📚 IB Edexcel Physics: Resistance | Exam Essentials | IB Edexcel 物理：电阻 考点精讲
Resistance is one of those topics that bridges all of electrical physics, from microscopic collisions of electrons to macroscopic circuit design. This essentials guide brings together the key ideas, definitions, equations and graph interpretations that IB and Edexcel examiners regularly test. Whether you are preparing for Paper 1 multiple‑choice or Paper 2 long‑form questions, mastering resistance will sharpen your ability to analyse any electric circuit.
电阻是连接所有电学物理学的桥梁之一，从微观的电子碰撞到宏观的电路设计，都离不开对电阻的理解。这份考点精讲汇集了 IB 和 Edexcel 考官经常考查的核心思想、定义、方程和图表解读。无论你正在准备 Paper 1 选择题还是 Paper 2 长答题，真正掌握电阻都会让你分析任何一个电路时更加从容自信。
1. What Is Resistance? | 电阻的定义
Resistance is a measure of how difficult it is for charge carriers (usually electrons) to flow through a material. For a given potential difference V across a component, the resistance R is defined by R = V ÷ I, where I is the current through it. The SI unit is the ohm (Ω). A component has a resistance of 1 ohm if a potential difference of 1 volt drives a current of 1 ampere through it.
电阻衡量的是电荷载流子（通常是电子）流过某种材料的难易程度。对于元件两端的给定电势差 V，电阻 R 定义为 R = V ÷ I，其中 I 是流经它的电流。国际单位是欧姆（Ω）。若 1 伏的电势差能驱动 1 安的电流，则该元件的电阻为 1 欧姆。
It is helpful to think of resistance in a hydraulic analogy: a narrow pipe offers strong resistance to water flow, just as a thin or resistive wire restricts charge flow. This definition is always true, even when the component does not obey Ohm’s law. Resistance tells us about the ratio V/I at any point, not necessarily a constant ratio.
我们可以借助水流类比来理解电阻：狭窄的管道对水流产生强大的阻碍，就像细导线或高阻材料限制电荷流动一样。这一定义始终成立，即使元件不遵循欧姆定律也是如此。电阻告诉我们的是电压与电流的比值 V/I，而不一定是固定不变的比值。
2. Ohm’s Law | 欧姆定律
Ohm’s law states that, for a metallic conductor at constant temperature, the current through it is directly proportional to the potential difference across it. This is expressed as V = I × R, where R is constant. A component that follows this rule is called an ohmic conductor; a fixed resistor is a classic example. In an I–V graph, an ohmic conductor gives a straight line passing through the origin, with slope 1/R.
欧姆定律指出，对于温度保持恒定的金属导体，流经它的电流与它两端的电势差成正比。由此得出 V = I × R，其中 R 为常数。遵循这一规律的元件称为欧姆导体；固定电阻器就是一个典型例子。在 I–V 图上，欧姆导体表现为过原点的直线，斜率为 1/R。
Bear in mind that Ohm’s law is a special case, not a universal law. Many components – filament lamps, diodes, thermistors – are non‑ohmic. For these, V/I still defines the resistance at any moment, but the ratio changes with current or temperature. In exam questions, always check whether you are told to assume constant temperature.
请记住，欧姆定律是一个特例，而非普适定律。许多元件——灯丝、二极管、热敏电阻——都是非欧姆性的。对它们而言，V/I 仍然定义着某一时刻的电阻，但这一比值会随电流或温度变化。在考试题中，一定要留意是否要求你假设温度恒定。
3. Resistivity – A Material Property | 电阻率——材料的本性
While resistance tells us about a specific piece of wire, resistivity tells us about the material itself. Resistivity ρ is defined by R = ρ × (L ÷ A), where L is the length of the conductor and A is its cross‑sectional area. The unit of resistivity is the ohm‑metre (Ω·m). A material with low resistivity, like copper, is a good conductor; one with high resistivity, like nichrome, is often used in heating elements.
电阻描述的是某一根导线的性质，而电阻率描述的是材料本身。电阻率 ρ 由 R = ρ × (L ÷ A) 定义，其中 L 为导体的长度，A 为其横截面积。电阻率的单位是欧姆·米（Ω·m）。电阻率低的材料（如铜）是良导体；电阻率高的材料（如镍铬合金）常用于发热元件。
Experiments to determine resistivity typically involve measuring the resistance of a wire of known diameter for several lengths, plotting R against L, and using the gradient = ρ/A. Don’t forget that the cross‑sectional area is calculated from A = πd²/4 when you measure the diameter d with a micrometer. Always use the same wire under the same temperature to keep ρ constant.
测定电阻率的实验通常包括：测量已知直径的导线在不同长度下的电阻，绘制 R 对 L 的图线，并利用斜率 = ρ/A 来求得。别忘了，当你用千分尺测得直径 d 后，要用 A = πd²/4 计算横截面积。务必使用同一根导线并在相同温度下测量，以保证 ρ 不变。
4. Temperature Dependence of Resistance | 电阻的温度依赖性
In metallic conductors, resistance increases with temperature. As the temperature rises, the positive metal ions vibrate more intensely, increasing the frequency of collisions with free electrons and thus reducing the drift velocity. This is why a filament lamp’s resistance rises dramatically as it heats up, producing the curved I–V characteristic typical of the component.
在金属导体中，电阻随温度升高而增大。温度升高时，带正电的金属离子振动加剧，增加了与自由电子碰撞的频率，从而降低了漂移速度。这就是为什么灯丝的电阻在加热过程中急剧上升，从而形成该元件特有的弯曲 I–V 特性曲线。
In contrast, semiconductor materials such as thermistors usually show a decrease in resistance with rising temperature. More charge carriers are liberated as thermal energy breaks covalent bonds. IB and Edexcel specifications often ask you to describe the behaviour of NTC (negative temperature coefficient) thermistors and their use in temperature‑sensing circuits.
相比之下，半导体材料（如热敏电阻）的电阻通常随温度升高而减小。热能使共价键断裂，从而释放出更多的电荷载流子。IB 和 Edexcel 大纲常常要求你描述 NTC（负温度系数）热敏电阻的特性及其在温度传感电路中的应用。
5. I–V Characteristics of Common Components | 常见元件的 I–V 特性
Being able to sketch, recognise and explain I–V graphs is a core exam skill. The main ones are:
能够绘制、识别并解释 I–V 图线是一项核心考试技能。主要曲线如下：
- Fixed resistor (ohmic): straight line through the origin; slope = 1/R.
固定电阻（欧姆导体）：过原点的直线；斜率 = 1/R。 - Filament lamp: curve with decreasing slope as V increases, because resistance rises with temperature.
灯丝：随着 V 增大，斜率越来越小的曲线，因为电阻随温度升高而增大。 - Diode: negligible current for reverse bias (V < 0) and forward bias only above a threshold voltage (≈0.7 V for silicon); thereafter current rises steeply.
二极管：反向偏压（V < 0）下电流可忽略不计，正向偏压只有超过阈值电压（硅管约 0.7 V）后才开始导通；此后电流急剧上升。
In practical exams, you may be asked to wire up a circuit to collect data for these graphs. Always include a protective resistor in series with the diode to prevent excessive current once it becomes forward‑biased.
在实验考试中，你可能会被要求连接电路来采集这些图线的数据。务必在二极管支路中串联一个保护电阻，以防止其正向导通后出现过大的电流。
6. Resistors in Series and Parallel | 串联与并联电阻
Combining resistors correctly is essential and regularly appears in mixed circuit problems.
正确组合电阻是必不可少的技能，常在混联电路题中出现。
| Configuration 连接方式 | Rule 规律 | Key point 要点 |
|---|---|---|
| Series 串联 | Rtotal = R₁ + R₂ + R₃ … | Same current through all. Total p.d. divides in proportion to resistance. |
| Parallel 并联 | 1/Rtotal = 1/R₁ + 1/R₂ + 1/R₃ … | Same p.d. across each branch. Total current divides; total resistance is always less than the smallest individual resistance. |
For only two resistors in parallel, you can use the product‑over‑sum shortcut: Rtotal = (R₁ × R₂) ÷ (R₁ + R₂). Always double‑check your arithmetic – parallel combinations often yield decimals.
对于只有两个电阻并联的情况，可以使用“乘积除以和”的便捷公式：Rtotal = (R₁ × R₂) ÷ (R₁ + R₂)。要始终仔细核对计算——并联组合经常得出小数。
7. Potential Dividers | 分压器
A potential divider is simply two resistors in series across a voltage supply. The output voltage Vout is taken across one of the resistors. The fundamental equation is Vout = Vin × (R₂ ÷ (R₁ + R₂)), where R₂ is the resistor across which the output is measured. This circuit is widely used to supply a variable voltage from a fixed supply.
分压器就是串联在电源上的两个电阻，输出电压 Vout 取自其中一个电阻的两端。基本方程为 Vout = Vin × (R₂ ÷ (R₁ + R₂))，其中 R₂ 是你测量输出电压的那个电阻。这个电路广泛用于从固定电源中获得可调电压。
When one resistor is replaced by a sensor (LDR or thermistor), the output voltage changes with physical conditions. For instance, a thermistor in the R₂ position with a rising temperature (resistance falling) gives a decreasing Vout. Exams often ask you to explain why the output changes in a certain direction and to design a circuit that switches on a heater when it gets cold.
当其中一个电阻被传感器（LDR 或热敏电阻）替代时，输出电压会随物理条件变化。例如，热敏电阻放在 R₂ 位置，温度升高（电阻减小）会导致 Vout 减小。考试常要求你解释输出电压为什么朝某个方向变化，并设计一个在天冷时接通加热器的电路。
8. Internal Resistance and Terminal p.d. | 内阻与端电压
Every real battery or power supply possesses internal resistance r. When current I flows, the source loses some voltage internally: the terminal potential difference V = ε − I × r, where ε is the electromotive force (emf) of the source. This explains why a battery’s measured voltage drops under load.
每一个真实的电池或电源都具有内阻 r。当有电流 I 流过时，电源在内部损失一部分电压：端电压 V = ε − I × r，其中 ε 是电源的电动势（emf）。这就解释了为什么电池在带负载时测量到的电压会下降。
The classic experiment to find ε and r uses a variable resistor to vary the current, while recording terminal p.d. V and I. Plotting V against I gives a straight line of gradient −r and y‑intercept ε. In IB and Edexcel data‑analysis questions, you might be given a table of values and asked to determine r and ε graphically.
测定 ε 和 r 的经典实验使用可变电阻来改变电流，同时记录端电压 V 和 I。绘制 V 对 I 的图得到一条直线，其斜率为 −r，y 轴截距为 ε。在 IB 和 Edexcel 的数据分析题中，你可能会得到一组数据表，并被要求通过作图求出 r 和 ε。
9. Electrical Energy and Power | 电能与电功率
Power dissipated in a resistor can be written in three equivalent forms: P = V × I, P = I² × R, and P = V² ÷ R. The choice depends on which quantities are known. The unit of power is the watt (W). Energy transferred to heat is simply power multiplied by time: E = P × t.
电阻上消耗的功率可以用三种等效形式表示：P = V × I、P = I² × R 和 P = V² ÷ R。选择哪一种取决于已知哪些量。功率的单位是瓦特（W）。转化为热量的能量就是功率乘以时间：E = P × t。
When analysing circuits with internal resistance, the useful power delivered to the external load is maximised when the load resistance equals the internal resistance (maximum power theorem). Although this theorem is more qualitative in IB and Edexcel syllabuses, understanding it helps to explain the power curve shape in output‑versus‑load graphs.
在分析带内阻的电路时，当负载电阻等于内阻时，输送给外部负载的有用功率最大（最大功率定理）。尽管在 IB 和 Edexcel 大纲中这个定理更多是定性的，但理解它有助于解释输出功率随负载变化的曲线形状。
10. Practical Measurement and Common Pitfalls | 实验测量与常见误区
Measuring resistance accurately requires careful choice of meters. An ideal voltmeter has infinite resistance and is placed in parallel; an ideal ammeter has zero resistance and is placed in series. In reality, meters have finite resistances, and their placement can affect readings. For low‑resistance components, use the ‘voltmeter‑across‑component’ setup to avoid adding ammeter resistance in series.
准确测量电阻需要谨慎选择电表。理想电压表具有无穷大的电阻，应并联连接；理想电流表电阻为零，应串联连接。实际电表存在有限内阻，其连接位置会影响读数。对于低电阻元件，应采用“电压表直接跨接在元件上”的接法，以避免在回路中额外串入电流表的内阻。
Common exam errors include forgetting to convert milliamperes to amperes, confusing gradient and intercept, or misapplying the parallel formula. Also, when explaining the effect of temperature on resistance, always connect back to the atomic‑scale picture – more vigorous lattice vibrations for metals, more free carriers for semiconductors.
考试中常见的错误包括：忘记将毫安换算为安培、混淆斜率和截距、或误用并联公式。此外，在解释温度对电阻的影响时，永远要回到原子层面的图景——金属中晶格振动加剧，半导体中自由载流子增多。
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