📚 Oxford AQA International AS Physics Electricity Topic Test: Formula Derivations | 牛津AQA国际AS物理电学主题测试：公式推导
Electricity is a cornerstone of AS Physics, and a firm grasp of the underlying derivations is essential for success in your topic test. This article revisits the key equations encountered in the Oxford AQA International AS Electricity unit, breaking down each derivation step by step. From the microscopic model of current to the behaviour of potential dividers, mastering these proofs will deepen your understanding and help you tackle both calculation and explanation questions with confidence.
电学是 AS 物理的基石，扎实掌握背后的公式推导对于在主题测试中取得成功至关重要。本文重温牛津 AQA 国际 AS 电学单元中的关键方程，逐步拆解每个推导过程。从电流的微观模型到分压器的工作方式，掌握这些证明将加深你的理解，帮助你自信地应对计算题和解释题。
1. Charge, Current and Drift Velocity | 电荷、电流与漂移速度
Electric current is defined as the rate of flow of charge. If an amount of charge ΔQ passes through a cross‑section of a conductor in a time interval Δt, the current I is simply I = ΔQ / Δt. The SI unit of current is the ampere (A), equivalent to one coulomb per second.
电流定义为电荷流动的速率。如果在时间间隔 Δt 内有电荷量 ΔQ 通过导体的某一横截面，则电流 I 就是 I = ΔQ / Δt。电流的国际单位是安培 (A)，相当于每秒一库仑。
To relate current to the microscopic motion of charge carriers, consider a wire of cross‑sectional area A. Let n be the number density of free charge carriers (e.g. electrons in a metal), each carrying a charge q. If the carriers move with a mean drift velocity vd, then in a time t each carrier travels a distance vdt. The volume of carriers that pass through the cross‑section is A vdt, so the number of carriers is N = n A vdt. The total charge delivered is Q = N q = n A vdt q. Substituting into I = Q / t yields the fundamental equation I = n A vd q. This shows that current depends on the density of carriers, their drift speed and the charge they carry.
为了将电流与载流子的微观运动联系起来，考虑一根横截面积为 A 的导线。设 n 为自由载流子（如金属中的电子）的数密度，每个载流子的电荷为 q。如果载流子以平均漂移速度 vd 运动，则在时间 t 内每个载流子移动了 vdt 的距离。通过横截面的载流子所占体积为 A vdt，因此载流子数目 N = n A vdt。输送的总电荷量 Q = N q = n A vdt q。代入 I = Q / t 就得到基本方程 I = n A vd q。这表明电流取决于载流子密度、漂移速度以及它们所带的电荷。
2. Potential Difference and Electromotive Force | 电势差与电动势
Potential difference (p.d.) between two points is defined as the work done per unit charge in moving a charge between those points. If W is the electrical energy transferred when charge Q moves through a component, then V = W / Q. The unit of potential difference is the volt (V), where 1 V = 1 J C⁻¹. In a circuit, the p.d. across a component indicates how much energy each coulomb of charge gains or loses as it passes through.
两点之间的电势差（p.d.）定义为单位电荷在该两点间移动时所做的功。当电荷 Q 通过一个元件时，若转移的电能为 W，则 V = W / Q。电势差的单位是伏特 (V)，1 V = 1 J C⁻¹。在电路中，元件两端的电势差表示每库仑电荷通过时所获得或损失的能量。
Electromotive force (e.m.f., ε) is the energy supplied by a source (such as a cell or battery) to each coulomb of charge passing through it. Like p.d., it is measured in volts. While p.d. refers to energy transferred to or from the charge by a circuit component, e.m.f. refers to the total energy per unit charge provided by the source, including the energy wasted inside the source itself due to internal resistance.
电动势（e.m.f., ε）是电源（如电池）为通过它的每库仑电荷所提供的能量。与电势差一样，它的单位也是伏特。电势差指的是电路元件传递给电荷或从电荷取出的能量，而电动势指电源提供的每单位电荷的总能量，包括由于内阻而在电源内部损失的能量。
3. Resistance and Ohm’s Law | 电阻与欧姆定律
The resistance R of a component is defined as the ratio of the potential difference V across it to the current I flowing through it: R = V / I. Resistance is measured in ohms (Ω), where 1 Ω = 1 V A⁻¹. This definition is always true, but for many materials the ratio is not constant.
电阻 R 的定义是元件两端的电势差 V 与流过它的电流 I 之比：R = V / I。电阻的单位是欧姆 (Ω)，1 Ω = 1 V A⁻¹。这个定义始终成立，但对于许多材料，这一比值并不是常数。
Ohm’s law states that, provided the temperature and other physical conditions remain constant, the current through a conductor is directly proportional to the potential difference across it. Mathematically, V ∝ I, or V = I R with R constant. Devices that obey this relationship are described as ohmic. For an ohmic resistor, an I‑V graph is a straight line through the origin. Non‑ohmic components, such as filament lamps and diodes, have I‑V curves that are not linear because their resistance changes with current or temperature.
欧姆定律指出，在温度和其他物理条件保持不变的条件下，通过导体的电流与导体两端的电势差成正比。数学上表示为 V ∝ I，或 V = I R，其中 R 为常数。遵循这一关系的器件被称为欧姆器件。对于欧姆电阻，其 I‑V 图为一条通过原点的直线。非欧姆元件，如白炽灯和二极管，其 I‑V 特性曲线不是线性的，因为它们的电阻随电流或温度而变化。
4. Resistivity | 电阻率
The resistance of a uniform conductor depends on its length L and cross‑sectional area A, as well as on the material itself. Experiment shows that R ∝ L and R ∝ 1/A. Introducing a constant of proportionality ρ (the resistivity), we obtain R = ρ L / A. Resistivity is a property of the material and is measured in ohm metres (Ω·m). Rearranging gives ρ = R A / L, showing that resistivity is the resistance of a sample of unit length and unit cross‑sectional area.
均匀导体的电阻取决于其长度 L、横截面积 A 以及材料本身。实验表明 R ∝ L 且 R ∝ 1/A。引入比例常数 ρ（电阻率），我们得到 R = ρ L / A。电阻率是材料的一种属性，单位为欧姆·米 (Ω·m)。重新整理可得 ρ = R A / L，这表明电阻率就是单位长度、单位横截面积的样品所具有的电阻。
R = ρ L / A
Metals have very low resistivities (e.g. copper ≈ 1.7 × 10⁻⁸ Ω·m), insulators have extremely high resistivities, and semiconductors lie in between. Resistivity also depends on temperature; for metals, resistivity increases with temperature because ion vibrations impede the drift of electrons.
金属的电阻率非常低（例如铜约为 1.7 × 10⁻⁸ Ω·m），绝缘体的电阻率极高，半导体则介于两者之间。电阻率还取决于温度；对于金属，电阻率随温度升高而增大，因为离子振动阻碍了电子的漂移。
5. Electrical Power and Energy | 电功率与能量
Power is the rate of energy transfer. Using the definitions V = W / Q and I = Q / t, we can derive the electrical power P. Since the work done W = V Q, the power is P = W / t = V Q / t = V I. Hence, the power delivered to a component is the product of the p.d. across it and the current through it.
功率是能量转移的速率。利用 V = W / Q 和 I = Q / t，我们可以推导出电功率 P。因为做功 W = V Q，所以功率 P = W / t = V Q / t = V I。因此，传递给元件的功率等于其两端电势差与流过电流的乘积。
Using Ohm’s law (V = I R) we can obtain two alternative forms for power dissipated in a resistor: P = I R × I = I² R, and P = V × (V / R) = V² / R. These are equivalent for ohmic components, but the most useful form depends on which quantities are known. The electrical energy transferred in a time t is E = P t = V I t.
利用欧姆定律 (V = I R)，我们可以得到电阻上消耗功率的另外两种形式：P = I R × I = I² R，以及 P = V × (V / R) = V² / R。对于欧姆元件，这些公式是等价的，但最实用的形式取决于已知哪些物理量。在时间 t 内转移的电能为 E = P t = V I t。
P = V I = I² R = V² / R
| Form | Useful for |
| P = V I | General, always valid |
| P = I² R | When current and resistance are known (series circuits) |
| P = V² / R | When p.d. and resistance are known (parallel circuits) |
6. Resistors in Series | 串联电阻
When resistors are connected in series, the current I through each resistor is the same. The total potential difference across the combination is the sum of the individual p.d.s: Vtotal = V₁ + V₂ + … . Applying V = I R to each resistor and to the equivalent resistance Rtotal gives:
当电阻串联时，通过每个电阻的电流 I 相同。组合两端的总电势差等于各个电势差之和：Vtotal = V₁ + V₂ + … 。对每个电阻以及等效电阻 Rtotal 应用 V = I R，得到：
I Rtotal = I R₁ + I R₂ + …
Since the current I is common factor, it cancels, yielding the series resistance formula: Rtotal = R₁ + R₂ + … . Thus the equivalent resistance of resistors in series is simply the sum of their individual resistances.
由于电流 I 是公因子，可以消去，得到串联电阻公式：Rtotal = R₁ + R₂ + … 。因此，串联电阻的等效电阻就是各电阻之和。
7. Resistors in Parallel | 并联电阻
For resistors connected in parallel, the p.d. across each resistor is the same, V, but the current divides. The total current entering the parallel network is the sum of the currents in each branch: Itotal = I₁ + I₂ + … . Using I = V / R for each resistor and for the equivalent resistance Rtotal gives:
对于并联电阻，每个电阻两端的电势差 V 相同，但电流分流。进入并联网络的总电流等于各支路电流之和：Itotal = I₁ + I₂ + … 。对每个电阻及等效电阻 Rtotal 使用 I = V / R，得到：
V / Rtotal = V / R₁ + V / R₂ + …
Cancelling the common V results in the parallel resistance formula: 1 / Rtotal = 1 / R₁ + 1 / R₂ + … . The equivalent resistance is always less than the smallest individual resistance because more paths are available for current.
消去公因子 V，就得到了并联电阻公式：1 / Rtotal = 1 / R₁ + 1 / R₂ + … 。等效电阻总是小于最小的单个电阻，因为为电流提供了更多的路径。
8. Kirchhoff’s Laws | 基尔霍夫定律
Kirchhoff’s first law (the junction rule) is a consequence of charge conservation. It states that at any junction in a circuit, the total current entering the junction equals the total current leaving it: Σ Iin = Σ Iout. This law is essential for analysing parallel branches and complex networks.
基尔霍夫第一定律（节点定律）源于电荷守恒。它指出，在电路中任一节点处，流入节点的电流之和等于流出节点的电流之和：Σ Iin = Σ Iout。该定律对于分析并联支路和复杂网络至关重要。
Kirchhoff’s second law (the loop rule) follows from energy conservation. It states that around any closed loop in a circuit, the algebraic sum of the e.m.f.s is equal to the algebraic sum of the p.d.s across the components: Σ ε = Σ V, or equivalently Σ V = 0. In practice, the sum of all potential differences (including those across sources and resistors) around a closed loop is zero. This allows us to determine unknown voltages in a circuit.
基尔霍夫第二定律（回路定律）源于能量守恒。它指出，沿电路中的任一闭合回路，电动势的代数和等于各元件上电势差的代数和：Σ ε = Σ V，或等价地 Σ V = 0。在实际中，绕闭合回路一圈所有电势差（包括电源和电阻两端的）的代数和为零。这使我们能够求解电路中的未知电压。
9. The Potential Divider | 分压器
A potential divider is one of the most practical circuits you will encounter. It consists of two resistors, R₁ and R₂, in series across a supply voltage Vin. The output voltage Vout is taken across R₂. Since the resistors are in series, the current I is the same through both and is given by I = Vin / (R₁ + R₂). Therefore, the voltage across R₂ is:
分压器是你将遇到的最实用的电路之一。它由两个电阻 R₁ 和 R₂ 串联在电源电压 Vin 上组成，输出电压 Vout 取自 R₂ 两端。由于电阻是串联的，通过两者的电流 I 相同，且 I = Vin / (R₁ + R₂)。因此，R₂ 两端的电压为：
Vout = I R₂ = (R₂ / (R₁ + R₂)) Vin
This formula shows that the output voltage is a fraction of the input voltage determined by the ratio of the resistors. By replacing one of the resistors with a variable resistor or a sensor (e.g. an LDR or thermistor), the output voltage can be made to respond to changes in the environment, which is the basis of many control circuits.
该公式表明，输出电压是输入电压的一部分，比例由电阻的比值决定。将其中一个电阻替换为变阻器或传感器（如光敏电阻或热敏电阻），输出电压就可以响应环境的变化，这是许多控制电路的基础。
10. Internal Resistance and Terminal Potential Difference | 内电阻与端电压
All real sources of e.m.f. have some internal resistance, denoted r. When a current I flows, some of the energy supplied by the source is dissipated as heat inside the source itself. The terminal p.d. V across the source is then less than its e.m.f. ε. From energy considerations, the lost p.d. inside the source is I r, so:
所有实际的电动势源都具有一定的内阻，记作 r。当有电流 I 流过时，电源提供的部分能量会以热的形式在电源内部消耗掉。电源两端的端电压 V 因此小于其电动势 ε。从能量角度看，电源内部损失的电压为 I r，所以：
ε = V + I r, or V = ε − I r
This equation is often rearranged to V = − r I + ε, which has the form y = m x + c. If we measure the terminal p.d. V for different currents I and plot a graph of V against I, we obtain a straight line with gradient − r and y‑intercept ε. This is the standard experimental method for determining the e.m.f. and internal resistance of a cell.
这个方程常被改写为 V = − r I + ε，形式为 y = m x + c。如果我们测量不同电流 I 下的端电压 V，并绘制 V 对 I 的图像，就会得到一条斜率为 − 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实验方法。
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