📚 A-Level Physics: Circuit Analysis Key Points | A-Level 物理：电路分析 考点精讲
Circuit analysis is the backbone of electricity in A-Level Physics. It is built upon a few fundamental laws and principles that allow us to predict how currents and voltages behave in any network of components. Mastering these concepts will not only help you solve exam problems with confidence but also develop a physicist’s intuition for conservation of energy and charge. This revision guide walks you through the essential topics, from Ohm’s law and Kirchhoff’s rules to potential dividers and internal resistance, all explained step by step.
电路分析是 A-Level 物理电学部分的基石。掌握几个基本定律和原理，就能预测电流和电压在任何元件网络中的行为。透彻理解这些概念，不仅能让你在考试中轻松解题，还能培养基于能量守恒和电荷守恒的物理直觉。这份复习指南将带你逐层梳理欧姆定律、基尔霍夫定律、分压器、内阻等核心考点，每一步都配合清晰解释。
1. Charge, Current and Potential Difference | 电荷、电流与电势差
Electric current is the rate of flow of charge. In a metallic conductor, it is carried by free electrons moving in the opposite direction to conventional current. The unit of current is the ampere (A), where 1 A = 1 C s⁻¹. Potential difference (p.d.) between two points is the energy transferred per unit charge. It is measured in volts (V); 1 V means 1 J of energy is transferred for every coulomb of charge that passes.
电流是电荷流动的速率。在金属导体中，载流子是自由电子，其运动方向与常规电流方向相反。电流的单位是安培 (A)，1 A = 1 C s⁻¹。两点之间的电势差 (p.d.) 是单位电荷转移的能量，单位是伏特 (V)；1 V 意味着每通过 1 库仑电荷转移 1 J 的能量。
The relationship between energy, charge and p.d. is W = QV. Alongside the definition of current I = ΔQ/Δt, these equations form the starting point for all circuit calculations. Remember that potential difference is often called voltage, and it is measured across a component, while current is measured through it.
能量、电荷与电势差的关系为 W = QV。结合电流的定义 I = ΔQ/Δt，这些方程是所有电路计算的起点。注意电势差常被称为电压，测量时必须跨接在元件两端，而电流则是测量通过元件的量。
2. Ohm’s Law and Resistance | 欧姆定律与电阻
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. The constant of proportionality is resistance: R = V / I. The unit of resistance is the ohm (Ω). Many components, however, do not obey Ohm’s law; diodes and filament lamps are typical non-ohmic conductors whose I–V graphs are not straight lines.
欧姆定律指出，对于温度恒定的金属导体，通过它的电流与其两端的电势差成正比。比例常数就是电阻：R = V / I。电阻的单位是欧姆 (Ω)。然而很多元件并不服从欧姆定律；二极管和白炽灯就是典型的非欧姆导体，它们的 I–V 特性曲线不是直线。
Resistance arises from collisions between free electrons and the lattice ions. The resistance of a wire can be calculated from its resistivity ρ, length L and cross-sectional area A: R = ρL / A. Resistivity depends on the material and its temperature. For metals, resistivity increases with temperature because lattice vibrations intensify, causing more frequent collisions.
电阻源于自由电子与晶格离子之间的碰撞。导线的电阻可以通过电阻率 ρ、长度 L 和横截面积 A 计算：R = ρL / A。电阻率取决于材料和温度。金属的电阻率随温度升高而增加，因为晶格振动加剧，碰撞更加频繁。
3. I–V Characteristics of Circuit Elements | 电路元件的伏安特性
Drawing and interpreting I–V graphs is a key skill. For an ohmic resistor at constant temperature, the graph is a straight line through the origin. A filament lamp shows a curve that bends towards the voltage axis at higher currents because its resistance increases as it heats up. A diode allows current to flow easily in one direction (forward biased) but has very high resistance in the reverse direction, giving a characteristic that rises steeply after a threshold voltage (about 0.6 V for silicon).
绘制并解释 I–V 图像是一项核心技能。对于温度恒定的欧姆电阻，图像是一条过原点的直线。白炽灯的曲线在电流较大时会向电压轴弯曲，这是因为发热导致电阻增大。二极管在正向偏置时容易导通，反向时电阻极高，其特性曲线在阈值电压（硅管约 0.6 V）之后迅速上升。
Understanding these shapes helps in predicting how circuits behave when components are combined. For example, the non-linear resistance of a filament lamp can be used to stabilise current in some designs, but it complicates simple proportional reasoning. Always check whether the component is ohmic before applying R = V/I to a changing circuit.
理解这些曲线形状有助于预测元件组合后电路的行为。例如，白炽灯的非线性电阻虽可用于某些设计中的稳流，但会使得简单的比例推理失效。在变化的电路中，应用 R = V/I 前务必先判断元件是否为欧姆导体。
4. Series and Parallel Circuits | 串联与并联电路
In a series circuit, the same current flows through all components. The total resistance is the sum of individual resistances: Rₜₒₜₐₗ = R₁ + R₂ + R₃ + … . The supply voltage divides across the components in proportion to their resistances. If one component fails (open circuit), the entire circuit stops working.
在串联电路中，所有元件通过相同的电流。总电阻等于各个电阻之和：Rₜₒₜₐₗ = R₁ + R₂ + R₃ + … 。电源电压按电阻比例分配在各元件上。若某个元件断路，整个电路就会断开。
In a parallel circuit, the total current from the source splits at junctions, and the voltage across each branch is the same. The reciprocal of the total resistance is the sum of the reciprocals of the individual resistances: 1/Rₜₒₜₐₗ = 1/R₁ + 1/R₂ + 1/R₃ + … . This means the total resistance is always smaller than the smallest branch resistance. Parallel circuits are widely used in household wiring because each appliance operates independently. If one branch fails, the others continue to work.
在并联电路中，电源总电流在节点处分配，各个支路两端电压相等。总电阻的倒数等于各支路电阻倒数之和：1/Rₜₒₜₐₗ = 1/R₁ + 1/R₂ + 1/R₃ + … 。这意味着总电阻始终小于最小的支路电阻。家庭电路广泛采用并联方式，因为各电器独立工作，某一支路故障不影响其他支路。
5. Kirchhoff’s Laws | 基尔霍夫定律
Kirchhoff’s first law (junction rule) is a consequence of charge conservation: at any junction, the sum of currents entering equals the sum of currents leaving. Symbolically, ΣIᵢₙ = ΣIₒᵤₜ. This law is essential for analysing complex networks where currents divide and recombine.
基尔霍夫第一定律（节点定律）是电荷守恒的结果：在任意节点处，流入的电流之和等于流出的电流之和。即 ΣIᵢₙ = ΣIₒᵤₜ。这一定律对于分析电流分流和汇合的复杂网络至关重要。
Kirchhoff’s second law (loop rule) stems from conservation of energy: around any closed loop in a circuit, the sum of the e.m.f.s is equal to the sum of the potential drops (voltage across resistors). In other words, Σε = ΣIR. When applying this rule, you must assign consistent directions and signs, but the final magnitudes will be correct regardless of your initial guess.
基尔霍夫第二定律（回路定律）源于能量守恒：沿电路中的任意闭合回路，电动势的代数和等于电阻上电压降的代数和。即 Σε = ΣIR。应用时必须给电流和电动势设定一致的正方向并注意符号，但最终计算结果的大小与初始假设方向无关。
6. Potential Dividers | 分压电路
A potential divider is one of the most practical circuits. It consists of two or more resistors in series connected to a voltage source. The output voltage Vₒᵤₜ across one resistor is given by the ratio: Vₒᵤₜ = Vᵢₙ × R₂ / (R₁ + R₂), where R₂ is the resistor across which the output is taken. This simple formula is derived from the fact that the current is the same through both resistors and V = IR.
分压器是最实用的电路之一。它由两个或更多串联电阻连接到电压源构成。跨接在其中某个电阻上的输出电压 Vₒᵤₜ 由比例关系给出：Vₒᵤₜ = Vᵢₙ × R₂ / (R₁ + R₂)，其中 R₂ 是取输出电压的那个电阻。这一简单公式源自串联电流相等以及 V = IR。
Potential dividers are often used with sensors such as light-dependent resistors (LDRs) and thermistors. As the resistance of the sensor changes with light intensity or temperature, the output voltage varies, enabling control circuits. In such arrangements, linking the sensor’s behaviour to the physical quantity and to Vₒᵤₜ is a common exam task.
分压器常与光敏电阻 (LDR) 和热敏电阻等传感器结合使用。随着光照强度或温度变化，传感器电阻改变，输出电压也随之改变，从而实现控制功能。这类题目通常要求把传感器的物理特性与 Vₒᵤₜ 的变化联系起来，是考试常见内容。
7. Electromotive Force and Internal Resistance | 电动势与内阻
The electromotive force (e.m.f.) of a source is the energy supplied per unit charge. However, real sources like batteries and power supplies have internal resistance r. When current I flows, some energy is dissipated inside the source, so the terminal potential difference V is less than the e.m.f.: V = ε − I r. This equation is fundamental when investigating power delivery and efficiency.
电源的电动势 (e.m.f.) 是每单位电荷提供的能量。但真实的电源如电池和直流电源都有内阻 r。当电流 I 流过时，部分能量在电源内部损耗，因此路端电压 V 会小于电动势：V = ε − I r。这一方程是研究功率输出和效率的基础。
You can determine the e.m.f. and internal resistance of a cell experimentally by measuring terminal p.d. for different load currents and plotting a graph of V against I. The y-intercept gives ε, and the gradient is −r. Keep in mind that the open-circuit voltage (I = 0) equals the e.m.f., but it is often difficult to measure without drawing any current at all.
可以通过实验测量不同负载电流下的路端电压，并绘制 V–I 图像来确定电池的电动势和内阻。y 轴截距即为 ε，斜率为 −r。注意开路电压 (I = 0) 等于电动势，但要做到完全无电流测量其实很困难。
8. Power in DC Circuits | 直流电路中的功率
Electrical power P is the rate of energy transfer. For any component, P = IV. Using Ohm’s law, for ohmic resistors you can also write P = I²R or P = V² / R. The choice of formula depends on which quantities are known or kept constant. For example, in a series circuit, current is constant, so P ∝ R; in a parallel circuit, voltage is constant, so power is inversely proportional to resistance, P ∝ 1/R.
电功率 P 是能量转移的速率。对任何元件，P = IV。结合欧姆定律，对于欧姆电阻还可写成 P = I²R 或 P = V² / R。选择哪个公式取决于哪些量已知或保持不变。例如，串联电路中电流恒定，所以 P ∝ R；并联电路中电压恒定，功率与电阻成反比，P ∝ 1/R。
Conservation of energy means the total power supplied by the source equals the sum of the powers dissipated in all components, including the internal resistance. Questions often ask you to find the power wasted inside a battery or the efficiency η = (useful power output / total power input) × 100%. For a simple circuit with load R, efficiency is R / (R + r) × 100%.
能量守恒要求电源提供的总功率等于所有元件（包括内阻）消耗的功率之和。考题常要求计算电池内阻消耗的功率或效率 η = (有用输出功率 / 总输入功率) × 100%。对于由一个负载 R 构成的简单电路，效率为 R / (R + r) × 100%。
9. Maximum Power Transfer | 最大功率传输
When a source with e.m.f. ε and internal resistance r is connected to a variable load resistor R, the power delivered to the load is P = I²R = [ε² R] / [(R + r)²]. By differentiating or plotting a graph, we find that the load power reaches a maximum when R = r. The maximum power is Pₘₐₓ = ε² / (4r). This is a favourite derivation in A-Level physics.
当一个电动势为 ε、内阻为 r 的电源连接可变负载电阻 R 时，输送到负载的功率为 P = I²R = [ε² R] / [(R + r)²]。通过求导或作图可以发现，当 R = r 时负载功率达到最大值，最大功率为 Pₘₐₓ = ε² / (4r)。这是 A-Level 物理中常见的推导题。
It is important to understand that maximum power transfer does not imply maximum efficiency. At R = r, the efficiency is only 50%, because half the power is wasted in the internal resistance. In power transmission systems, we want high efficiency, so R >> r, but for signal circuits (like audio amplifiers) maximum power transfer is often desirable.
需要明确一点，最大功率传输并不意味着最高效率。当 R = r 时效率只有 50%，因为一半功率浪费在内阻上。在电力传输系统中我们希望效率高，因此要求 R >> r；但在信号电路（如音频放大器）中，往往追求最大功率传输。
10. Combining Cells in Series and Parallel | 电池的串联与并联
Identical cells can be connected in series to produce a larger e.m.f. The total e.m.f. of n cells in series is nε, and the total internal resistance is nr. This is useful when a higher voltage is required. When identical cells are connected in parallel, the total e.m.f. remains ε, but the combined internal resistance drops to r / n. This arrangement can supply a larger current without a severe voltage drop.
相同的电池可以串联起来获得更大的电动势。n 个电池串联的总电动势为 nε，总内阻为 nr。这在需要较高电压时很有用。当相同电池并联时，总电动势仍为 ε，但总内阻降为 r / n。这种连接方式能在较小电压跌落的情况下提供更大的电流。
Mixed configurations are possible for both higher voltage and lower internal resistance. Exam questions might ask you to redraw a battery pack or calculate the terminal p.d. when a load is connected. Always treat the whole group of cells as a single source with an equivalent e.m.f. and an equivalent internal resistance.
也可以采用混联方式同时获得较高的电压和较低的内阻。考题可能会让你重新绘制电池组或计算连接负载后的路端电压。始终要把整个电池组看作具有等效电动势和等效内阻的单一电源来处理。
11. Measurement and Practical Issues | 测量与实验问题
In practical circuit work, ammeters must be placed in series and have very low resistance so they do not affect the current. Voltmeters are connected in parallel and must have extremely high resistance so they draw negligible current. Digital multimeters typically have high input impedance, making them suitable for accurate voltage measurements.
在电路实验中，电流表必须串联接入且内阻要非常低，以免影响电流。电压表并联接入且内阻必须极高，从而只吸取可忽略不计的电流。数字万用表通常具有高输入阻抗，适合精确电压测量。
Systematic errors can arise from the meter’s own resistance. For example, a voltmeter with a resistance comparable to the component being measured will alter the effective resistance of the circuit. Using the potential divider concept, you can calculate the true voltage from the measured value if the meter’s resistance is known. Zero errors and parallax in analogue meters should also be considered.
仪表自身电阻会导致系统误差。例如，如果电压表的内阻与被测元件的电阻量级相当，就会改变电路的有效阻值。利用分压原理，可以在已知表头内阻的情况下由测量值计算出真实电压。此外还要考虑模拟电表的调零误差和读数视差。
12. Typical Exam Pitfalls and Tips | 常见考试陷阱与应对技巧
One common mistake is forgetting that a voltmeter measures the p.d. across itself, not some abstract point. Another is confusing total e.m.f. with terminal voltage when internal resistance is present. When applying Kirchhoff’s laws, students often mix up sign conventions; a reliable approach is to write the loop equation by going around in one direction and assigning a positive sign to a voltage when moving from − to + through the source, and a negative voltage drop across a resistor when moving with the assumed current.
一个常见错误是忘记电压表测量的是它自身两端的电势差，而不是某个抽象的点。另一个错误是当存在内阻时混淆总电动势和路端电压。应用基尔霍夫定律时，学生经常搞错符号惯例；一个可靠的方法是沿回路单向巡行，对电源从负极到正极取正电压，沿着假定的电流方向经过电阻时取负的电压降。
Always check whether the question asks for a quantity in standard units (e.g., Joules, Watts) or in derived forms. Deriving, rather than just memorising, the potential divider formula and maximum power condition will save you in case your memory fails. Drawing the circuit and labelling all currents and voltages before writing equations is an excellent habit that prevents most sign errors.
务必看清楚题目要求的单位是标准单位（如焦耳、瓦特）还是推导形式。不要死记硬背分压公式和最大功率条件，而是学会推导，这样即使一时忘记也能救急。先画出电路图并标出所有电流和电压，再动手列方程，这是一个能避免绝大多数符号错误的绝佳习惯。
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