📚 IB Physics: Circuit Analysis – Key Concepts and Exam Tips | IB 物理：电路分析 考点精讲
In IB Physics, circuit analysis forms the backbone of Topic 5 (Electricity and Magnetism). Understanding how charge moves, how potential difference drives current, and how components like resistors, cells, and measuring instruments behave in series and parallel configurations is essential for both Paper 1 and Paper 2. This guide breaks down the core concepts you must master, from Ohm’s law and Kirchhoff’s rules to potential dividers and internal resistance, helping you approach numerical problems and data‑based questions with confidence.
在 IB 物理中，电路分析是第五章（电与磁）的核心。理解电荷如何移动、电势差如何驱动电流，以及电阻、电池和测量仪表在串联与并联电路中的行为，对于应对卷一和卷二都至关重要。本文精讲欧姆定律、基尔霍夫定律、电位分压器、内阻等必考内容，助你从容应对计算题和实验数据分析题。
1. Electric Current and Charge | 电流与电荷
ic current is the rate of flow of charge. If an amount of charge ΔQ passes through a cross‑ rrent is carried by free electrons drifting in the opposite direction to conventional current, which is defined as the direction in which positive charges would move.
电流是电荷的流动速率。若在 Δt 时间内通过导体横截面的电荷量为 ΔQ，则电流 I = ΔQ / Δt。电流的单位是安培（A），1 A = 1 C s⁻¹。在金属导体中，载流子是自由电子，其漂移方向与规定的正电荷运动方向（即传统电流方向）相反。
Charge is conserved in any closed circuit: the total current entering a junction equals the total current leaving it. This conservation principle underpins Kirchhoff’s current law, which is crucial for analysing parallel branches.
任何闭合电路中电荷守恒：进入节点的总电流等于离开节点的总电流。这一守恒原理是基尔霍夫电流定律的基础，对分析并联支路至关重要。
2. Potential Difference (Voltage) and EMF | 电势差（电压）与电动势
/ q. It is measured in volts (V), where 1 V = 1 J C⁻¹. Across a resistor, potential difference drives a current and dissipates energy as heat. In contrast, the electromotive force (EMF) ε of a source is the total energy supplied per unit charge to drive charge around a complete circuit, including any energy lost to the internal resistance of the source.
两点间的电势差 V 是将单位正电荷从一点移动到另一点时所做的功，即 V = W / q。单位为伏特（V），1 V = 1 J C⁻¹。在电阻两端，电势差驱动电流并以热的形式耗散能量。而电源的电动势（EMF） ε 是电源将单位电荷绕完整电路驱动一周所提供的总能量，包含因内阻而损耗的能量。
When a voltmeter of very high (ideally infinite) resistance is connected across a component, it draws negligible current and therefore measures the true potential difference. An open‑switch reading of a cell’s terminal voltage is approximately equal to its EMF if the voltmeter resistance is far larger than the internal resistance.
当高阻值（理想为无限大）电压表并联在元件两端时，几乎不分流，因而能测量真实的电势差。若电压表内阻远大于电源内阻，则开关开路时测得的端电压近似等于电动势。
3. Resistance and Ohm’s Law | 电阻与欧姆定律
he ohm (Ω), with 1 Ω = 1 V A⁻¹. Ohm’s law states that, for many materials at constant temperature, the current through a conductor is directly proportional to the potential difference across it, so R is constant. A resistor that obeys Ohm’s law is called an ohmic component; its I‑V graph is a straight line through the origin.
电阻 R 的定义为 R = V / I，其中 V 为导体两端的电势差，I 为通过它的电流。电阻的 SI 单位是欧姆（Ω），1 Ω = 1 V A⁻¹。欧姆定律指出，在温度恒定的条件下，许多材料中的电流与两端电势差成正比，因此 R 为定值。遵守欧姆定律的电阻器称为欧姆型元件，其 I‑V 图像是一条通过原点的直线。
tage. For a filament lamp, the resistance increases as the filament heats up, causing the I‑V graph to curve. The resistance of a component at a specific point on a curved I‑ n by R = V / I, and the instantaneous gradient can be used to find the dynamic resistance.
如白炽灯灯丝和半导体二极管等元件不遵循欧姆定律，因为它们的电阻随温度或所加电压变化。白炽灯的电阻随灯丝升温而增大，导致 I‑V 曲线弯曲。在弯曲的 I‑V 图上，某一点对应的电阻仍可用 R = V / I 求出，而该点切线的斜率可用于确定动态电阻。
4. Resistivity | 电阻率
The resistance of a uniform wire depends on its length L, cross‑ property that quantifies how strongly a material opposes current flow. It has units of ohm‑ have low resistivity (≈ 1.7 × 10⁻⁸ Ω·m), while insulators have extremely high resistivity.
均匀导线的电阻取决于其长度 L、横截面积 A 以及材料的电阻率 ρ：R = ρ L / A。电阻率是反映材料阻碍电流流动能力的本征属性，单位为欧姆·米（Ω·m）。铜等良导体的电阻率很低（≈ 1.7 × 10⁻⁸ Ω·m），而绝缘体的电阻率极高。
ρ₀ [1 + α(T − T₀)], where α is the temperature coefficient of resistivity. This temperature dependence is used in resistance thermometers and must be considered when designing circuits that operate over a wide temperature range.
许多金属的电阻率随温度近似线性增大，可表示为 ρ = ρ₀ [1 + α(T − T₀)]，其中 α 为电阻率温度系数。这种温度依赖性被用于电阻温度计中，在宽温度范围工作的电路设计中也需加以考虑。
5. Series Circuits | 串联电路
In a series circuit, components are connected end‑to‑ viding a single path for current. The current I is the same at every point: I₁ = I₂ = I₃ = … al difference across the combination is the sum of the individual potential differences: Vtotal = V₁ + V₂ + V₃ + … . Consequently, the equivalent resistance of resistors in series is Req = R₁ + R₂ + R₃ + … .
在串联电路中，元件依次首尾相连，提供唯一的电流通路。电路中各处的电流相同：I₁ = I₂ = I₃ = … 。组合两端的总电势差等于各个电势差之和：Vtotal = V₁ + V₂ + V₃ + … 。因此，串联电阻的等效电阻为 Req = R₁ + R₂ + R₃ + … 。
The potential divider principle follows directly from series connections: the voltage across a particular resistor in a series chain is proportional to its resistance, V₁ / Vtotal = R₁ / Req. This allows a fraction of the total voltage to be tapped off for sensor circuits or reference voltages.
分压器原理直接源于串联连接：串联链中某一电阻两端的电压与其阻值成正比，V₁ / Vtotal = R₁ / Req。这样就能从总电压中取出一个比例电压，用于传感器电路或参考电压。
6. Parallel Circuits | 并联电路
ependent branches. The potential difference across each branch is the same: V₁ = V₂ = V₃ = … . The total current supplied by the source is the sum of the currents in the branches: Itotal = I₁ + I₂ + I₃ + … . The reciprocal equivalent resistance for resistors in parallel is given by 1/Req = 1/R₁ + 1/R₂ + 1/R₃ + … .
在并联电路中，所有元件连接在相同的两个节点之间，形成多条独立支路。各支路两端的电势差相等：V₁ = V₂ = V₃ = … 。电源供出的总电流等于各支路电流之和：Itotal = I₁ + I₂ + I₃ + … 。并联电阻的等效电阻倒数满足 1/Req = 1/R₁ + 1/R₂ + 1/R₃ + … 。
For two resistors in parallel, the equivalent resistance simplifies to Req nce of a parallel combination is always less than the smallest individual resistance, because adding more paths allows more current to flow for the same applied voltage.
两电阻并联时，等效电阻简化为 Req = (R₁ R₂) / (R₁ + R₂)。并联组合的总电阻始终小于最小的单个电阻，因为增加并联路径可在相同电压下使总电流增大。
7. Kirchhoff’s Circuit Laws | 基尔霍夫定律
ving. For a junction with three wires, I₁ = I₂ + I₃. Kirchhoff’s voltage law (KVL) follows from energy conservation: in any closed loop, the algebraic sum of the EMFs and potential differences is zero, i.e., Σ ε = Σ IR.
基尔霍夫电流定律（KCL）源于电荷守恒：在任何节点，流入电流之和等于流出电流之和。对于有三根导线的节点，I₁ = I₂ + I₃。基尔霍夫电压定律（KVL）源于能量守恒：在任一闭合回路中，电动势与电势差的代数和为零，即 Σ ε = Σ IR。
When applying KVL, choose a direction to traverse the loop. If a resistor is traversed in the direction of the current, the potential change is −IR; if a cell is traversed from the negative to the positive terminal, the change is +ε. These sign conventions allow you to set up simultaneous equations for unknown currents in multi‑loop circuits.
应用 KVL 时需选定回路绕行方向。若沿电流方向经过电阻，电势变化为 −IR；若从电池负极到正极，电势变化为 +ε。利用这些正负号规则，可为多回路电路中的未知电流列出联立方程组。
8. Potential Divider Circuits | 电位分压器
A potential divider uses two resistors in series to provide a fraction of the input voltage. If two resistors R₁ and R₂ are connected in series across a supply voltage Vin, the output voltage across R₂ is Vout = Vin widely used to produce a variable voltage from a fixed supply, especially when one of the resistors is a sensor such as a thermistor or an LDR.
电位分压器通过两个串联电阻来提供输入电压的一部分。若 R₁ 和 R₂ 串联后接在供电电压 Vin 两端，则 R₂ 两端的输出电压为 Vout = Vin × [R₂ / (R₁ + R₂)]。这种安排广泛用于从固定电源获得可变电压，特别是当其中一个电阻是热敏电阻或光敏电阻等传感器时。
When a load is connected across the output, it effectively places itself in parallel with R₂, reducing the combined resistance and therefore altering the output voltage. The design of potential dividers must account for the load resistance to ensure the output remains within the desired range.
当在输出端接上负载时，负载实际上与 R₂ 并联，降低了组合电阻，从而改变输出电压。设计电位分压器时必须考虑负载电阻，以保证输出电压留在期望范围内。
9. Internal Resistance and Terminal Voltage | 内阻与端电压
A real source of EMF, such as a battery, has internal resistance r. When a current I is drawn from the source, the terminal voltage VT across its terminals is less than the EMF: VT = ε − I r. The "lost volts" (I r) increase with current, and if the battery is short‑ sistance R = 0), the terminal voltage becomes zero and the current is limited only by r: Imax = ε / r.
真实的电动势源（如电池）具有内阻 r。当从电源吸取电流 I 时，其端电压 VT 低于电动势：VT = ε − I r。“损失的电压”（I r）随电流增大而增加。若电池被短路（外电阻 R = 0），端电压变为零，电流仅由 r 限制：Imax = ε / r。
The relationship VT riment, plotting terminal voltage against current gives a straight line with gradient −r and y‑ This is a standard IB practical for determining the EMF and internal resistance of a cell.
关系式 VT = ε − I r 是线性方程。在实验中，画出端电压随电流变化的图像，可得一条斜率为 −r、y 轴截距为 ε 的直线。这是测定电池电动势和内阻的标准 IB 实验。
10. Electrical Power and Energy | 电功率与电能
cross it. For a resistor, using Ohm’s law, the power can also be expressed as P = I² R or P = V² / R. The SI unit of power is the watt (W), and energy E transferred in time t is E = P t, measured in joules (J).
电路元件耗散的功率 P 是其能量转换的速率。对任何元件，P = I V，其中 V 为它两端的电势差。对于电阻，结合欧姆定律，功率还可表示为 P = I² R 或 P = V² / R。功率的 SI 单位是瓦特（W），在时间 t 内转换的能量为 E = P t，单位为焦耳（J）。
In IB problems, you may be asked to compare the brightness of identical lamps connected in different configurations or to calculate the heating effect in transmission cables. Remember that the power rating specifies the power consumed when the component operates at its rated voltage. Exceeding this voltage leads to overheating and possible damage.
在 IB 题目中，你可能需比较相同灯泡在不同连接方式下的亮度，或计算输电线的热效应。记住，额定功率表示元件在额定电压下工作时的功耗。超过该电压会导致过热甚至损坏。
11. Ammeters, Voltmeters and Safety | 电流表、电压表与安全
An ammeter measures current and must be connected in series so that the current to be measured passes through it. To avoid significantly altering the circuit current, an ammeter should have very low (ideally zero) resistance. A voltmeter measures potential difference and is connected in parallel with the component. It must have very high (ideally infinite) resistance to draw negligible current.
电流表测量电流，必须串联在电路中以使待测电流通过它。为尽量减少对电路电流的影响，电流表的电阻应极低（理想为零）。电压表测量电势差，并联在元件两端。它必须具有极高内阻（理想为无限大），以便分流极小。
Fuses and circuit breakers are safety devices that prevent excessive currents. A fuse contains a thin wire that melts and breaks the circuit if the current exceeds its rated value. The earth wire, insulation and double‑insulation are additional safety features that protect users from electric shock. In IB questions, you should be able to explain why a fuse must be placed in the live wire and why appliances with metal cases require earthing.
熔断器和断路器是防止电流过大的安全装置。熔断器中有一条细丝，当电流超过额定值时会熔断并断开电路。地线、绝缘和双重绝缘等安全设计可保护使用者免于触电。在 IB 试题中，应能解释为何熔断器必须接在火线上，以及为何金属外壳的电器需要接地。
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