Circuit Analysis for IB and Edexcel Physics | IB Edexcel 物理：电路分析考点精讲

📚 Circuit Analysis for IB and Edexcel Physics | IB Edexcel 物理：电路分析考点精讲

Circuit analysis forms a cornerstone of both IB Physics and Edexcel A Level Physics. Whether you are dealing with simple resistive networks or more advanced concepts like internal resistance, potential dividers, and RC time constants, a solid command of Ohm’s law, Kirchhoff’s laws, and energy considerations is essential. This article systematically covers the key syllabus points, offers clear explanations, and equips you with problem-solving strategies typical of exam-style questions.

电路分析是 IB 物理和 Edexcel A Level 物理的共同基石。无论是处理简单的电阻网络，还是应对内阻、分压器、RC 时间常数等进阶概念，扎实掌握欧姆定律、基尔霍夫定律以及能量分析都不可缺少。本文系统梳理了核心考点，给出清晰解释，并为你提供应对考试典型题目的解题策略。

1. Ohm’s Law and Resistance | 欧姆定律与电阻

For an ohmic conductor at constant temperature, the current I flowing through it is directly proportional to the potential difference V across it. This relationship is described by Ohm’s law: V = I R, where R is the resistance measured in ohms (Ω). The resistance of a component determines how much it opposes the flow of charge.

对于温度恒定的欧姆导体，流过它的电流 I 与两端的电势差 V 成正比。这一关系由欧姆定律描述：V = I R，其中 R 为电阻，单位是欧姆 (Ω)。电阻值反映元件对电荷流动的阻碍程度。

V = I × R

The I-V graph of an ohmic resistor is a straight line passing through the origin, with slope equal to 1/R. Doubling the applied voltage doubles the current, provided temperature remains unchanged. In many metals, resistance increases with temperature because lattice vibrations scatter the drifting electrons more frequently.

欧姆电阻的 I-V 特性曲线是一条过原点的直线，斜率等于 1/R。若温度不变，电压加倍则电流也加倍。许多金属的电阻随温度升高而增大，原因是晶格振动更剧烈地散射漂移电子。

2. Resistors in Series and Parallel | 电阻的串联与并联

In a series circuit, the same current passes through each resistor. The total potential difference is the sum of individual p.d.s, and the equivalent resistance is R_total = R₁ + R₂ + R₃ + … The p.d. splits in direct proportion to the resistances: V₁ / V₂ = R₁ / R₂.

在串联电路中，各电阻流过相同电流。总电压等于各元件电压之和，等效电阻为 R_总 = R₁ + R₂ + R₃ + …。电压按电阻正比分配：V₁ / V₂ = R₁ / R₂。

In a parallel circuit, the voltage across each branch is identical. The total current from the source equals the sum of the branch currents. The reciprocal of the equivalent resistance is the sum of the reciprocals: 1/R_total = 1/R₁ + 1/R₂ + … This arrangement always produces a total resistance smaller than the smallest individual resistor.

在并联电路中，各支路两端电压相等。干路总电流等于各支路电流之和。等效电阻的倒数等于各电阻倒数之和：1/R_总 = 1/R₁ + 1/R₂ + …。这种连接方式得到的总电阻总是小于最小的单个电阻。

Feature / 特性	Series / 串联	Parallel / 并联
Current / 电流	Same everywhere 处处相同	Splits among branches 分支分流
Voltage / 电压	Divided across resistors 各电阻分压	Same across each branch 各支路电压相同
Equivalent resistance / 等效电阻	R_total = sum of all R R_总 = 各电阻之和	1/R_total = sum of 1/R 1/R_总 = 各电阻倒数之和

3. Kirchhoff’s Circuit Laws | 基尔霍夫电路定律

Kirchhoff’s first law (the junction rule) arises from conservation of charge: at any junction, the total current entering equals the total current leaving. Written in symbols, Σ I_in = Σ I_out. This law allows you to relate currents in complex networks.

基尔霍夫第一定律（节点定律）源于电荷守恒：在任意节点，流入的总电流等于流出的总电流。用符号表示为 Σ I_入 = Σ I_出。该定律帮助你在复杂网络中建立电流关系。

Kirchhoff’s second law (the loop rule) is a consequence of energy conservation. The sum of the electromotive forces (emfs) around any closed loop equals the sum of the products of current and resistance (potential drops): Σ E = Σ (I R). In other words, the algebraic sum of all potential differences around a closed loop is zero.

基尔霍夫第二定律（回路定律）源于能量守恒。任一闭合回路中，电动势的代数和等于各段电流与电阻乘积（电势降）的代数和：Σ E = Σ (I R)。换而言之，沿闭合回路一周，所有电势差的代数和为零。

When applying the loop rule, choose a consistent direction for each loop, assign positive signs to p.d.s that increase potential (such as emfs from negative to positive terminal) and negative signs to p.d.s across resistors in the direction of the loop current.

应用回路定律时，要为每个回路选定一致方向，对使电势升高的元件（例如从负极到正极的电动势）赋正号，对顺着回路电流方向的电阻两端电势降赋负号。

4. Electromotive Force and Internal Resistance | 电动势与内阻

Every real power source, such as a cell or battery, possesses some internal resistance r. The electromotive force (emf) E, measured in volts, is the energy supplied per unit charge when no current is drawn. Once a current I flows, the terminal potential difference V_t across the source is less than E due to the voltage drop across the internal resistance: V_t = E – I r.

任何实际电源，如电池，都具有内阻 r。电动势 E 的单位为伏特，表示断路时每单位电荷获得的能量。当有电流 I 流过时，电源两端的路端电压 V_t 会比电动势小，因为内阻上存在电压降：V_t = E – I r。

E = I (R + r) and V_t = E – I r

The standard experimental method to determine E and r uses a variable load resistor. By recording terminal voltage V and current I, you plot a graph of V against I. The y-intercept gives E, and the negative gradient equals the internal resistance r.

测定 E 与 r 的标准实验采用可变负载电阻。记录路端电压 V 和电流 I，绘制 V-I 图像：纵轴截距为 E，斜率绝对值为内阻 r。

5. Potential Dividers and Sensor Circuits | 分压器与传感器电路

A potential divider typically consists of two resistors in series connected to a voltage supply. The output voltage V_out taken across one resistor R₂ is given by:

分压器通常由两个电阻串联并连接到电源上构成。取自电阻 R₂ 两端的输出电压 V_out 由下式给出：

V_out = V_in × R₂ / (R₁ + R₂)

Replacing one resistor with a sensing component such as a light-dependent resistor (LDR) or thermistor turns the divider into a sensor circuit. As the physical condition changes, the resistance of the sensor varies, causing V_out to shift. For instance, an LDR’s resistance drops under bright light, so V_out across a fixed resistor in series with the LDR will rise in darkness.

将其中一个电阻换成光敏电阻 (LDR) 或热敏电阻等传感元件，分压器就变成了传感器电路。当物理条件改变时，传感器的电阻会变化，使 V_out 发生偏移。例如，LDR 在强光下电阻下降，因此与 LDR 串联的固定电阻两端的 V_out 在黑暗时升高。

This principle is heavily examined in both IB and Edexcel specifications. You are expected to design or analyse circuits that employ a thermistor or LDR to switch on a warning light, trigger a heating system, or control a logic gate input.

这一原理在 IB 和 Edexcel 考纲中都是重点考查内容。你需要能够设计或分析利用热敏电阻或 LDR 点亮警示灯、触发加热系统或控制逻辑门输入的电路。

6. Electrical Power and Energy Dissipation | 电功率与能量耗散

The power P transferred to a circuit component is the product of the current through it and the potential difference across it: P = I V. Using Ohm’s law, we obtain two alternative forms for resistive components: P = I² R and P = V² / R. The energy E dissipated in time t is simply E = P t.

传递给电路元件的电功率 P 等于流过电流与两端电压的乘积：P = I V。结合欧姆定律，纯电阻元件还有两种等效形式：P = I² R 和 P = V² / R。时间 t 内耗散的能量 E = P t。

The unit of power is the watt (W, equivalent to J s^-1). When comparing bulbs in series or parallel, remember that a higher resistance bulb dissipates more power in a series circuit (since current is fixed), while in a parallel circuit a lower resistance bulb draws more current and thus dissipates more power.

功率的单位是瓦特 (W, 相当于 J s^-1)。比较串联或并联电路中的灯泡时，需要记住：串联电路中电流相同，阻值较大的灯泡耗散功率更大；并联电路中电压相同，阻值较小的灯泡电流更大，因此耗散功率也更大。

7. Resistivity and Conductivity of Materials | 材料的电阻率与电导率

The resistance R of a uniform conductor depends on its length L, cross-sectional area A, and a material property called resistivity ρ (unit: Ω m). The defining equation is R = ρ L / A. Longer wires and thinner wires have higher resistance. Conductivity σ is the reciprocal of resistivity: σ = 1/ρ.

均匀导体的电阻 R 取决于长度 L、横截面积 A 以及材料性质——电阻率 ρ（单位：Ω m）。定义式为 R = ρ L / A。导线越长、越细，电阻越大。电导率 σ 是电阻率的倒数：σ = 1/ρ。

In an experiment to determine resistivity, you typically measure the resistance of a wire for several lengths, plot R against L, and extract ρ from the gradient using the known cross-sectional area. This investigation is a required practical in both IB and Edexcel courses.

在测定电阻率的实验中，通常要测量同一导线在不同长度下的电阻，绘制 R-L 图，利用已知截面积从斜率求出 ρ。这项探究是 IB 和 Edexcel 课程中的必修实验。

8. I-V Characteristics of Common Components | 常见元器件的伏安特性

Different components exhibit distinctive current-voltage relationships:

Fixed resistor (ohmic): straight line through origin.
Filament lamp: curve that flattens at higher voltages because its resistance rises with temperature.
Diode: negligible current for reverse bias, then a sharp exponential rise above the threshold forward voltage (typically ~0.7 V for silicon).

不同元器件的伏安特性各异：

固定电阻（欧姆元件）：过原点的直线。
白炽灯灯丝：电压较高时曲线趋于平缓，因为温度升高使电阻增大。
二极管：反偏时电流几乎为零；正向电压超过阈值（硅管约 0.7 V）后电流呈指数急升。

You must be able to sketch these I-V curves and explain the underlying physics. For the filament lamp, the key is that the metal lattice vibrations intensify with temperature, increasing collisions and thus resistance. For the diode, the p-n junction allows current only when external voltage overcomes the built-in potential barrier.

你需要能够画出这些 I-V 曲线并解释背后的物理原理。白炽灯灯丝的关键在于金属晶格振动随温度增强，碰撞加剧使电阻上升。二极管中，只有当外加电压克服内建电势垒时，p-n 结才允许大量载流子通过。

9. Capacitors and RC Time Constants | 电容器与RC时间常数

Capacitance C is defined as the charge stored per unit potential difference: C = Q / V, measured in farads (F). When a capacitor is charged through a resistor, the voltage across it grows according to V = V₀ (1 – e^{–t / (RC)}). During discharge, the voltage decays as V = V₀ e^{–t / (RC)}.

电容 C 定义为每单位电势差储存的电荷量：C = Q / V，单位是法拉 (F)。当电容器通过电阻充电时，两端电压按 V = V₀ (1 – e^{–t / (RC)}) 增长；放电时则按 V = V₀ e^{–t / (RC)} 衰减。

The product RC is called the time constant τ (tau). It represents the time taken for the voltage to rise to 63% of the supply voltage during charging, or to fall to 37% of the initial voltage during discharging. In one time constant, the change is always about 63% of the remaining gap.

乘积 RC 称为时间常数 τ。它表示充电过程中电压升至电源电压 63% 所需的时间，或放电过程中降至初始电压 37% 所需的时间。经过一个时间常数，变化量总是剩余差距的约 63%。

Examiners often ask you to interpret exponential decay graphs, calculate τ from a graph, or design a circuit to switch something at a preset voltage. Remember that the energy stored in a capacitor is E = ½ C V².

考官常要求解释指数衰减图像、从图像计算 τ，或设计在某预设电压触发的电路。电容器储存的能量 E = ½ C V² 也需要牢记。

10. Wheatstone Bridge and Measurement Techniques | 惠斯通电桥与测量技术

The Wheatstone bridge is an arrangement of four resistors used to measure an unknown resistance precisely. When the bridge is balanced, no current flows through the central galvanometer, and the ratio of the two known resistors equals that of the unknown pair: R₁ / R₂ = R₃ / R_x.

惠斯通电桥由四个电阻构成，可用于精确测量未知电阻。当电桥平衡时，中间检流计无电流通过，两个已知电阻的比值等于未知对臂的比值：R₁ / R₂ = R₃ / R_x。

Balance is independent of the supply voltage, making the method highly accurate. Practical forms of the bridge include the metre bridge, where a slide wire of uniform resistance provides the variable ratio. The unknown resistance is found by R_x = (L₂ / L₁) R_known, where L₁ and L₂ are lengths of wire segments.

平衡条件与电源电压无关，因此该方法精度很高。实用电桥如滑线电桥，利用一段均匀电阻丝提供可变比例。未知电阻由 R_x = (L₂ / L₁) R_已知 求得，其中 L₁、L₂ 为电阻丝两段长度。

11. Circuit Analysis Strategy and Problem Solving | 电路分析策略与解题技巧

Effective circuit analysis follows a logical sequence. First, simplify the network by identifying series and parallel combinations. Next, assign currents and label unknown potential differences. Apply Kirchhoff’s laws to generate simultaneous equations, then solve for the unknowns using algebraic methods.

高效的电路分析遵循逻辑顺序。先识别串联与并联部分，简化网络。随后，设定电流并标出未知电势差。利用基尔霍夫定律列出联立方程，再用代数方法求解未知量。

When a circuit includes more than one emf or multiple loops, you usually need both the junction rule and the loop rule. Count the number of unknowns and ensure you have at least as many independent equations. Practice with combinations of batteries, internal resistances, and capacitors is essential.

当电路含有多个电动势或多个回路时，往往需要同时使用节点定律和回路定律。数清未知量数目，并确保独立方程数目至少与之相等。多练习包含电池组、内阻与电容的电路组合非常重要。

Watch out for common pitfalls: forgetting that ammeters have very low resistance and voltmeters very high resistance; ignoring internal resistance when it is present; and misapplying the potential divider formula when the output terminal draws significant current. Always check whether your calculated values satisfy power and energy conservation.

小心常见误区：忘记安培表内阻极低、伏特表内阻极高；存在内阻时却忽略了它；以及当输出端汲取较大电流时仍直接套用分压器公式。始终检查计算值是否满足功率与能量守恒。

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Circuit Analysis for IB and Edexcel Physics | IB Edexcel 物理：电路分析 考点精讲