A-Level CIE Physics: Resistance – Key Points | A-Level CIE 物理：电阻考点精讲

📚 A-Level CIE Physics: Resistance – Key Points | A-Level CIE 物理：电阻考点精讲

Resistance is a fundamental concept in CIE A-Level Physics, appearing in paper 2 theory, paper 3 practical skills, and paper 4 structured questions. This article consolidates everything you need to know for the exam: from Ohm’s law and resistivity to potential dividers and internal resistance, with clear explanations, typical I-V graphs, and key formulas expressed in accessible notation.

电阻是 CIE A-Level 物理中的一个基础概念，出现在卷二理论、卷三实验技能和卷四结构化问题中。本文整合了考试所需的全部内容：从欧姆定律和电阻率到分压器和内阻，配以清晰的解释、典型的 I-V 图，以及用通俗符号表示的关键公式。

1. Defining Resistance and Ohm’s Law | 电阻的定义与欧姆定律

Resistance (R) is the ratio of potential difference (V) across a component to the current (I) flowing through it. The defining equation is R = V/I, and the SI unit is the ohm (Ω). A component obeys Ohm’s law if the current through it is directly proportional to the potential difference, provided physical conditions (such as temperature) remain constant. Ohm’s law is a special property, not a universal law for all materials.

电阻 (R) 是元件两端的电势差 (V) 与流过它的电流 (I) 之比。定义式为 R = V/I，国际单位是欧姆 (Ω)。如果通过一个元件的电流与电势差成正比，且物理条件（如温度）保持不变，则该元件遵守欧姆定律。欧姆定律是一种特殊性质，并非所有材料都遵守的普遍规律。

R = V / I

In this equation, V is measured in volts (V), I in amperes (A), and R in ohms (Ω). When a graph of V against I is a straight line through the origin, the component is ohmic; the gradient of the line equals its resistance.

在此式中，V 的单位是伏特 (V)，I 的单位是安培 (A)，R 的单位是欧姆 (Ω)。当 V 对 I 的图像是一条过原点的直线时，该元件为欧姆元件；直线的斜率等于其电阻值。

2. Resistivity and Conductivity | 电阻率与电导率

The resistance of a uniform wire depends on its length L, cross-sectional area A, and the material’s resistivity ρ. The relationship is given by R = ρL/A. Resistivity is a material property measured in ohm-metres (Ω m), and it varies with temperature. Conductivity σ is the reciprocal of resistivity: σ = 1/ρ, with units of siemens per metre (S m⁻¹).

一段均匀导线的电阻取决于其长度 L、横截面积 A 以及材料的电阻率 ρ。关系式为 R = ρL/A。电阻率是材料属性，单位为欧姆·米 (Ω m)，并随温度变化。电导率 σ 是电阻率的倒数：σ = 1/ρ，单位为西门子每米 (S m⁻¹)。

R = ρ × L / A

For a given material, longer wires have higher resistance, while thicker wires (larger A) have lower resistance. Resistivity explains why metals are good conductors (low ρ), and insulators have extremely high ρ.

对于给定材料，导线越长电阻越大，而导线越粗（A 越大）电阻越小。电阻率解释了为什么金属是良导体（ρ 低），而绝缘体具有极高的 ρ。

3. Temperature Dependence of Resistance | 电阻的温度依赖性

For most metallic conductors, resistance increases with temperature. The approximate linear relation is R = R₀[1 + α(T − T₀)], where R₀ is the resistance at a reference temperature T₀, and α is the temperature coefficient of resistance (unit: K⁻¹). For pure metals, α is positive and of order 10⁻³ K⁻¹. For thermistors (semiconductors), resistance typically decreases as temperature rises, giving a negative temperature coefficient.

对于大多数金属导体，电阻随温度升高而增大。近似的线性关系为 R = R₀[1 + α(T − T₀)]，其中 R₀ 是参考温度 T₀ 时的电阻，α 是电阻温度系数（单位：K⁻¹）。对于纯金属，α 为正，数量级为 10⁻³ K⁻¹。对于热敏电阻（半导体），电阻通常随温度升高而减小，因此温度系数为负。

R = R₀ [1 + α (T − T₀)]

This temperature dependence is crucial in questions involving the calibration of resistance thermometers or the explanation of the filament lamp’s I-V curve. You are expected to explain the change in resistance using the lattice vibration model: higher temperature causes more vigorous ion vibrations, impeding the drift of free electrons.

这种温度依赖性在涉及电阻温度计校准或解释灯丝灯 I-V 曲线的问题中至关重要。你需要用晶格振动模型解释电阻的变化：温度升高导致离子振动更剧烈，阻碍了自由电子的漂移。

4. Superconductivity | 超导电性

Superconductivity is a phenomenon where certain materials have exactly zero electrical resistivity below a critical temperature (Tc). The transition is abrupt. Once a current is established in a superconducting loop, it persists indefinitely without any applied potential difference. Superconductors have applications in powerful electromagnets (e.g. MRI scanners) and low-loss power transmission, but they require cooling with liquid nitrogen or helium, which is energy-intensive.

超导电性是指某些材料在临界温度 (Tc) 以下电阻率恰好为零的现象。该转变是突变的。一旦在超导回路中建立起电流，电流就会在没有外加电势差的情况下持久存在。超导体可用于强电磁铁（如核磁共振成像扫描仪）和低损耗电力传输，但需要用液氮或液氦冷却，这需要消耗大量能源。

CIE questions often ask you to interpret a resistance-temperature graph, showing a sudden drop to zero at Tc. Note that superconductivity is not simply extremely low resistance, but exactly zero resistance.

CIE 考题经常要求你解读电阻-温度图，图中显示在 Tc 处电阻突然降至零。请注意，超导电性不是极低的电阻，而是恰好为零的电阻。

5. I-V Characteristics of Ohmic and Non-Ohmic Components | 欧姆和非欧姆元件的 I-V 特性

The current-voltage relationship reveals whether a component is ohmic. An ohmic conductor (e.g. a metal wire at constant temperature) gives a straight line through the origin. Non-ohmic components include filament lamps, diodes, and thermistors. You must be able to sketch and interpret their I-V graphs, explaining the shape using resistance change arguments.

电流-电压关系揭示了元件是否为欧姆元件。欧姆导体（例如恒温下的金属导线）的图像是一条过原点的直线。非欧姆元件包括灯丝灯、二极管和热敏电阻。你必须能够绘制并解读它们的 I-V 图，并用电阻变化来解释曲线的形状。

Filament lamp: As current increases, the filament heats up, the metal ions vibrate more, increasing resistance. The I-V graph curves towards the V axis (gradient = 1/R decreases).
灯丝灯：随着电流增大，灯丝发热，金属离子振动加剧，电阻增大。I-V 图像偏向 V 轴（斜率 = 1/R 减小）。
Diode: Allows current in forward bias only (after threshold voltage ~0.6 V for silicon). Resistance is extremely high in reverse bias until breakdown. The graph is not symmetric.
二极管：仅在正向偏置（硅管阈值电压约 0.6 V）时导通。反向偏置时电阻极高，直至击穿。图像不对称。
Ohmic resistor: Straight line, constant resistance.
欧姆电阻：直线，电阻恒定。

In paper 3, you may have to plot an I-V graph from experimental data and calculate resistance at a point using R = V/I, not the gradient unless the component is ohmic.

在卷三中，你可能需要根据实验数据绘制 I-V 图，并利用 R = V/I 计算某一点的电阻，除非元件是欧姆元件，否则不能使用斜率。

6. Resistors in Series and Parallel | 电阻的串联与并联

For resistors in series, the total resistance is the sum: R_total = R₁ + R₂ + R₃ + … The same current passes through each resistor, and the potential difference divides in proportion to the resistances. For resistors in parallel, the reciprocal of the total equals the sum of reciprocals: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + … The potential difference across each branch is the same, while the current divides.

对于串联电阻，总电阻为各电阻之和：R_total = R₁ + R₂ + R₃ + … 流过每个电阻的电流相同，电势差按电阻比例分配。对于并联电阻，总电阻的倒数等于各电阻倒数之和：1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + … 每并联支路两端的电势差相同，电流则分流。

Series: R_total = R₁ + R₂ + … Parallel: 1/R_total = 1/R₁ + 1/R₂ + …

For two resistors in parallel, a shortcut formula is R_total = (R₁ × R₂) / (R₁ + R₂). Remember that the total resistance of a parallel combination is always less than the smallest individual resistance.

对于两个并联电阻，简捷公式为 R_total = (R₁ × R₂) / (R₁ + R₂)。请记住，并联组合的总电阻始终小于各个电阻中最小的一个。

7. Potential Dividers | 分压器

A potential divider consists of two or more resistors in series across a supply voltage. The output voltage V_out is taken across one of the resistors. If two resistors R₁ and R₂ are connected with V_in across the series combination, the output across R₂ is V_out = V_in × R₂ / (R₁ + R₂). This circuit can provide a variable output if one resistor is replaced by a variable resistor or a sensor such as a thermistor or LDR.

分压器由跨接在电源电压上的两个或多个串联电阻组成。输出电压 V_out 取自其中一个电阻的两端。若将两个电阻 R₁ 和 R₂ 串联并施以总电压 V_in，则 R₂ 两端的输出电压为 V_out = V_in × R₂ / (R₁ + R₂)。如将其中一个电阻换成可变电阻器或传感器（如热敏电阻或 LDR），此电路可提供可变的输出电压。

V_out = V_in × R₂ / (R₁ + R₂)

In an LDR-light sensor potential divider, as light intensity increases, the LDR’s resistance falls, so V_out across the LDR decreases. In a thermistor-based temperature sensor, a rise in temperature lowers the thermistor resistance, decreasing V_out across it. Make sure you can predict how V_out changes when the sensor resistance changes.

在 LDR 光传感器分压器中，随着光照强度增加，LDR 的电阻下降，因此 LDR 两端的 V_out 减小。在基于热敏电阻的温度传感器中，温度升高会使热敏电阻阻值下降，从而降低其两端的 V_out。务必确保你能预测传感器电阻变化时 V_out 如何变化。

8. Internal Resistance and Terminal PD | 内阻与端电压

All real sources of emf (ε) have internal resistance (r). When a current I flows, the terminal potential difference V is less than the emf: V = ε − I r. The lost volts are the energy dissipated inside the source per coulomb. A graph of V against I gives a straight line with y-intercept ε and gradient −r. This is a frequent experiment question in paper 3.

所有实际的电动势 (ε) 电源都具有内阻 (r)。当有电流 I 流过时，端电势差 V 小于电动势：V = ε − I r。损失的电压是每库仑电荷在电源内部耗散的能量。V 对 I 的图线是一条斜率为 −r、y 轴截距为 ε 的直线。这是卷三中常见的实验题。

V = ε − I r

The total power supplied by the source is P = ε I. The useful power delivered to the external circuit is P_useful = V I. Power wasted in the internal resistance is I² r. Efficiency = (V/ε) × 100%.

电源提供的总功率为 P = ε I。输送到外电路的有用功率为 P_useful = V I。在内阻上浪费的功率为 I² r。效率 = (V/ε) × 100%。

9. Electrical Power and Energy Dissipated | 电功率与能量耗散

The power dissipated in a resistor can be expressed in three ways: P = I V, P = I² R, and P = V² / R. These equations assume that the component is purely resistive (all electrical energy converted to heat). The unit of power is the watt (W). Energy transferred is E = P t = I V t, often expressed in joules or kilowatt-hours (1 kWh = 3.6 × 10⁶ J) for domestic contexts.

电阻器中消耗的功率可用三种方式表达：P = I V、P = I² R 和 P = V² / R。这些公式假设元件是纯电阻性的（所有电能均转化为热能）。功率的单位是瓦特 (W)。转移的能量为 E = P t = I V t，常以焦耳或千瓦时 (1 kWh = 3.6 × 10⁶ J) 表示，用于家庭用电场景。

P = I V P = I² R P = V² / R

Questions often ask you to compare the power dissipated by resistors in series and parallel. Because total resistance differs, the current and power distribution change. In a parallel circuit, the resistor with the smallest R dissipates the most power if the voltage across each branch is constant.

考题经常要求比较串联和并联电路中电阻消耗的功率。由于总电阻不同，电流和功率分配也随之改变。在并联电路中，若各支路电压恒定，电阻最小的支路消耗的功率最大。

10. Practical Applications and Exam Tips | 实际应用与考试技巧

Resistance concepts appear in many real-life contexts: use of high-resistivity materials (nichrome) in heating elements, selection of appropriate thickness for power cables to minimize power loss, and the design of sensors using potential dividers. For exam success, always convert units to base SI (m² not cm² for area), recall that the gradient of a V–I graph for an ohmic conductor gives R, but for non-ohmic components, resistance at a point is found by V/I, not the gradient.

电阻概念出现在许多实际生活中：高热阻材料（镍铬合金）用于加热元件，为减少功率损耗而选择适当粗细的电缆，以及利用分压器设计传感器。为了在考试中取得成功，一定要将单位转换为国际基本单位（面积用 m² 而非 cm²），牢记欧姆导体的 V-I 图的斜率给出 R，而对于非欧姆元件，某一点的电阻应通过 V/I 求得，而非斜率。

Always state the physical condition needed for Ohm’s law: constant temperature.
在讲述欧姆定律时务必说明物理条件：恒温。
When describing superconductor graphs, use ‘resistance drops to zero’ not ‘decreases to zero’.
在描述超导体图线时，使用“电阻降为零”而不是“减小到零”。
For potential divider calculations, identify which resistor V_out is across; the larger that resistor, the larger V_out.
进行分压器计算时，要明确 V_out 是跨接在哪个电阻上；该电阻越大，V_out 越大。
In internal resistance experiments, the graph’s gradient is negative (−r).
在内阻实验中，图线的斜率为负值 (−r)。

Use these revision notes alongside past paper questions to strengthen your understanding of resistance and gain confidence in applying the formulas. Remember to explain the physics behind temperature effects and I-V shapes—examiners look for detailed reasoning.

将这些复习笔记与历年真题结合使用，以加深你对电阻的理解，并增强运用公式的信心。记得要解释温度效应和 I-V 曲线形状背后的物理原理——考官看重的是详尽的推理过程。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

A-Level CIE Physics: Resistance – Key Points | A-Level CIE 物理：电阻 考点精讲