势？
Electromotive force, abbreviated EMF and denoted by the symbol ε, is the energy supplied by a source to each unit of charge that passes through it. Despite its name, EMF is not a force; it is measured in volts (V) and represents the maximum potential difference a source can provide when no current flows.
电动势，缩写为EMF，用符号ε表示，是电源对通过它的每单位电荷所提供的能量。尽管名称中有“力”，但电动势并不是力；它以伏特（V）为单位，表示在无电流流动时电源能够提供的最大电势差。
When a battery is not connected to any external circuit, the terminal potential difference equals the EMF. However, once current flows, the terminal voltage drops below the EMF due to internal resistance.
当电池未连接到任何外部电路时，端电势差等于电动势。然而，一旦有电流流动，由于内阻的存在，端电压将降至电动势以下。
2. Internal Resistance: The Hidden Resistor | 内阻：隐藏的电阻
A real battery is not an ideal voltage source. Inside every battery, chemical reactions and ionic conduction create resistance to current flow. This is called internal resistance, denoted by r, and it behaves as if a small resistor is connected in series with an ideal EMF source.
真实电池并非理想电压源。在每个电池内部，化学反应和离子传导会对电流产生阻力。这被称为内阻，用r表示，其行为类似于一个与理想电动势源串联的小电阻。
Typical internal resistance values are small: a fresh AA alkaline battery may have r ≈ 0.1-0.3 Ω, while a car battery might be around 0.01 Ω. As batteries age or are discharged rapidly, internal resistance increases, reducing performance.
典型的内阻值很小：一节新的AA碱性电池的内阻约为0.1-0.3 Ω，而汽车电池的内阻约为0.01 Ω。随着电池老化或快速放电，内阻会增大，从而降低性能。
3. The Circuit Model: EMF, r, and Load | 电路模型：电动势、内阻与负载
The simplest model of a real battery connected to an external resistor R is a series circuit consisting of an ideal EMF source ε, the internal resistance r, and the load resistance R.
真实电池连接到外部电阻R的最简单模型是：由理想电动势源ε、内阻r和负载电阻R组成的串联电路。
Using Ohm’s law for the complete circuit:
对完整电路应用欧姆定律：
ε = I(R + r)
Rearranging for the current I:
整理后得到电流I：
I = ε / (R + r)
This shows that the current is determined by the total resistance in the circuit, not just the external load.
这表明电流由电路中的总电阻决定，而不仅仅由外部负载决定。
4. Terminal Voltage vs EMF | 端电压与电动势的区别
The terminal voltage V (or terminal potential difference) is the voltage actually measured across the battery terminals when current is flowing. It is always less than the EMF when the battery is supplying current.
端电压V（或端电势差）是在有电流流动时实际测量到的电池两极之间的电压。当电池输出电流时，端电压总是小于电动势。
The relationship is:
其关系为：
V = ε – Ir
Here, Ir is the “lost volts” — the potential difference consumed inside the battery due to its internal resistance.
其中Ir是“损失电压”——因内阻而在电池内部消耗的电势差。
When the external circuit is open (I = 0), V = ε. As current increases, V decreases linearly with slope -r.
当外部电路断开时（I = 0），V = ε。随着电流增大，V呈线性下降，斜率为-r。
5. Graphical Determination of EMF and r | 通过图像测定电动势和内阻
A common IB experiment involves varying the external resistance and measuring terminal voltage and current. By plotting V against I, a straight line is obtained:
一个常见的IB实验是改变外部电阻并测量端电压和电流。通过绘制V随I变化的图像，可以得到一条直线：
V = -rI + ε
From the graph:
从图像中可以得出：
- The y-intercept equals the EMF ε (when I = 0).
- The slope equals -r (negative of the internal resistance).
- y轴截距等于电动势ε（当I = 0时）。
- 斜率等于-r（内阻的负值）。
This graphical method is preferred in IB assessments because it uses all data points and averages out random errors. When using a voltmeter and ammeter, remember the voltmeter is placed across the battery terminals, and the ammeter in series with the external resistor.
这种图像法在IB评估中更受青睐，因为它利用所有数据点并平均了随机误差。使用电压表和电流表时，请记住电压表并联在电池两端，电流表与外部电阻串联。
6. Power Delivered to the Load | 输送到负载的功率
The power delivered to the external load resistor R is:
输送到外部负载电阻R的功率为：
P = I²R = (ε / (R + r))² × R
As R varies, the power output changes. For a fixed EMF and internal resistance, maximum power transfer occurs when the load resistance equals the internal resistance:
随着R变化，功率输出也发生变化。对于固定的电动势和内阻，最大功率传输发生在负载电阻等于内阻时：
R = r ⟹ P_max = ε² / (4r)
This condition is known as the maximum power transfer theorem. However, it is important to note that at maximum power transfer, the efficiency is only 50%, because half the power is dissipated inside the battery.
这个条件被称为最大功率传输定理。然而，需要注意的是，在最大功率传输时，效率仅为50%，因为有一半的功率消耗在电池内部。
For maximum efficiency, on the other hand, R should be as large as possible, approaching the behaviour of an open circuit.
另一方面，若追求最大效率，R应尽可能大，接近开路状态的行为。
7. Efficiency of a Battery | 电池的效率
The efficiency of a battery is defined as the ratio of power delivered to the load to total power produced by the EMF:
电池的效率定义为输送到负载的功率与电动势产生的总功率之比：
η = P_load / P_total = I²R / (I²(R + r)) = R / (R + r)
Alternatively, using terminal voltage: η = V / ε.
或者，使用端电压表示：η = V / ε。
Thus, a battery with very small internal resistance has high efficiency because most of the energy is transferred to the external circuit. In real power systems, such as national grids, internal resistance of generators is kept extremely low to achieve efficiencies above 95%.
因此，内阻非常小的电池效率很高，因为大部分能量被传递到外部电路。在真实的电力系统中，例如国家电网，发电机的内阻被保持在极低水平，以实现超过95%的效率。
8. Short Circuit and Overload | 短路与过载
When the external resistance is reduced to zero (short circuit), the current reaches its maximum value:
当外部电阻减小到零（短路）时，电流达到最大值：
I_short = ε / r
In this situation, the terminal voltage V = 0, no power is delivered to an external load, and all energy is dissipated inside the battery, potentially causing overheating or damage. This is why batteries should never be short-circuited.
在这种情况下，端电压V = 0，没有功率输送到外部负载，所有能量都耗散在电池内部，可能导致过热或损坏。这就是为什么电池绝不能短路的缘故。
A related concept is overloading: drawing too much current from a battery (low external resistance) causes the terminal voltage to collapse and may damage the battery or connected devices.
相关的概念还有过载：从电池中提取过多电流（外部电阻过低）会导致端电压骤降，并可能损坏电池或连接的设备。
9. Experimental Measurement: Key IB Skills | 实验测量：关键IB技能
In the IB physics laboratory, you may be asked to determine ε and r experimentally. A typical setup includes a battery, a variable resistor, an ammeter, and a voltmeter. The procedure is as follows:
在IB物理实验室中，你可能会被要求通过实验测定ε和r。典型装置包括电池、滑动变阻器、电流表和电压表。步骤如下：
- Connect the circuit: battery in series with the ammeter and variable resistor; voltmeter connected across the battery terminals.
- Adjust the variable resistor to several different values and record both ammeter and voltmeter readings.
- Plot a graph of V (y-axis) against I (x-axis).
- Determine the y-intercept (ε) and the negative slope (r) from the best-fit line.
- 连接电路：电池与电流表和滑动变阻器串联；电压表并联在电池两端。
- 将滑动变阻器调到多个不同位置，记录电流表和电压表的读数。
- 绘制V（y轴）随I（x轴）变化的图像。
- 从最佳拟合线确定y轴截距（ε）和负斜率（r）。
Important experimental considerations: use a switch to avoid continuous current drain; allow the battery to rest between readings to avoid temperature changes; and use a high-resistance voltmeter and low-resistance ammeter to reduce measurement errors.
重要的实验注意事项：使用开关以避免持续耗电；每次读数之间让电池休息以防温度变化；使用高内阻电压表和低内阻电流表以减少测量误差。
10. Common IB Exam Problems | 常见IB考题类型
The following are typical problem types you may encounter in IB Paper 1 and Paper 2:
以下是在IB试卷1和试卷2中可能遇到的典型题型：
- Calculating the current in a circuit given ε, r, and R.
- Finding terminal voltage when a known current is drawn.
- Using a V-I graph to determine ε and r.
- Comparing power and efficiency for different load resistances.
- 已知ε、r和R，计算电路中的电流。
- 已知输出电流，求端电压。
- 利用V-I图像确定ε和r。
- 比较不同负载电阻下的功率和效率。
For example: A battery of EMF 12 V and internal resistance 2 Ω is connected to a 4 Ω resistor. Calculate (a) the current, (b) the terminal voltage, (c) the power dissipated in the external resistor, and (d) the efficiency.
例如：一个电动势为12 V、内阻为2 Ω的电池连接到4 Ω的电阻上。计算(a)电流，(b)端电压，(c)外部电阻消耗的功率，以及(d)效率。
(a) I = ε / (R + r) = 12 / (4 + 2) = 2 A
(b) V = ε – Ir = 12 – 2×2 = 8 V
(c) P = I²R = 2² × 4 = 16 W
(d) η = R / (R + r) = 4 / (4 + 2) = 66.7%
Notice that the power dissipated inside the battery is I²r = 8 W, and total power produced is εI = 24 W, confirming 16 + 8 = 24 W.
注意电池内部消耗的功率为I²r = 8 W，总功率为εI = 24 W，验证16 + 8 = 24 W。
11. Distinction between EMF and Potential Difference | 电动势与电势差的区别
Many students confuse EMF with terminal potential difference. The key distinctions are:
许多学生容易混淆电动势与端电势差。关键区别如下：
| EMF (ε) | Terminal Voltage (V) |
| Total energy supplied per unit charge | Energy delivered to the external circuit per unit charge |
| Independent of current | Decreases with increasing current |
| Measured with no external circuit (open circuit) | Measured while current is flowing |
| Always greater than V for a discharging battery | Always less than ε for a discharging battery |
EMF can be thought of as the “ideal” voltage, while terminal voltage is the “real” voltage available to the external circuit.
电动势可以被理解为“理想”电压，而端电压是外部电路实际可用的“真实”电压。
12. Summary and Key Equations | 总结与关键公式
To master EMF and internal resistance, keep the following equations and concepts at the forefront:
要掌握电动势和内阻，请牢记以下公式和概念：
ε = I(R + r)
V = ε – Ir
P_max = ε² / (4r) when R = r
η = R / (R + r) = V / ε
In IB exams, always check whether the question asks for EMF or terminal voltage. Remember: if current is zero, terminal voltage equals EMF. If current flows, terminal voltage is reduced by the lost volts Ir.
在IB考试中，务必检查题目问的是电动势还是端电压。记住：如果电流为零，端电压等于电动势；如果有电流流动，端电压会因损失电压Ir而降低。
Finally, practice drawing and interpreting V-I graphs, as this is one of the most frequently assessed skills in the IB physics syllabus. Understanding internal resistance not only helps you score marks but also explains phenomena like why phone batteries drain faster when the device is running demanding applications — higher current means greater lost volts and more heating.
最后，练习绘制和解读V-I图像，这是IB物理大纲中最常考查的技能之一。理解内阻不仅帮助你得分，还能解释为什么手机在运行高耗能应用时电池掉电更快——更大的电流意味着更大的损失电压和更多的发热。
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