Current & Circuits: Ohm's Law and Circuit Analysis | 电流与电路：欧姆定律与电路分析

📚 Current & Circuits: Ohm’s Law and Circuit Analysis | 电流与电路：欧姆定律与电路分析

Electric current and circuits form the foundation of classical electromagnetism and are essential topics in IB Physics. In this article, we explore the definitions of current and voltage, the origin of resistance, Ohm’s law, and the analytical tools used to solve circuits, including series/parallel combinations and Kirchhoff’s rules. We also address non-ohmic behaviour and the practical role of internal resistance.

电流与电路是经典电磁学的基础，也是 IB 物理的核心内容。本文将系统讲解电流与电压的定义、电阻的起源、欧姆定律，以及串联/并联组合和基尔霍夫法则等电路分析工具。同时我们也会讨论非欧姆元件的行为和电源内阻在实际问题中的作用。

1. Electric Current | 电流

Electric current \( I \) is the rate of flow of electric charge through a cross-section of a conductor. The SI unit is the ampere (A), where \(1\ \text{A} = 1\ \text{C s}^{-1}\). For a steady current, the charge \( q \) passing in time \( t \) is related by the equation:

电流 \( I \) 是电荷通过导体横截面的速率，国际单位是安培（A），即 \(1\ \text{A} = 1\ \text{C s}^{-1}\)。对于恒定电流，时间 \( t \) 内通过的电量 \( q \) 满足关系式：

I = q / t or q = I × t

In metals, the charge carriers are free electrons, which drift slowly under an electric field. Conventional current is defined as the direction in which positive charges would flow, which is opposite to the electron drift direction. Current is measured with an ammeter placed in series with the circuit element.

金属中的自由电子在电场作用下缓慢漂移形成电流。所谓“常规电流方向”被定义为正电荷流动的方向，与电子漂移方向相反。电流用电流表测量，电流表应与被测元件串联连接。

2. Potential Difference and Resistance | 电势差与电阻

The potential difference (p.d.) \( V \) between two points is the work done per unit charge to move charge between those points. Its SI unit is the volt (V), with \(1\ \text{V} = 1\ \text{J C}^{-1}\). A voltmeter measures p.d. and must be connected in parallel across the component.

两点之间的电势差 \( V \) 是将单位正电荷从一点移到另一点所做的功，单位是伏特（V），即 \(1\ \text{V} = 1\ \text{J C}^{-1}\)。伏特表测量电势差，必须与被测元件并联连接。

Resistance \( R \) is the ratio of the potential difference across a component to the current flowing through it:

电阻 \( R \) 是元件两端电势差与通过它的电流之比：

R = V / I

The ohm (Ω) is defined as \(1\ \Omega = 1\ \text{V A}^{-1}\). Resistance arises from collisions between charge carriers and the lattice of the conductor, which convert electrical energy into internal energy.

欧姆（Ω）定义为 \(1\ \Omega = 1\ \text{V A}^{-1}\)。电阻的微观本质是载流子与导体晶格的碰撞，这种碰撞将电能转化为内能。

3. Ohm’s Law | 欧姆定律

Ohm’s law states that, provided physical conditions such as temperature remain constant, the current through an ohmic conductor is directly proportional to the potential difference across it:

欧姆定律指出：在温度等物理条件保持恒定的情况下，通过欧姆导体的电流与它两端的电势差成正比：

V = I × R

The \(I\)-\(V\) characteristic of an ohmic resistor is a straight line through the origin, and its gradient is \(1/R\). Metals at constant temperature obey Ohm’s law closely. The condition of constant temperature is crucial, because the resistance of a metal generally increases with temperature.

欧姆电阻的 \(I\)-\(V\) 特性曲线是一条过原点的直线，其斜率为 \(1/R\)。在恒定温度下，金属近似严格遵守欧姆定律。温度恒定这一条件至关重要，因为金属的电阻通常随温度升高而增大。

4. Resistivity and Temperature | 电阻率与温度

For a uniform conductor of length \(L\) and cross-sectional area \(A\), its resistance is determined by the material property called resistivity \(\rho\):

对于长度为 \(L\)、横截面积为 \(A\) 的均匀导体，其电阻由材料的电阻率 \(\rho\) 决定：

R = ρ × L / A

Resistivity is measured in ohm-metres (Ω m) and depends only on the material and its temperature. For metals, resistivity rises with temperature because lattice vibrations scatter electrons more strongly. For semiconductors such as silicon, resistivity usually falls as temperature rises because more charge carriers are released.

电阻率的单位是欧姆·米（Ω m），只取决于材料种类和温度。金属的电阻率随温度升高而增大，因为晶格振动加剧、对电子的散射增强；对硅等半导体而言，电阻率通常随温度升高而下降，因为温度升高释放出更多载流子。

5. Ohmic and Non-Ohmic Components | 欧姆与非欧姆元件

Components whose \(I\)-\(V\) graphs are straight lines through the origin are called ohmic. Filament lamps, diodes, thermistors and light-dependent resistors (LDRs) are non-ohmic because their resistance changes with current, voltage or light intensity.

若元件的 \(I\)-\(V\) 图是一条过原点的直线，则称为欧姆元件。白炽灯、二极管、热敏电阻和光敏电阻（LDR）是非欧姆元件，因为它们的电阻随电流、电压或光照强度而改变。

Filament lamp: as current increases, the filament heats up, so resistance increases and the \(I\)-\(V\) curve bends downwards.
白炽灯：随电流增大，灯丝温度升高，电阻增大，\(I\)-\(V\) 曲线向下弯曲。
Diode: conducts strongly in the forward bias direction but very little in reverse bias; it has a threshold voltage about 0.6-0.7 V for silicon.
二极管：正向偏压时导通良好，反向偏压时几乎不导通；硅二极管的开启电压约为 0.6-0.7 V。
Thermistor: its resistance decreases rapidly when temperature increases (negative temperature coefficient).
热敏电阻：温度升高时电阻迅速减小（负温度系数）。
LDR: its resistance decreases when light intensity increases.
光敏电阻：光照增强时电阻减小。

6. Series and Parallel Circuits | 串联与并联电路

In a series circuit, the same current flows through every component, and the total potential difference across the network is the sum of the p.d.s across each component. The equivalent resistance is:

在串联电路中，通过每个元件的电流相同，网络两端的总电势差等于各元件两端电势差之和。等效电阻为：

Rₛ = R₁ + R₂ + R₃ + …

In a parallel circuit, every component has the same potential difference across it, while the total current entering the junction equals the sum of the currents in each branch. The equivalent resistance satisfies:

在并联电路中，每个元件两端的电势差相同，而流入节点的总电流等于各支路电流之和。等效电阻满足：

1/Rₚ = 1/R₁ + 1/R₂ + 1/R₃ + …

Property	Series	Parallel
Current	Same everywhere	Divides among branches
Voltage	Divides among components	Same across each branch
Equivalent resistance	Larger than any single resistor	Smaller than the smallest branch
用途	电流相同、需分压	电压相同、需分流

When analysing mixed circuits, reduce successive parallel or series combinations step by step until a single equivalent resistor remains.

分析混联电路时，应逐步将串联或并联组合化简，最终得到一个等效电阻。

7. Kirchhoff’s Laws | 基尔霍夫定律

Kirchhoff’s two laws are powerful tools for solving complex circuits.

基尔霍夫定律是分析复杂电路的有力工具。

Kirchhoff’s current law (KCL): the sum of currents entering a junction equals the sum of currents leaving the junction, which follows from charge conservation. Mathematically, \(\sum I_{\rm in} = \sum I_{\rm out}\).

基尔霍夫电流定律（KCL）：流入节点的总电流等于流出节点的总电流，这是电荷守恒的必然结果。数学表达式为 \(\sum I_{\rm in} = \sum I_{\rm out}\)。

Kirchhoff’s voltage law (KVL): in any closed loop, the sum of all electromotive forces equals the sum of all potential drops, which follows from energy conservation. Mathematically, \(\sum \varepsilon = \sum I R\).

基尔霍夫电压定律（KVL）：沿任一闭合回路，所有电动势之和等于所有电势降之和，这是能量守恒的必然结果。数学表达式为 \(\sum \varepsilon = \sum I R\)。

To apply KVL, choose a direction around the loop. A current that goes through a resistor in the chosen direction gives a drop \(IR\); a cell whose positive terminal is reached first gives a rise in potential.

应用 KVL 时，先选定绕行方向。沿绕行方向经过电阻产生电势降 \(IR\)；若先到达电源正极，则该电源产生电势升。

8. Potential Dividers | 电位分压器

A potential divider uses two resistors in series to produce a desired output voltage between 0 and the supply voltage. For two resistors \(R_1\) and \(R_2\) connected across a supply \(V_{\rm in}\), the voltage across \(R_2\) is:

电位分压器利用两个串联电阻，从电源电压 \(V_{\rm in}\) 中分得一个介于 0 和电源电压之间的输出电压。对于串联的 \(R_1\) 和 \(R_2\)，\(R_2\) 两端的电压为：

Vₒᵤₜ = Vᵢₙ × R₂ / (R₁ + R₂)

This arrangement is extremely useful for sensors. For example, a thermistor or LDR can replace one of the resistors so that the output voltage changes with temperature or light intensity, enabling automatic switching circuits.

该结构在传感器电路中非常实用。例如，用热敏电阻或光敏电阻替换其中一个电阻，输出电压就会随温度或光照强度变化，可用于自动控制开关电路。

9. Real Batteries and Internal Resistance | 实际电池与内阻

A real battery is modelled as an ideal electromotive force (emf) source \(\varepsilon\) in series with an internal resistance \(r\). When a current \(I\) is delivered to an external circuit, the terminal potential difference \(V_{\rm terminal}\) is less than the emf because energy is lost inside the battery:

实际电池可以看作一个理想电动势 \(\varepsilon\) 与一个内阻 \(r\) 串联。当电池向外电路提供电流 \(I\) 时，由于电池内部消耗能量，路端电压 \(V_{\rm terminal}\) 小于电动势：

V_terminal = ε − I × r

When the switch is open, no current flows, so the voltmeter reading equals the emf. When current flows, the voltage across the external load \(R\) is:

当开关断开、没有电流通过时，伏特表读数等于电动势。当有电流时，外部负载 \(R\) 两端电压为：

V = ε × R / (R + r)

If a short circuit occurs, the current is limited by the internal resistance: \(I_{\rm short} = \varepsilon / r\). A smaller internal resistance produces a more stable output voltage, which is why high-quality batteries are designed with very low \(r\).

如果发生短路，电流受内阻限制：\(I_{\rm short} = \varepsilon / r\)。内阻越小，输出电压越稳定，因此优质电池的内阻设计得很小。

10. Electric Power and Energy | 电功率与电能

The rate at which electrical energy is converted by a component is the electric power:

元件将电能转化为其他形式能量的速率称为电功率：

P = V × I

Combining this with Ohm’s law gives two equivalent forms for a resistor:

结合欧姆定律，可以得到电阻上功率的两种等价表达式：

P = I² × R and P = V² / R

The total energy dissipated in time \(t\) is \(E = P \times t\), measured in joules. In practical contexts, electrical energy is often quoted in kilowatt-hours (kWh), where \(1 \text{ kWh} = 3.6 \times 10^6 \text{ J}\).

在时间 \(t\) 内消耗的总电能为 \(E = P \times t\)，单位是焦耳。实际生活中常用“千瓦时”作为电能单位，\(1\ \text{kWh} = 3.6 \times 10^6\ \text{J}\)。

In summary, mastering these fundamental ideas — current, potential difference, resistance, Ohm’s law, circuit laws and power — gives you a complete toolkit for solving problems in DC circuits. Practice drawing circuit diagrams, labelling currents and applying KVL/KCL to build your confidence for exam questions.

总而言之，掌握电流、电势差、电阻、欧姆定律、电路定律与电功率这些核心概念，将为你提供解决直流电路问题的完整工具。多做练习，勤画电路图并标注电流方向，熟练运用基尔霍夫定律，考试时就会游刃有余。

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Current & Circuits: Ohm’s Law and Circuit Analysis | 电流与电路：欧姆定律与电路分析