📚 Circuit Analysis in IB & OCR Physics: Key Points Explained | IB OCR 物理：电路分析 考点精讲
Circuit analysis forms the backbone of electricity and electronics in both IB Physics and OCR A-Level Physics. Mastering the behaviour of charges, currents, voltages and resistances in direct‑current (DC) circuits is essential for tackling examination questions confidently. This guide consolidates the most important concepts, laws and problem‑solving strategies, explained step by step with paired English‑Chinese explanations.
电路分析是 IB 物理和 OCR A-Level 物理中电学与电子学的基础。熟练掌握电荷、电流、电压和电阻在直流电路中的行为，对于自信地应对考题至关重要。本指南将最重要的概念、定律和解题策略集中起来，通过英中对照讲解逐一说明。
1. Charge, Current, and Potential Difference | 电荷、电流与电势差
Electric charge (Q) is a fundamental property of matter, measured in coulombs (C). The elementary charge e = 1.60 × 10⁻¹⁹ C is the magnitude of charge on a proton or electron. Current (I) is the rate of flow of charge: I = ΔQ / Δt, where ΔQ passes through a cross‑section in time Δt. The unit is the ampere (A).
电荷（Q）是物质的基本属性，单位为库仑（C）。元电荷 e = 1.60 × 10⁻¹⁹ C 是一个质子或电子所带电荷的大小。电流（I）是电荷流动的速率：I = ΔQ / Δt，其中 ΔQ 是在时间 Δt 内通过某截面的电荷量，单位为安培（A）。
Potential difference (p.d.) or voltage (V) between two points is the work done per unit charge to move a charge between those points: V = W / Q. It is measured in volts (V). The electromotive force (emf) of a source is the energy supplied per unit charge, also in volts.
两点之间的电势差（p.d.）或电压（V）是移动单位电荷所做的功：V = W / Q，单位为伏特（V）。电源的电动势（emf）是每单位电荷所供应的能量，单位也是伏特。
I = ∆Q / ∆t ; V = W / Q
2. Ohm’s Law and Resistance | 欧姆定律与电阻
For an ohmic conductor at constant temperature, the current through it is directly proportional to the potential difference across it: V = IR. The constant R is the resistance, measured in ohms (Ω). The I–V characteristic graph for an ohmic resistor is a straight line through the origin.
对于温度恒定的欧姆导体，通过它的电流与它两端的电势差成正比：V = IR。常数 R 是电阻，单位为欧姆（Ω）。欧姆电阻的 I–V 特性曲线是一条通过原点的直线。
Resistance limits the flow of charge. Components such as a filament lamp or a diode are non‑ohmic; their I–V graphs are curved because resistance changes with temperature or voltage direction.
电阻限制电荷的流动。如灯丝灯泡或二极管等元件是非欧姆性的；其 I–V 图是弯曲的，因为电阻随温度或电压方向变化。
3. 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: R = ρL / A. Resistivity has units Ω·m. A low ρ means the material is a good conductor. Conductivity σ is the reciprocal of resistivity: σ = 1 / ρ.
一根均匀导线的电阻取决于其长度 L、横截面积 A 和材料的电阻率 ρ：R = ρL / A。电阻率的单位是 Ω·m。ρ 值低表示材料是良导体。电导率 σ 是电阻率的倒数：σ = 1 / ρ。
Resistivity increases with temperature for most metals, which explains why resistance rises when a filament gets hot. In a resistance‑temperature graph, a positive gradient indicates a positive temperature coefficient.
大多数金属的电阻率随温度升高而增大，这就是灯丝变热后电阻升高的原因。在电阻‑温度图上，正梯度表示正温度系数。
R = ρ × L / A
4. Series and Parallel Circuits | 串联与并联电路
In a series circuit, the current is the same at all points. The total resistance is the sum: R_total = R₁ + R₂ + … . The supply voltage is divided across components, and the sum of p.d.s equals the supply voltage.
在串联电路中，各处电流相同。总电阻为各电阻之和：R_total = R₁ + R₂ + … 。电源电压被分配到各个元件上，各部分电压之和等于电源电压。
In a parallel circuit, the voltage across each branch is the same. The total current is the sum of branch currents. The total resistance is found from: 1 / R_total = 1 / R₁ + 1 / R₂ + … . For two resistors in parallel, the product‑over‑sum shortcut R_total = (R₁ × R₂) / (R₁ + R₂) can be used.
在并联电路中，各支路两端电压相同。总电流等于各支路电流之和。总电阻由下式求得：1 / R_total = 1 / R₁ + 1 / R₂ + … 。对于两个电阻并联，可使用乘积除以和的口诀 R_total = (R₁ × R₂) / (R₁ + R₂)。
| Quantity | Series | Parallel |
|---|---|---|
| Current I | Same everywhere | Divides; I_total = I₁ + I₂ + … |
| Voltage V | Divides; V_total = V₁ + V₂ + … | Same across each branch |
| Resistance R | R_total = R₁ + R₂ + … | 1/R_total = 1/R₁ + 1/R₂ + … |
5. Potential Dividers and Potentiometers | 分压器与电位器
A potential divider uses two resistors in series to produce a fraction of the input voltage. The output voltage V_out across R₂ is given by: V_out = V_in × (R₂ / (R₁ + R₂)). This is derived from the fact that the current I = V_in / (R₁ + R₂) is the same through both, so V_out = I × R₂.
分压器利用两个串联电阻来产生输入电压的一部分。R₂ 两端的输出电压为：V_out = V_in × (R₂ / (R₁ + R₂))。这是基于流过两个电阻的电流相同 I = V_in / (R₁ + R₂)，因此 V_out = I × R₂。
A potentiometer is a variable potential divider. By adjusting the slider, the output voltage can vary continuously from 0 V to the full supply voltage. It is often used in sensor circuits, volume controls, and to compare emfs without drawing current.
电位器是一个可变的分压器。通过调节滑动端，输出电压可以从 0 V 连续变化到满电源电压。它常用于传感器电路、音量控制以及在不抽取电流的情况下比较电动势。
V_out = V_in × R₂ / (R₁ + R₂)
6. Kirchhoff’s Laws | 基尔霍夫定律
Kirchhoff’s current law (KCL) states that the sum of currents entering a junction equals the sum of currents leaving it: Σ I_in = Σ I_out. This is a consequence of charge conservation.
基尔霍夫电流定律（KCL）指出，流入一个节点的电流之和等于流出该节点的电流之和：Σ I_in = Σ I_out。这是电荷守恒的结果。
Kirchhoff’s voltage law (KVL) states that around any closed loop in a circuit, the algebraic sum of the emfs and the potential differences across all components is zero: Σ ε + Σ (IR) = 0. In practice, the sum of the emfs equals the sum of the p.d.s in that loop.
基尔霍夫电压定律（KVL）指出，沿电路中任一回路的代数和，电动势与各元件两端电势差的总和为零：Σ ε + Σ (IR) = 0。实际应用中，回路中电源电动势的总和等于电阻上电压降的总和。
These two laws allow the analysis of complex circuits with multiple loops and sources, which cannot be reduced to simple series‑parallel combinations.
这两个定律可以用来分析含有多个回路和电源、无法简化为简单串并联组合的复杂电路。
7. Internal Resistance and EMF | 内阻与电动势
A real power source (cell, battery) has an internal resistance r. The terminal potential difference V across its terminals when a current I flows is less than the emf ε: V = ε – I r. Lost volts = I r represent energy dissipated inside the source.
真实的电源（电池）具有内阻 r。当电流 I 流过时，电源两端的端电压 V 小于电动势 ε：V = ε – I r。损失的电压 I r 代表在电源内部耗散的能量。
The graph of terminal p.d. against current is a straight line with equation V = –r I + ε. Its gradient is –r and its y‑intercept is ε. The maximum power delivered to an external load occurs when the load resistance equals the internal resistance (R = r).
端电压随电流变化的图线是一条直线，方程为 V = –r I + ε。斜率为 –r，y 轴截距为 ε。当外部负载电阻等于内阻（R = r）时，输出的功率最大。
V = ε – I r ; P_max when R = r
8. Electrical Power | 电功率
Power P is the rate of energy transfer in a circuit, measured in watts (W). For any component, P = I V. For a resistor, using V = I R, we can write: P = I² R = V² / R. These forms help determine power dissipation and heating effects.
功率 P 是电路中能量转换的速率，单位为瓦特（W）。对任何元件，P = I V。对于电阻，利用 V = I R，可写为：P = I² R = V² / R。这些形式有助于确定功率耗散和热效应。
In series circuits, the larger resistance dissipates more power (P = I² R). In parallel circuits, the smaller resistance dissipates more power (P = V² / R). The kilowatt‑hour (kWh) is a unit of energy: 1 kWh = 3.6 × 10⁶ J.
串联电路中，较大的电阻耗散更多的功率（P = I² R）。并联电路中，较小的电阻耗散更多的功率（P = V² / R）。千瓦时（kWh）是能量单位：1 kWh = 3.6 × 10⁶ J。
P = I V = I² R = V² / R
9. Capacitors and Capacitance | 电容器与电容
A capacitor stores charge and energy in an electric field. Its capacitance C is defined by C = Q / V, where Q is the magnitude of charge on one plate and V is the p.d. across the plates. The unit is the farad (F), often used with microfarads (µF) and picofarads (pF).
电容器在电场中储存电荷和能量。其电容 C 定义为 C = Q / V，其中 Q 是一片极板上的电荷量，V 是两极板间的电势差。单位是法拉（F），常用微法（µF）和皮法（pF）。
For a parallel‑plate capacitor, capacitance is proportional to the plate area A and inversely proportional to the plate separation d: C = ε₀ ε_r A / d, where ε₀ is the permittivity of free space and ε_r is the relative permittivity of the dielectric.
对于平行板电容器，电容与极板面积 A 成正比，与极板间距 d 成反比：C = ε₀ ε_r A / d，其中 ε₀ 是真空介电常数，ε_r 是介质的相对介电常数。
In series, total capacitance decreases: 1/C_total = 1/C₁ + 1/C₂ + … . In parallel, total capacitance adds: C_total = C₁ + C₂ + … . Energy stored in a capacitor is E = ½ QV = ½ C V² = ½ Q²/C.
串联时总电容减小：1/C_total = 1/C₁ + 1/C₂ + … 。并联时总电容相加：C_total = C₁ + C₂ + … 。电容器储存的能量为 E = ½ QV = ½ C V² = ½ Q²/C。
C = Q / V ; E = ½ C V²
10. RC Time Constant and Charging/Discharging | RC 时间常数与充放电
When a capacitor charges or discharges through a resistor, the voltage changes exponentially. The time constant τ = R × C indicates how quickly the capacitor charges or discharges. After time τ, the voltage has reached about 63% of its final value during charging, or fallen to 37% during discharging.
当电容器通过电阻充电或放电时，电压呈指数变化。时间常数 τ = R × C 表示电容器充放电的快慢。经过时间 τ 后，充电时电压约达到最终值的 63%，或放电时降至原来的 37%。
Charging equation: V(t) = V₀ (1 – e^(–t / RC)). Discharging equation: V(t) = V₀ e^(–t / RC). The current follows a similar exponential decay. In practice, a capacitor is considered fully charged after about 5 time constants (5τ).
充电方程：V(t) = V₀ (1 – e^(–t / RC))。放电方程：V(t) = V₀ e^(–t / RC)。电流遵循类似的指数衰减。实际中，经过约 5 个时间常数（5τ）后，可认为电容器已充满。
Logarithmic analysis can be used to verify the exponential relationship. For discharge, plotting ln V against t gives a straight line with gradient –1/RC, which enables determination of C if R is known.
可以用对数分析来验证指数关系。对于放电过程，绘制 ln V 对 t 的图得到一条直线，斜率为 –1/RC，据此可在已知 R 时求出 C。
τ = R C ; V_discharge(t) = V₀ e^(–t / τ)
11. Circuit Analysis Strategies | 电路分析策略
Start by simplifying the circuit where possible: combine series and parallel resistors step by step to find the total resistance. For complex networks, apply Kirchhoff’s laws to set up simultaneous equations for unknown currents and voltages.
首先尽可能简化电路：逐步合并串联和并联电阻以求出总电阻。对于复杂网络，应用基尔霍夫定律为未知电流和电压建立联立方程。
Always draw arrows to indicate assumed current directions and label loops. If a calculated current turns out negative, it simply means the actual direction is opposite to the assumed one. Double‑check that every power source is accounted for in voltage loops.
一定要画箭头标示假定的电流方向，并给回路编号。如果算出的电流为负值，那仅仅表示实际方向与假定方向相反。仔细核对每条电压回路是否包含了所有电源。
When dealing with RC circuits, identify the closed loop with the capacitor and resistor and write differential equations only if required; at A‑Level and IB, exponential formulas are usually sufficient. For internal resistance experiments, use the graph of V against I to extract ε and r accurately.
处理 RC 电路时，要找出电容器与电阻构成的闭合回路，仅当需要时才列微分方程；在 A‑Level 和 IB 中，指数公式通常就足够了。对于内阻实验，利用 V 对 I 的图线精确求出 ε 和 r。
Remember that voltmeters have very high (ideally infinite) resistance and ammeters have very low (ideally zero) resistance, which affects their placement. Practice converting wordy problems into clear circuit diagrams to avoid confusion.
记住，电压表具有极高（理想为无限大）电阻，电流表具有极低（理想为零）电阻，这会影响其连接位置。练习将文字描述的问题转换成清晰的电路图，以避免混淆。
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