Resistance in IB WJEC Physics | IB WJEC 物理：电阻考点精讲

📚 Resistance in IB WJEC Physics | IB WJEC 物理：电阻考点精讲

Understanding resistance is fundamental to mastering electric circuits in IB and WJEC Physics. This article breaks down key concepts, formulas, and practical insights to help you excel in your exams.

理解电阻是掌握 IB 和 WJEC 物理电路部分的基础。本文拆解核心概念、公式与实践要点，助你备考无忧。

1. Defining Resistance | 电阻的定义

Resistance (R) is a measure of the opposition to the flow of electric current. It is defined as the ratio of potential difference (V) across a conductor to the current (I) flowing through it: R = V / I. The SI unit of resistance is the ohm (Ω), where 1 Ω = 1 V A⁻¹.

电阻 (R) 衡量对电流流动的阻碍作用。它被定义为导体两端电势差 (V) 与流过电流 (I) 的比值：R = V / I。电阻的国际单位是欧姆 (Ω)，1 Ω = 1 V A⁻¹。

This definition holds for ohmic materials where R remains constant under constant physical conditions. However, for non-ohmic components like filament lamps or diodes, the ratio V/I is not constant and resistance depends on the applied voltage.

这一定义适用于欧姆材料，即物理条件不变时 R 保持恒定。但对于灯丝灯泡或二极管等非欧姆元件，V/I 比值并不恒定，电阻随所加电压变化。

2. Ohm’s Law in Detail | 欧姆定律详解

Ohm’s law states that the current through a metallic conductor at constant temperature is directly proportional to the potential difference across its ends. The I–V graph for an ohmic conductor is a straight line through the origin, indicating constant resistance. The slope gives the conductance (1/R).

欧姆定律指出，恒温下通过金属导体的电流与其两端电势差成正比。欧姆导体的 I–V 曲线是过原点的直线，表明电阻恒定。斜率给出电导 (1/R)。

It is crucial to remember that Ohm’s law is a special case, not a universal law. Semiconductors, electrolytes, and gases often show non-linear behaviour. WJEC and IB exam questions frequently ask you to distinguish ohmic from non-ohmic behaviour using I–V characteristics.

必须牢记，欧姆定律是一个特例而非普适定律。半导体、电解液和气体常表现出非线性行为。WJEC 和 IB 考题常要求利用 I–V 特性曲线区分欧姆与非欧姆行为。

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

Resistance depends on the material’s intrinsic property called resistivity (ρ). The relationship is R = ρL / A, where L is the length and A is the cross-sectional area. Resistivity has units of Ω m. A low ρ means the material easily allows charge flow.

电阻取决于材料的内禀属性——电阻率 (ρ)。关系式为 R = ρL / A，其中 L 为长度，A 为横截面积。电阻率单位为 Ω m。低 ρ 意味着材料容易让电荷通过。

Conductivity (σ) is the reciprocal of resistivity: σ = 1/ρ. It is measured in S m⁻¹ (siemens per metre). In IB Data Booklet and WJEC formula sheets, you will see both quantities. Pay attention to conversions: 1 Ω m = 1 m / S.

电导率 (σ) 是电阻率的倒数：σ = 1/ρ。单位为 S m⁻¹ (西门子每米)。在 IB 数据手册和 WJEC 公式表中你会看到这两个量。注意换算：1 Ω m = 1 m / S。

Material	Resistivity (Ω m) at 20°C
Silver	1.59 × 10⁻⁸
Copper	1.68 × 10⁻⁸
Graphite	(3−60) × 10⁻⁵
Glass	10¹⁰ − 10¹⁴

4. Factors Affecting Resistance | 影响电阻的因素

Resistance is influenced by four primary factors:

电阻受四个主要因素影响：

Length (L): R ∝ L. Doubling the wire length doubles its resistance (assuming constant area and temperature).
长度 (L)： R ∝ L。导线长度加倍，电阻加倍（假设面积和温度不变）。
Cross-sectional Area (A): R ∝ 1/A. A thicker wire has less resistance. Doubling the area halves the resistance.
横截面积 (A)： R ∝ 1/A。较粗导线电阻较小。面积加倍，电阻减半。
Material (ρ): Different materials have different resistivities due to number density of free electrons and crystal structure.
材料 (ρ)： 不同材料因自由电子数密度和晶体结构不同而有不同电阻率。
Temperature: For metals, resistance increases with temperature (positive temperature coefficient). For semiconductors and insulators, resistance usually decreases.
温度： 对金属而言，电阻随温度升高而增大（正温度系数）。半导体和绝缘体的电阻通常减小。

5. Temperature Dependence and the Resistor Model | 温度依赖性与电阻模型

In metals, as temperature rises, lattice ions vibrate more vigorously, increasing the frequency of collisions with drifting electrons. This reduces the mean free time between collisions, thus increasing resistivity. The approximate linear relation is ρ_T = ρ₀[1 + α(T − T₀)], where α is the temperature coefficient of resistivity (for copper α ≈ 3.9×10⁻³ K⁻¹).

金属中，温度升高时晶格离子振动加剧，与漂移电子的碰撞频率增加。这缩短了碰撞间平均自由时间，从而使电阻率增大。近似线性关系为 ρ_T = ρ₀[1 + α(T − T₀)]，其中 α 为电阻温度系数（铜的 α ≈ 3.9×10⁻³ K⁻¹）。

In pure semiconductors, thermal agitation releases more charge carriers (electrons and holes), so resistance drops dramatically with temperature. Thermistors exploit this negative temperature coefficient (NTC) for temperature sensing.

纯半导体中，热激发释放更多载流子（电子和空穴），因此电阻随温度显著下降。热敏电阻利用这种负温度系数 (NTC) 进行温度传感。

Superconductivity is a state where certain materials below a critical temperature (T_c) have exactly zero resistivity. In WJEC, you study the properties and applications, while IB includes BCS theory and Meissner effect. High-T_c superconductors above 77 K use liquid nitrogen cooling.

超导是某些材料在临界温度 (T_c) 以下电阻率完全为零的状态。WJEC 学习中会涉及性质和应用，而 IB 包括 BCS 理论和迈斯纳效应。高于 77 K 的高温超导体使用液氮冷却。

6. I–V Characteristics of Components | 元件的 I–V 特性曲线

Exam boards require sketching and interpreting current–voltage graphs for:

考试要求绘制和解读以下元件的电流-电压图：

Ohmic resistor: Straight line through origin; constant resistance.
欧姆电阻器： 过原点直线；电阻恒定。
Filament lamp: Curve bending towards voltage axis; resistance increases due to heating (PTC).
灯丝灯泡： 弯向电压轴的曲线；因发热电阻增大 (PTC)。
Silicon diode: Negligible current in reverse bias; sharp increase in forward bias after threshold voltage (~0.7 V).
硅二极管： 反向偏压下电流极小；正向偏压超过阈值电压 (~0.7 V) 后电流激增。

WJEC practical assessments often involve plotting such graphs from experimental data. IB requires analysis of gradient to find resistance at specific points (tangent method for non-linear).

WJEC 实验评估常要求根据实验数据绘制这些图。IB 要求通过斜率分析某一点的电阻（非线性曲线的切线法）。

7. Resistors in Circuits: Series and Parallel | 电路中的电阻：串联与并联

For resistors in series: Equivalent resistance R_total = R₁ + R₂ + R₃ + … The current is the same through all resistors, and the total p.d. is the sum of the individual p.d.s.

串联电阻：等效电阻 R_total = R₁ + R₂ + R₃ + … 。通过各电阻的电流相同，总电势差为各电势差之和。

For resistors in parallel: The reciprocal of the total resistance equals the sum of the reciprocals: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + … . The p.d. across each branch is the same, but the total current splits.

并联电阻：总电阻的倒数等于各倒数之和：1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + … 。各支路两端电势差相同，总电流分流。

Common mistakes include applying series formula to parallel circuits and forgetting that for two parallel resistors, product over sum works only for two: R_total = (R₁R₂)/(R₁+R₂). For more than two, use reciprocal method.

常见错误有将串联公式用于并联电路，以及忘记两个电阻并联时可用积/和：R_total = (R₁R₂)/(R₁+R₂)。多于两个时必须使用倒数法。

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

A real power source (battery or cell) has internal resistance (r). The terminal p.d. V = ε − Ir, where ε is the electromotive force (emf). When no current flows (open circuit), V = ε. Under load, the lost volts (Ir) reduce the terminal p.d.

实际电源（电池）具有内阻 (r)。端电压 V = ε − Ir，其中 ε 为电动势 (emf)。无电流时（开路），V = ε。带负载时，内压降 (Ir) 使端电压降低。

The maximum power transfer theorem states that power delivered to an external load is maximum when load resistance equals internal resistance (R = r). This is derived in IB from P = I²R with I = ε/(R+r).

最大功率传输定理表明，负载电阻等于内阻 (R = r) 时，输送到外负载的功率最大。IB 中利用 P = I²R 和 I = ε/(R+r) 推导该结论。

Measuring internal resistance: Use variable resistor, measure V and I pairs. Plot V (y-axis) vs I (x-axis); straight line with gradient = −r and y-intercept = ε.

测量内阻：使用可变电阻器，测量 V 和 I 数据对。绘制 V (纵轴) 对 I (横轴) 图像；直线斜率为 −r，纵截距为 ε。

9. Electrical Power and Heating Effect | 电功率与热效应

The power dissipated in a resistor is P = IV = I²R = V²/R. The energy dissipated is E = Pt, often measured in joules or kilowatt-hours. Resistance heating is used in kettles, toasters, and electric heaters. Unwanted heating causes energy loss in transmission lines.

电阻器耗散的功率为 P = IV = I²R = V²/R。耗散能量 E = Pt，常用焦耳或千瓦时计量。电阻加热用于水壶、烤面包机和电热器。不必要的发热导致输电线路能量损耗。

Joule’s law (also known as Joule heating) quantitatively describes this: heat produced per second = I²R. In IB, you may need to combine this with calorimetry (mcΔθ) to find specific heat capacity or energy transfer.

焦耳定律定量描述这一过程：每秒产生的热量 = I²R。在 IB 中，你可能需要结合量热学 (mcΔθ) 求比热容或能量传递。

WJEC requires calculations of efficiency when electrical energy is converted to other forms, e.g., E_output / E_input × 100%.

WJEC 要求计算电能转化为其他形式能量时的效率，如 E_output / E_input × 100%。

10. Potential Dividers and Sensing Circuits | 分压器与传感电路

A potential divider uses two resistors in series to produce a fraction of the input voltage. V_out = V_in × [R₂/(R₁+R₂)]. This principle is widely used with sensors (LDRs, thermistors) to create circuits that respond to light or temperature changes.

分压器利用两个串联电阻产生输入电压的一部分。V_out = V_in × [R₂/(R₁+R₂)]。这一原理广泛应用于传感器（光敏电阻、热敏电阻）电路，以响应光照或温度变化。

If R₁ is a fixed resistor and R₂ an LDR, V_out increases when light intensity decreases (LDR resistance goes up). Replacing R₁ with an LDR gives the opposite behaviour. WJEC exams include designing such circuits and predicting V_out.

若 R₁ 是固定电阻，R₂ 是光敏电阻，则光照强度降低时（LDR 电阻增大）V_out 升高。把 R₁ 换成 LDR 则得到相反行为。WJEC 考试包括设计此类电路并预测 V_out。

IB extends this to bridge circuits like the Wheatstone bridge for accurate resistance measurement. When the bridge is balanced, R₁/R₂ = R₃/Rₓ, allowing calculation of unknown Rₓ.

IB 将之扩展到惠斯通电桥等桥式电路，用于精密电阻测量。电桥平衡时，R₁/R₂ = R₃/Rₓ，可计算未知电阻 Rₓ。

11. Experimental Determination of Resistance | 电阻的实验测定

Standard method: Voltmeter-ammeter method. Connect voltmeter in parallel with the component and ammeter in series. Vary the supply voltage (or use a variable resistor) and record multiple I–V pairs. Plot V vs I (or I vs V) and determine resistance from the graph.

标准方法：伏安法。将电压表并联在元件两端，电流表串联。改变电源电压（或用可变电阻器），记录多组 I–V 数据。绘制 V–I 或 I–V 图像，从图中求出电阻。

For low resistance values, use a four-point (Kelvin) probe method to eliminate contact resistance and lead resistance. WJEC may discuss simple circuits; IB includes the use of a potentiometer to measure emf without drawing current.

对于低电阻值，使用四点（开尔文）探针法消除接触电阻和引线电阻。WJEC 会讨论简单电路；IB 包括使用电位差计在无电流情况下测量电动势。

Uncertainty analysis is essential: combine percentage uncertainties from voltage and current readings to find uncertainty in R. IB requires rigorous uncertainty calculations using ΔR/R = ΔV/V + ΔI/I for division.

不确定度分析必不可少：合并电压和电流读数的百分不确定度以求得 R 的不确定度。IB 要求严格的除法不确定度计算：ΔR/R = ΔV/V + ΔI/I。

12. Superconductivity and Modern Applications | 超导与现代应用

When certain materials are cooled below their critical temperature T_c, they undergo a phase transition where electrical resistance drops exactly to zero. Persistent currents can flow indefinitely without energy loss. Superconducting magnets generate intense magnetic fields for MRI scanners and particle accelerators (e.g., LHC).

当某些材料冷却至临界温度 T_c 以下时，发生相变，电阻完全降至零。持续电流可无限流动而无能量损耗。超导磁体为 MRI 扫描仪和粒子加速器（如 LHC）产生强磁场。

The Meissner effect – expulsion of magnetic flux from a superconductor – leads to magnetic levitation, which is a key application in maglev trains. Both IB and WJEC highlight the environmental and technological potential of superconductivity, while acknowledging challenges like cryogenic cooling.

迈斯纳效应——超导体排出磁通——导致了磁悬浮，这是磁悬浮列车的关键应用。IB 和 WJEC 均强调超导的环境与技术潜力，同时承认低温冷却等挑战。

Understanding resistance thus stretches from microscopic electron scattering to quantum coherent phenomena, reinforcing the depth of physics in the IB and WJEC syllabus.

对电阻的理解因此从微观电子散射延伸到量子相干现象，深化了 IB 和 WJEC 课程中物理的深度。

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Resistance in IB WJEC Physics | IB WJEC 物理：电阻 考点精讲