📚 IB WJEC Physics: Circuit Analysis Essentials | IB WJEC 物理：电路分析考点精讲
Circuit analysis forms the backbone of electricity and magnetism in both IB and WJEC A‑Level Physics. Understanding how current, voltage, and resistance interact in a network is essential for tackling both theoretical problems and practical investigations. This revision guide walks you through the core concepts, key equations, and exam techniques you need to master circuit analysis, presented in a clear bilingual format.
电路分析是 IB 和 WJEC A‑Level 物理电磁学部分的基石。理解电流、电压和电阻在电路网络中的相互作用，对于解决理论问题和实践探究都至关重要。这份复习指南将以清晰的双语形式，带你梳理必须掌握的核心概念、关键公式和应试技巧。
1. Ohm’s Law and Resistivity | 欧姆定律与电阻率
Ohm’s Law states that the current through a conductor is directly proportional to the potential difference across it, provided temperature remains constant. The relationship is expressed as V = IR, where V is the voltage (volts), I is the current (amperes), and R is the resistance (ohms). This linear behaviour characterises ohmic conductors such as metal wires at constant temperature.
欧姆定律指出，在温度保持不变的条件下，通过导体的电流与导体两端的电势差成正比。关系式为 V = IR，其中 V 表示电压（伏特），I 表示电流（安培），R 表示电阻（欧姆）。这种线性关系描述了欧姆导体（例如恒温下的金属导线）的特性。
Resistance of a wire is determined by its resistivity ρ, length L, and cross-sectional area A: R = ρL / A. Resistivity is a material property that depends on temperature; for metals, resistivity increases with temperature, which explains why resistance rises when a filament lamp heats up. In IB and WJEC exams, you may be asked to design an experiment to determine the resistivity of a wire using a voltmeter, ammeter, and a metre rule.
导线的电阻由其电阻率 ρ、长度 L 和横截面积 A 决定：R = ρL / A。电阻率是材料的固有属性，会随温度变化；对于金属，电阻率随温度升高而增大，这就解释了为什么灯丝灯泡发热后电阻会上升。在 IB 和 WJEC 考试中，你可能需要设计实验，利用电压表、电流表和米尺来测量导线的电阻率。
2. Series Circuits | 串联电路
In a series circuit, components are connected end‑to‑end, so the same current flows through each element. The total resistance Rtotal is the sum of individual resistances: Rtotal = R₁ + R₂ + … + Rₙ. The supply voltage is shared among the components; the sum of the potential differences across each resistor equals the source EMF.
在串联电路中，元件首尾相连，因此每个元件中流过的电流相同。总电阻 Rtotal 等于各电阻之和：Rtotal = R₁ + R₂ + … + Rₙ。电源电压在元件之间分配；各电阻两端电压之和等于电源电动势。
A key consequence is the potential divider principle: the voltage across a resistor in series is proportional to its resistance. For two resistors R₁ and R₂ in series, V₁ = [R₁/(R₁+R₂)] × Vsupply. This is frequently tested in sensor circuits, where a thermistor or LDR changes resistance and alters the output voltage.
一个重要的推论是分压原理：串联电路中，电阻两端的电压与其电阻值成正比。对于两个串联电阻 R₁ 和 R₂，V₁ = [R₁/(R₁+R₂)] × V电源。这一点在传感器电路中经常考察，例如热敏电阻或光敏电阻改变阻值，从而改变输出电压。
3. Parallel Circuits | 并联电路
Parallel circuits provide alternative paths for current. The voltage across each branch is the same as the source voltage. The total current supplied is the sum of the currents in each branch. The reciprocal of the total resistance is the sum of the reciprocals of the individual resistances: 1/Rtotal = 1/R₁ + 1/R₂ + … + 1/Rₙ. For two resistors in parallel, the convenient product‑over‑sum formula can be used: Rtotal = (R₁R₂)/(R₁+R₂).
并联电路为电流提供了多条路径。各支路两端的电压均等于电源电压。电源提供的总电流等于各支路电流之和。总电阻的倒数等于各电阻倒数之和：1/Rtotal = 1/R₁ + 1/R₂ + … + 1/Rₙ。对于两个并联电阻，可以使用简便的“积除以和”公式：Rtotal = (R₁R₂)/(R₁+R₂)。
Adding more resistors in parallel decreases the total resistance, which increases the total current drawn from the source. This is why household appliances are connected in parallel – each device receives the full mains voltage and can operate independently.
并联更多电阻会降低总电阻，从而增大从电源汲取的总电流。这就是家用电器采用并联连接的原因——每台设备都能获得完整的市电电压，并且可以独立工作。
4. Kirchhoff’s Laws | 基尔霍夫定律
Kirchhoff’s Current Law (KCL) states that the total current entering a junction equals the total current leaving it. This is simply a statement of charge conservation and is a powerful tool for analysing complex networks.
基尔霍夫电流定律指出，流入一个节点的总电流等于流出该节点的总电流。这本质上就是电荷守恒定律的表述，是分析复杂电路网络的强有力工具。
Kirchhoff’s Voltage Law (KVL) states that the sum of the EMFs around any closed loop equals the sum of the potential drops across the resistances in that loop. When writing loop equations, it is crucial to assign consistent current directions and to take account of the sign of each voltage change.
基尔霍夫电压定律指出，沿任意闭合回路，电动势的代数和等于该回路中电阻上电势降落的代数和。在列写回路方程时，务必规定一致的电流方向，并考虑每一步电压变化的符号。
Both IB HL and WJEC papers often present multi‑loop circuits with two sources or a mix of series and parallel components. Systematic application of Kirchhoff’s laws enables you to determine unknown currents and voltages.
IB 高水平课程和 WJEC 试卷中经常出现含有两个电源或串并联混合的多回路电路。系统性地应用基尔霍夫定律，可以帮助你求出未知的电流和电压。
5. Internal Resistance and EMF | 内阻与电动势
A real power source has electromotive force (EMF, ε) and internal resistance (r). When a current I flows, the terminal voltage V is less than the EMF: V = ε − Ir. This voltage drop occurs because some of the electrical energy is dissipated as heat inside the source.
实际电源具有电动势（ε）和内阻（r）。当有电流 I 流过时，端电压 V 会小于电动势：V = ε − Ir。这个电压降是因为部分电能以热能的形式在电源内部耗散了。
The EMF and internal resistance can be found experimentally by varying an external load and recording terminal voltage against current. A graph of V against I yields a straight line with gradient −r and y‑intercept ε. This is a classic required practical in WJEC specification and a common IA topic in IB Physics.
通过改变外部负载并记录端电压随电流的变化，可以用实验的方法求出电动势和内阻。绘制 V–I 图像可以得到一条直线，斜率为 −r，纵轴截距为 ε。这是 WJEC 考纲中的经典必做实验，也是 IB 物理内部评估（IA）中常见的选题。
6. Potential Dividers | 分压器
A potential divider is a simple circuit that uses two or more resistors to produce a fraction of the input voltage. The output voltage Vout is taken across one of the resistors. For a divider with resistors R₁ and R₂, Vout = Vin × R₂/(R₁+R₂) when the output is across R₂.
分压器是一种利用两个或多个电阻来产生输入电压一部分的简单电路。输出电压 Vout 取自其中一个电阻的两端。对于由电阻 R₁ 和 R₂ 组成的分压器，若输出取在 R₂ 两端，则 Vout = Vin × R₂/(R₁+R₂)。
Potential dividers are used extensively in sensor circuits. Replacing one fixed resistor with a thermistor creates a temperature‑sensitive voltage source; replacing it with a light‑dependent resistor (LDR) allows light intensity to be monitored. The output can then be fed into a comparator or transistor switch to trigger an alarm or activate a relay.
分压器被广泛用于传感器电路中。用一个热敏电阻替换其中一个固定电阻，就可以获得对温度敏感的电压源；换成光敏电阻（LDR）则可以监测光照强度。输出的电压信号可以输入到比较器或晶体管开关，用来触发警报或驱动继电器。
7. Electrical Power and Energy | 电功率与电能
The power P dissipated in a circuit component is given by P = IV, P = I²R, or P = V²/R. These forms are derived from Ohm’s law and are equivalent for ohmic components. Power is measured in watts (W), where 1 W = 1 J s⁻¹.
电路元件消耗的电功率 P 可用 P = IV、P = I²R 或 P = V²/R 来表示。这些公式由欧姆定律推导而来，对于欧姆元件它们是等价的。功率的单位是瓦特（W），1 W = 1 J s⁻¹。
Energy E transferred is power multiplied by time: E = Pt. The kilowatt‑hour (kWh) is a common practical unit: 1 kWh = 3.6 × 10⁶ J. Questions often involve calculating the cost of electricity or comparing the efficiency of different appliances.
消耗的电能 E 等于功率乘以时间：E = Pt。千瓦时（kWh）是常用的实际单位：1 kWh = 3.6 × 10⁶ J。考题中经常涉及计算电费或比较不同电器的效率。
Understanding power dissipation also links to heating effects and the choice of appropriate resistors. Exceeding a resistor’s power rating can cause failure, so you may need to calculate the maximum safe current or voltage.
理解功率耗散还与热效应以及选择合适电阻密切相关。超过电阻的额定功率会导致损坏，因此你可能需要计算最大安全电流或电压。
8. Ammeters and Voltmeters | 电流表与电压表
Ammeters are connected in series and must have very low internal resistance so that they do not significantly alter the current being measured. Voltmeters are connected in parallel and should have extremely high resistance to minimise the current drawn away from the circuit.
电流表串联在电路中，其内阻必须非常小，以免显著改变待测的电流。电压表并联在元件两端，其内阻应当极高，以尽量减少从电路中分走的电流。
In an ideal world, an ammeter has zero resistance and a voltmeter infinite resistance. Real instruments are close approximations, but in precise measurements you may need to account for their loading effects. This concept appears in both IB data‑analysis tasks and WJEC practical exams.
理想情况下，电流表的电阻为零，电压表的电阻为无穷大。实际的仪表可以很好地接近这一理想状态，但在精密测量中可能仍需考虑它们的负载效应。这一概念在 IB 的数据分析任务和 WJEC 的实验考试中均有涉及。
9. Circuit Analysis Strategies | 电路分析策略
When faced with a complex circuit, start by simplifying series and parallel combinations where possible. Redraw the circuit so that the connections become clearer. Label all known and unknown quantities, including assumed current directions. Apply Kirchhoff’s laws to generate a set of simultaneous equations if the circuit cannot be reduced.
面对复杂电路时，首先尽可能简化串并联组合。重新绘制电路图，使连接关系更加清晰。标出所有已知和未知量，包括假设的电流方向。如果电路无法化简，则应用基尔霍夫定律列出一组联立方程。
Consistency is vital. If an assumed current direction turns out to be negative in the solution, it simply means the current flows opposite to your initial guess. Always check that your answers obey conservation of energy and charge – that is, power in equals power out, and currents at junctions sum correctly.
一致性至关重要。如果在求解中发现某个假设的电流方向为负值，那仅表示实际电流方向与初始假设相反。务必检查你的答案是否满足能量和电荷守恒——即输入功率等于输出功率，以及节点处电流之和正确。
10. Practical Skills and Graphs | 实验技能与图像处理
Both syllabi emphasise experimental skills. You should be comfortable setting up circuits from a diagram, correctly placing ammeters and voltmeters, and using variable resistors or potential dividers to obtain a range of readings. Key graphs include I‑V characteristics for ohmic and non‑ohmic components, and V‑I plots to determine EMF and internal resistance.
两套教学大纲都强调实验技能。你应当能够根据电路图搭建实际电路，正确地放置电流表和电压表，并利用变阻器或分压器获取一系列读数。关键图像包括欧姆和非欧姆元件的 I‑V 特性曲线，以及用来测定电动势和内阻的 V‑I 图像。
Uncertainty analysis is required in IB. You must be able to calculate percentage uncertainties in multimeter readings, identify the largest source of error, and discuss improvements such as using longer wires or digital meters with higher precision. WJEC practical assessments similarly reward careful consideration of experimental technique.
IB 要求进行不确定度分析。你必须能够计算多用表读数的百分比不确定度，找出最主要的误差来源，并讨论改进措施，比如使用更长导线或更高精度的数字仪表。WJEC 的实验评价同样重视对实验技术的细致考量。
11. Safety in Circuit Investigations | 电路探究中的安全事项
Always check that your circuit is correctly wired before connecting power. Use a protective resistor in series with LEDs to prevent excessive current. When measuring internal resistance, avoid shorting the battery for long periods – this can cause overheating and damage. Keep current levels within the ratings of your components and cables.
接通电源前，务必检查电路连接是否正确。在 LED 上串联保护电阻，以防止电流过大。在测量内阻时，避免长时间短路电池——这可能导致过热和损坏。将电流水平控制在元件和导线额定值之内。
In mains circuits, never work with live equipment. Although most school experiments use low‑voltage DC supplies, building an awareness of safe practices is part of the learning outcomes.
对于市电电路，切勿操作带电设备。尽管大多数学校实验使用低压直流电源，但培养安全操作的意识也是学习目标的一部分。
12. Quick‑Fire Equations Summary | 核心公式速览
Below is a concise table of the most frequently used equations. Use it as a last‑minute reference.
下表汇总了最常用的公式，可作为考前快速回顾的参考。
| Quantity (量) | Equation (公式) |
|---|---|
| Ohm’s Law | V = IR |
| Resistivity | R = ρL / A |
| Series resistance | Rtotal = R₁ + R₂ + … |
| Parallel resistance | 1/Rtotal = 1/R₁ + 1/R₂ + … |
| Internal resistance | V = ε − Ir |
| Potential divider | Vout = Vin × R₂/(R₁+R₂) |
| Power | P = IV = I²R = V²/R |
| Energy | E = Pt |
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