📚 IB CCEA Physics Circuit Analysis Key Points | IB CCEA 物理：电路分析 考点精讲
Mastering circuit analysis is fundamental to success in both the IB and CCEA A‑level Physics specifications. This article revisits the core principles — from Ohm’s law to Kirchhoff’s rules — with clear explanations, essential equations and practical examples to reinforce your understanding of DC circuits.
掌握电路分析是 IB 和 CCEA A‑level 物理考试取得成功的基础。本文从欧姆定律到基尔霍夫定则，重温核心原理，通过清晰解释、关键方程和实例精讲，帮助你巩固对直流电路的理解。
1. Ohm’s Law and Resistance | 欧姆定律与电阻
Ohm’s law states that the potential difference V across an ohmic conductor is directly proportional to the current I flowing through it, provided the temperature remains constant. The constant of proportionality is the resistance R, measured in ohms (Ω).
欧姆定律指出，只要温度保持恒定，流过欧姆导体的电流 I 与导体两端的电势差 V 成正比。比例常数即为电阻 R，单位为欧姆 (Ω)。
V = IR ; R = V / I
If the resistance is constant, a graph of V against I is a straight line through the origin. Conductors that follow this linear relationship are called ohmic; components like diodes and filament lamps are non‑ohmic because their resistance changes with voltage or temperature.
如果电阻恒定，V 随 I 变化的图像是一条通过原点的直线。遵循这一线性关系的导体称为欧姆导体；二极管和灯丝灯泡等元件则是非欧姆导体，因为它们的电阻会随电压或温度改变。
Resistance depends on both the material and geometry of the conductor. It also dissipates electrical energy as heat when current flows through it.
电阻取决于导体的材料和几何形状。当电流流过导体时，电阻还会将电能以热能形式耗散。
2. Resistivity and Conductivity | 电阻率与导电性
The resistance R of a uniform wire is directly proportional to its length L and inversely proportional to its cross‑sectional area A. The proportionality constant is the resistivity ρ of the material.
均匀导线的电阻 R 与其长度 L 成正比，与其横截面积 A 成反比。比例常数即为材料的电阻率 ρ。
R = ρ × (L / A)
Resistivity has units of ohm‑metre (Ω·m) and is a property of the material at a given temperature. Good conductors have very low resistivity (e.g. copper ≈ 1.68 × 10⁻⁸ Ω·m); insulators have extremely high resistivity. Resistivity increases with temperature for most metals, which is essential for explaining the temperature dependence of resistance in conductors.
电阻率的单位是欧姆·米 (Ω·m)，它是材料在给定温度下的固有属性。良导体的电阻率很低（例如铜约为 1.68 × 10⁻⁸ Ω·m）；绝缘体的电阻率极高。对大多数金属而言，电阻率随温度升高而增大，这是解释导体电阻温度依赖性的关键。
Conductivity σ is the reciprocal of resistivity: σ = 1/ρ. It quantifies how easily a material allows the flow of electric current.
电导率 σ 是电阻率的倒数：σ = 1/ρ。它定量描述了材料允许电流通过的难易程度。
3. Series Circuits | 串联电路
In a series circuit, components are connected end‑to‑end, providing a single path for current. The current is the same through every component, while the total potential difference is the sum of the individual p.d.s across each component.
在串联电路中，元件首尾相连，只为电流提供一条通路。流过每个元件的电流都相同，而总电势差等于各个元件两端电势差之和。
The equivalent (total) resistance for resistors in series is simply the sum of their individual resistances.
串联电阻的等效（总）电阻等于各个电阻值之和。
R_total = R₁ + R₂ + R₃ + …
Because the same current flows through all resistors, the voltage across each resistor is proportional to its resistance (V₁ : V₂ = R₁ : R₂). Series circuits are therefore useful as voltage dividers, but if one component fails, the entire circuit becomes open.
由于所有电阻流过相同的电流，每个电阻两端的电压与其电阻值成正比 (V₁ : V₂ = R₁ : R₂)。因此串联电路可用作分压器，但如果其中一个元件发生故障，整个电路便会开路。
4. Parallel Circuits | 并联电路
In a parallel circuit, components are connected across common points, so the potential difference across each branch is the same. The total current drawn from the supply is the sum of the currents in the individual branches.
在并联电路中，元件跨接在公共节点之间，因此每条支路两端的电势差都相同。从电源流出的总电流等于各支路电流之和。
The reciprocal of the equivalent resistance for resistors in parallel is the sum of the reciprocals of the individual resistances.
并联电阻的等效电阻倒数等于各个电阻倒数之和。
1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + …
The total resistance of a parallel combination is always less than the smallest individual resistance. This is because adding more parallel branches provides additional paths for current, reducing the overall opposition to flow. Household wiring uses parallel connections so that appliances operate independently at the same voltage.
并联组合的总电阻总是小于其中最小的单个电阻。这是因为增加并联支路为电流提供了更多路径，从而降低了整体阻碍作用。家庭电路采用并联连接，这样电器可以在相同电压下独立工作。
5. Kirchhoff’s Laws | 基尔霍夫定律
Kirchhoff’s current law (KCL) arises from the conservation of charge: at any junction in a circuit, the sum of currents entering equals the sum of currents leaving.
基尔霍夫电流定律 (KCL) 源于电荷守恒：在电路的任一节点，流入的电流之和等于流出的电流之和。
Σ I_in = Σ I_out
Kirchhoff’s voltage law (KVL) arises from the conservation of energy: the sum of all electromotive forces around any closed loop equals the sum of all potential drops (IR drops) in that loop.
基尔霍夫电压定律 (KVL) 源于能量守恒：沿任一闭合回路，所有电动势的代数和等于该回路中所有电势降落 (IR 降落) 的代数和。
Σ ε = Σ IR or Σ V = 0 around a closed loop
These two laws are powerful tools for analysing circuits with multiple loops and branches. When solving circuits, assign a direction to each current, then write a system of equations based on KCL and KVL. Consistent sign conventions are essential — for example, a current entering a resistor in the direction of the loop is taken as a voltage drop.
这两个定律是分析多回路、多支路电路的强大工具。在求解电路时，先为每条支路的电流设定方向，再依据 KCL 和 KVL 写出方程组。符号规定必须一致——例如，沿回路行进方向，电流流入电阻时记作电压降落。
6. Potential Divider Circuit | 分压电路
A potential divider consists of two or more resistors in series connected across a voltage supply. It is used to obtain a variable output voltage that is a fraction of the input voltage.
分压电路由两个或多个电阻串联后跨接在电源上组成。它能获得一个可变的输出电压，该电压是输入电压的一部分。
For two resistors R₁ and R₂ in series, with the output taken across R₂, the output voltage is
对于两个串联的电阻 R₁ 和 R₂，若输出取自 R₂ 两端，则输出电压为
V_out = V_in × [ R₂ / (R₁ + R₂) ]
If R₂ is a variable resistor (or a thermistor / light‑dependent resistor), the output voltage changes in response to resistance variation. This principle is widely used in sensor circuits, such as temperature alarms and light‑activated switches.
如果 R₂ 是可变电阻（或热敏电阻、光敏电阻），输出电压就会随电阻变化而改变。这一原理广泛应用于传感器电路，例如温度报警器和光控开关。
The current drawn from the output must be negligibly small for the divider to behave ideally; otherwise, a load resistor connected across R₂ will alter the effective resistance and thus the output voltage.
为了使分压器达到理想效果，从输出端汲取的电流必须极小；否则，跨接在 R₂ 上的负载电阻将改变等效电阻，进而影响输出电压。
7. Electromotive Force (emf) and Internal Resistance | 电动势与内阻
A real source of electrical energy, such as a cell or battery, has an internal resistance r. The electromotive force ε is the energy supplied per unit charge when no current is drawn — it is the terminal voltage when the circuit is open.
真实的电能来源（如电池）具有内阻 r。电动势 ε 是在无电流输出时单位电荷获得的能量，即电路开路时的端电压。
When a current I flows, the terminal voltage V is less than the emf due to the internal voltage drop Ir.
当有电流 I 流过时，由于内阻上的电压降落 Ir，端电压 V 会小于电动势。
V = ε − I r
This linear relationship can be investigated by varying an external load resistor and measuring the terminal p.d. and current. A graph of V against I is a straight line with gradient −r and y‑intercept ε. The condition for maximum power transfer to a load is when the load resistance equals the internal resistance of the source.
这一线性关系可以通过改变外接负载电阻并测量端电压和电流进行研究。V 随 I 变化的图像是一条直线，斜率为 −r，纵截距为 ε。负载获得最大功率的条件是负载电阻等于电源的内阻。
8. Electrical Power and Energy | 电功率与电能
The rate at which electrical energy is transferred in a circuit component is the power P. For any component, power is the product of the current through it and the potential difference across it.
电路元件中电能转换的速率即为功率 P。对任何元件而言，功率等于流过它的电流与它两端电势差的乘积。
P = I V
For a resistor, where V = IR, we can also express power as
对于电阻，利用 V = IR，我们还可以将功率表示为
P = I² R or P = V² / R
Electrical energy E transferred over time t is then E = P t = I V t. The SI unit of energy is the joule (J); in practical electricity billing, the kilowatt‑hour (kW·h) is used, where 1 kW·h = 3.6 × 10⁶ J.
在时间 t 内转换的电能 E 为 E = P t = I V t。能量的国际单位是焦耳 (J)；在实际电费计算中则常用千瓦时 (kW·h)，1 kW·h = 3.6 × 10⁶ J。
Heating elements exploit the I² R (Joule heating) effect, while electric motors convert electrical energy into both mechanical work and internal heat. Efficiency in energy transfer is always an important consideration in circuit design.
加热元件利用 I² R（焦耳热）效应工作，而电动机则将电能转换为机械功和内能。在电路设计中，能量转换效率始终是一个重要的考量因素。
9. The Potentiometer | 电势计
A potentiometer is a precision instrument that uses a uniform resistance wire and a sliding contact to compare or measure emfs without drawing any current from the source being tested. It works on the principle that the potential drop across a segment of uniform wire is proportional to its length.
电势计是一种精密仪器，它利用均匀电阻丝和一个滑动触头来比较或测量电动势，且不会从待测源汲取任何电流。其工作原理是均匀电阻丝上一段的电势降落与其长度成正比。
To compare an unknown emf ε_unk with a known standard emf ε_std, the sliding contact is adjusted until the galvanometer reads zero (balanced condition). At balance, ε_unk / ε_std = L_unk / L_std, where L represents the corresponding lengths of wire.
为了比较未知电动势 ε_unk 与已知标准电动势 ε_std，需调节滑动触头直到检流计读数为零（平衡状态）。在平衡时，ε_unk / ε_std = L_unk / L_std，其中 L 表示对应的电阻丝长度。
The potentiometer can also be used to measure the internal resistance of a cell by comparing the open‑circuit p.d. with the terminal p.d. when a known load is connected. It provides more accurate results than a conventional voltmeter because it eliminates the loading effect.
电势计还可以通过比较开路电势和连接已知负载时的端电压来测量电池内阻。由于它消除了负载效应，因此比普通电压表测量更加精确。
10. Solving Complex Circuits | 复杂电路的分析
Complex circuits that cannot be reduced to simple series or parallel combinations require the systematic application of Kirchhoff’s laws. Begin by clearly labelling all known and unknown currents and choosing a consistent direction for each.
对于无法简化为简单串联或并联组合的复杂电路，需要系统性地应用基尔霍夫定律。首先清晰标出所有已知和未知电流，并为每条支路选定一致的方向。
Apply KCL at the junctions to write current equations, then apply KVL around independent loops to write voltage equations. You will often end up with a set of simultaneous linear equations that can be solved algebraically for the unknown currents.
在节点处应用 KCL 写出电流方程，再沿独立回路应用 KVL 写出电压方程。通常会得到一组线性联立方程，可用代数方法求解未知电流。
A useful check is the power balance: the total power supplied by the sources should equal the total power dissipated as heat in all the resistors plus any other energy conversions. This confirms whether your solution is consistent with energy conservation.
一个有效的检验方法是功率平衡：各电源提供的总功率应等于所有电阻上以热量形式耗散的总功率加上任何其他形式的能量转换。这可以确认你的解是否符合能量守恒。
Both IB and CCEA specifications often include multi‑loop circuit problems requiring you to set up and solve these equations. Practice with a variety of networks, including those with two batteries, to build confidence in systematic circuit analysis.
IB 和 CCEA 的考试大纲都经常要求建立并求解这类多回路电路方程。要多练习含有一个或多个电池的各种网络，以建立系统性分析电路的信心。
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