📚 A-Level AQA Science: Mastering Electricity and Magnetism | A-Level AQA 科学：电与磁 考点精讲
This comprehensive revision guide covers the essential topics of electricity and magnetism for the AQA A-Level Science specification. From static charges to electromagnetic induction, every concept is broken down with clear explanations, key equations, and exam-focused insights to help you master the subject and boost your confidence.
这份全面的复习指南涵盖了 AQA A-Level 科学大纲中电与磁的核心内容。从静电荷到电磁感应，每个概念都通过清晰的解释、关键方程和紧扣考点的分析逐一拆解，帮助你掌握知识，提升应试信心。
1. Electric Charge and Coulomb’s Law | 电荷与库仑定律
Electric charge is a fundamental property of matter, existing in two types: positive (+) and negative (−). Like charges repel each other, while opposite charges attract. The unit of charge is the coulomb (C). Coulomb’s law quantifies the force between two point charges: F = kQ₁Q₂/r², where k = 1/(4πε₀) ≈ 8.99×10⁹ N·m²/C².
电荷是物质的基本属性，分为正电荷（+）和负电荷（−）两种。同种电荷相互排斥，异种电荷相互吸引。电荷的单位是库仑（C）。库仑定律量化了两个点电荷之间的作用力：F = kQ₁Q₂ / r²，其中 k = 1/(4πε₀) ≈ 8.99×10⁹ N·m²/C²。
In AQA exams, you must use ε₀ = 8.85×10⁻¹² F/m. The force is a vector directed along the line joining the charges. For multiple charges, the principle of superposition applies: the net force is the vector sum of individual forces.
在 AQA 考试中，必须使用 ε₀ = 8.85×10⁻¹² F/m。力是沿两电荷连线方向的矢量。对于多个电荷，叠加原理适用：合力是各个力的矢量和。
- Attractive force between a proton and an electron separated by 5.3×10⁻¹¹ m: F = (8.99×10⁹)(1.6×10⁻¹⁹)²/(5.3×10⁻¹¹)² ≈ 8.2×10⁻⁸ N.
- 相距 5.3×10⁻¹¹ m 的质子和电子之间的吸引力：F = (8.99×10⁹)(1.6×10⁻¹⁹)²/(5.3×10⁻¹¹)² ≈ 8.2×10⁻⁸ N。
2. Electric Fields | 电场
An electric field is a region where a charged particle experiences a force. The electric field strength E is defined as the force per unit positive charge: E = F/q, measured in N/C or equivalently V/m. Field lines point away from positive charges and toward negative charges.
电场是带电粒子受到力的区域。电场强度 E 定义为单位正电荷所受的力：E = F/q，单位是 N/C 或等效的 V/m。电场线从正电荷发出，指向负电荷。
For a uniform field, such as between two parallel plates, E = V/d, where V is the potential difference and d is the plate separation. The field is constant in magnitude and direction. For a point charge, E = kQ/r², radial and inversely proportional to r².
对于均匀电场，例如两平行板之间，E = V/d，其中 V 是电势差，d 是板间距。场的大小和方向处处相同。对于点电荷，E = kQ/r²，呈辐射状且与 r² 成反比。
Potential difference between two points in a uniform field: V = Ed. Work done moving a charge q through a potential difference V: W = qV. An electron accelerated through 1 V gains 1 eV of kinetic energy (1 eV = 1.6×10⁻¹⁹ J).
均匀电场中两点间的电势差：V = Ed。将电荷 q 移动通过电势差 V 所做的功：W = qV。电子通过 1 V 加速获得 1 eV 的动能（1 eV = 1.6×10⁻¹⁹ J）。
3. Current, Resistance, and Ohm’s Law | 电流、电阻与欧姆定律
Electric current I is the rate of flow of charge: I = ΔQ/Δt, unit ampere (A). Conventional current direction is the flow of positive charge, opposite to electron flow. In metallic conductors, current is due to the drift of free electrons.
电流 I 是电荷流动的速率：I = ΔQ/Δt，单位安培（A）。约定电流方向为正电荷的流动方向，与电子流动方向相反。在金属导体中，电流由自由电子的漂移形成。
Resistance R measures opposition to current: R = V/I, unit ohm (Ω). Ohm’s law states that for many conductors at constant temperature, V ∝ I, so resistance is constant. Ohmic conductors include metals; non-ohmic examples include filament lamps and diodes.
电阻 R 衡量对电流的阻碍作用：R = V/I，单位欧姆（Ω）。欧姆定律指出，在恒温下，许多导体的 V ∝ I，因此电阻恒定。欧姆导体包括金属；非欧姆导体的例子有白炽灯丝和二极管。
Resistivity ρ is a material property: R = ρL/A, where L is length and A is cross-sectional area. Resistivity increases with temperature for metals (due to increased lattice vibrations) and decreases for semiconductors.
电阻率 ρ 是材料属性：R = ρL/A，其中 L 是长度，A 是横截面积。金属的电阻率随温度升高而增大（晶格振动加剧），半导体则减小。
4. Circuit Rules and Potential Divider | 电路规则与分压器
Kirchhoff’s laws are essential. First law (junction rule): total current entering a junction equals total current leaving (conservation of charge). Second law (loop rule): sum of emf in a closed loop equals sum of pd across components (conservation of energy).
基尔霍夫定律至关重要。第一定律（节点电流定律）：流入节点的总电流等于流出节点的总电流（电荷守恒）。第二定律（回路电压定律）：闭合回路中电动势的代数和等于各元件上电压降的代数和（能量守恒）。
In series circuits, current is the same; resistances add (Rₜₒₜ = R₁ + R₂ + …). In parallel, pd is the same across branches; combined resistance: 1/Rₜₒₜ = 1/R₁ + 1/R₂ + … This leads to a smaller total resistance than the smallest individual resistor.
串联电路中，电流处处相等；电阻相加（Rₜₒₜ = R₁ + R₂ + …）。并联电路中，各支路两端电压相等；总电阻：1/Rₜₒₜ = 1/R₁ + 1/R₂ + …，这使得总电阻小于最小的单个电阻。
A potential divider circuit uses two resistors in series to provide a fraction of the input voltage: Vₒᵤₜ = Vᵢₙ × R₂/(R₁ + R₂). It is widely used in sensor circuits with thermistors or LDRs (light-dependent resistors).
分压电路使用两个串联电阻提供输入电压的一部分：Vₒᵤₜ = Vᵢₙ × R₂/(R₁ + R₂)。它广泛应用于带有热敏电阻或光敏电阻 LDR 的传感器电路。
5. Electromotive Force and Internal Resistance | 电动势与内阻
Electromotive force (emf, ε) is the energy transferred per unit charge in a source (e.g., battery), measured in volts. The terminal pd V is less than ε when current flows due to internal resistance r: V = ε − Ir. This linear relationship allows determination of ε and r from a graph of V against I.
电动势（ε）是电源（如电池）中每个单位电荷所转换的能量，单位伏特。当有电流流过时，由于内阻 r，路端电压 V 小于 ε：V = ε − Ir。这种线性关系可以通过 V-I 图像求出 ε 和 r。
Maximum power transfer occurs when the load resistance R equals the internal resistance r. Power delivered to the external load: P = I²R = V²R/(R+r)². Efficiency improves when R ≫ r.
当外接负载电阻 R 等于内阻 r 时，输出功率最大。提供给外部负载的功率：P = I²R = V²R/(R+r)²。当 R ≫ r 时效率提高。
Typical exam question: “A battery of emf 9.0 V and internal resistance 1.2 Ω is connected to a 10 Ω resistor. Calculate terminal pd.” Answer: I = 9.0/(1.2+10) = 0.80 A, V = ε − Ir = 9.0 − 0.80×1.2 = 8.04 V.
典型试题：“一个电动势为 9.0 V、内阻为 1.2 Ω 的电池连接到一个 10 Ω 的电阻。计算路端电压。”解答：I = 9.0/(1.2+10) = 0.80 A，V = ε − Ir = 9.0 − 0.80×1.2 = 8.04 V。
6. Magnetic Fields and Forces on Moving Charges | 磁场与运动电荷受力
Magnetic fields are produced by moving charges (currents). A magnetic field can be represented by field lines which point from north to south. The magnetic flux density B, measured in tesla (T), indicates the strength of the field. Force on a current-carrying conductor in a uniform field: F = BIL sinθ, where θ is the angle between conductor and field.
磁场由运动电荷（电流）产生。磁场可用从北指向南的磁感线表示。磁通量密度 B 的单位是特斯拉（T），表示磁场强度。均匀磁场中载流导体所受的力：F = BIL sinθ，其中 θ 是导体与磁场方向的夹角。
For a single moving charge: F = Bqv sinθ, given by Fleming’s left-hand rule (force, field, current). The direction of force on a positive charge is given by the rule, but for electrons (negative), the force direction is reversed.
对于单个运动电荷：F = Bqv sinθ，由弗莱明左手定则确定（力、场、电流）。正电荷受力方向遵循该定则，但电子（负电荷）受力方向相反。
Circular motion of charged particles in a magnetic field: When velocity is perpendicular to B, the force provides centripetal acceleration: Bqv = mv²/r. Hence radius r = mv/(Bq). Applications include mass spectrometers and particle accelerators.
带电粒子在磁场中的圆周运动：当速度垂直于 B 时，洛伦兹力提供向心力：Bqv = mv²/r。因此半径 r = mv/(Bq)。应用包括质谱仪和粒子加速器。
7. Magnetic Flux and Faraday’s Law | 磁通量与法拉第定律
Magnetic flux Φ through a coil is B A cosθ, where θ is the angle between the field and the normal to the area. Unit is weber (Wb). Flux linkage NΦ = BAN cosθ. Faraday’s law: induced emf equals the rate of change of flux linkage: ε = −N ΔΦ/Δt. The negative sign indicates Lenz’s law.
穿过线圈的磁通量 Φ = B A cosθ，其中 θ 是磁场与面积法线间的夹角。单位是韦伯（Wb）。磁链 NΦ = BAN cosθ。法拉第定律：感应电动势等于磁链的变化率：ε = −N ΔΦ/Δt。负号表示楞次定律。
Lenz’s law states that the induced current flows in a direction that opposes the change in magnetic flux producing it. This is a consequence of conservation of energy. For a straight conductor moving through a field, ε = Blv (where l is length, v is speed).
楞次定律指出，感应电流的方向总是使其产生的磁通量反抗引起感应电流的磁通量变化。这是能量守恒的结果。对于直导体切割磁感线，ε = Blv（其中 l 是长度，v 是速度）。
8. Transformers and Alternating Current | 变压器与交流电
An alternating current (AC) produces a continuously changing magnetic flux, allowing efficient induction. A simple transformer consists of two coils wound on a soft iron core. For an ideal transformer, power is conserved: VₚIₚ = VₛIₛ. The turns ratio equation: Vₛ/Vₚ = Nₛ/Nₚ.
交流电产生持续变化的磁通量，从而实现高效感应。简单变压器由绕在软铁芯上的两个线圈组成。对于理想变压器，功率守恒：VₚIₚ = VₛIₛ。匝数比公式：Vₛ/Vₚ = Nₛ/Nₚ。
Step-up transformers increase voltage (Nₛ > Nₚ) and step-down transformers decrease voltage. Transformers are used in the National Grid to reduce power losses (P = I²R) during transmission. Efficiency is very high (>99%) but not perfect due to coil resistance, eddy currents, and hysteresis.
升压变压器提高电压（Nₛ > Nₚ），降压变压器降低电压。变压器用于国家电网以减少输电过程中的功率损耗（P = I²R）。效率极高（>99%），但由于线圈电阻、涡流和磁滞，并非完美。
Root mean square (rms) values for sinusoidal AC: Iᵣₘₛ = I₀/√2, Vᵣₘₛ = V₀/√2. These are used to calculate average power. In the UK, mains supply is 230 V rms at 50 Hz.
正弦交流电的方均根值：Iᵣₘₛ = I₀/√2，Vᵣₘₛ = V₀/√2。它们用于计算平均功率。在英国，市电为 230 V 有效值、50 Hz。
9. Motors and Generators | 电动机与发电机
The electric motor effect: a current-carrying coil in a magnetic field experiences a torque that rotates the coil. The split-ring commutator in a DC motor reverses current every half turn to maintain rotation direction. Torque = BIA N sinθ, maximum when the plane is parallel to the field.
电动机效应：磁场中的载流线圈受到力矩作用而转动。直流电动机中的换向器每半圈反转电流以保持转动方向。力矩 = BIA N sinθ，当线圈平面平行于磁场时力矩最大。
In a simple AC generator (alternator), a coil rotates in a magnetic field, inducing an alternating emf. Slip rings allow continuous connection. The generated emf follows ε = BANω sin(ωt), where ω is angular frequency. The peak emf is BANω.
在简单的交流发电机中，线圈在磁场中旋转，产生交变电动势。滑环实现持续连接。产生的电动势遵循 ε = BANω sin(ωt)，其中 ω 是角频率。峰值电动势为 BANω。
Comparison: DC generator uses a split-ring commutator to produce pulsating direct current. Back emf in motors opposes the applied voltage and limits current when the motor runs fast.
对比：直流发电机使用换向器产生脉动直流电。电动机中的反电动势对抗外加电压，并在高速运转时限制电流。
10. Practical Techniques and Graph Analysis | 实验技巧与图像分析
AQA exam papers frequently test practical skills. For resistivity: measure resistance of a wire at different lengths using a multimeter or voltmeter-ammeter method, plot R vs L, gradient = ρ/A. Avoid heating the wire to keep temperature constant.
AQA 试卷经常考查实验技能。测定电阻率：用万用表或伏安法测量不同长度导线的电阻，绘制 R-L 图像，斜率 = ρ/A。避免导线发热以保持温度恒定。
For internal resistance: vary external resistance, measure terminal pd and current, plot V vs I. The y-intercept gives ε, negative gradient gives r. Use a switch to avoid draining battery when no readings are taken.
测量内阻：改变外接电阻，测量路端电压和电流，绘制 V-I 图像。y 轴截距给出 ε，负斜率给出 r。不读数时使用开关防止电池耗尽。
Magnetic field experiments: use a Hall probe to measure magnetic flux density; investigate force on a current-carrying conductor using a top pan balance. Analysis of Faraday’s law often involves dropping a magnet through a coil and observing the induced emf on a data logger.
磁场实验：使用霍尔探头测量磁通量密度；利用托盘天平研究作用在载流导体上的力。法拉第定律分析常涉及让磁铁穿过线圈，并用数据记录器观察感应电动势。
11. Common Mistakes and How to Avoid Them | 常见错误与规避方法
One frequent mistake is confusing electric field strength E and potential V. Remember, E is gradient of potential, and for a point charge, E is vector, V is scalar. Do not mix up the formulas for force (F = qE) and work (W = qV).
一个常见错误是混淆电场强度 E 与电势 V。记住，E 是电势的梯度，而且对于点电荷，E 是矢量，V 是标量。不要将力公式 (F = qE) 和功公式 (W = qV) 弄混。
In circuits, students often misapply the voltage divider formula without considering the load resistance. The formula Vₒᵤₜ = Vᵢₙ × R₂/(R₁+R₂) holds only when no significant current is drawn from the output. Use Kirchhoff’s laws for complex networks.
在电路中，学生常在不考虑负载电阻的情况下错误应用分压公式。公式 Vₒᵤₜ = Vᵢₙ × R₂/(R₁+R₂) 仅在输出端没有明显电流时成立。处理复杂网络时使用基尔霍夫定律。
For electromagnetic induction, a common error is forgetting that induced emf depends on the rate of change of flux, not the value of flux. Lenz’s law must be used to determine direction, not just Faraday’s law magnitude.
在电磁感应中，一个常见错误是忘记了感应电动势取决于磁通量的变化率，而不是磁通量本身。必须使用楞次定律确定方向，而不仅仅用法拉第定律计算大小。
12. Key Equations Summary | 核心公式总结
| Equation 公式 | Meaning 含义 |
|---|---|
| F = kQ₁Q₂/r² | Coulomb’s law 库仑定律 |
| E = F/q, E = V/d, E = kQ/r² | Electric field strength 电场强度 |
| V = W/q | Potential difference 电势差 |
| I = ΔQ/Δt, R = V/I | Current, Resistance 电流, 电阻 |
| ρ = RA/L | Resistivity 电阻率 |
| V = ε − Ir | Terminal pd with internal resistance 内阻路端电压 |
| F = BIL sinθ, F = Bqv sinθ | Magnetic force on conductor / charge 安培力 / 洛伦兹力 |
| ε = −N ΔΦ/Δt | Faraday’s law 法拉第定律 |
| Vₛ/Vₚ = Nₛ/Nₚ | Ideal transformer 理想变压器 |
| Iᵣₘₛ = I₀/√2, Vᵣₘₛ = V₀/√2 | AC rms values 交流电方均根值 |
| F = Bqv = mv²/r | Charged particle circular motion 带电粒子圆周运动 |
Refer back to this table while attempting past paper questions to ensure all quantities are understood and units are correct.
在练习历年真题时，可以回顾此表格，确保所有物理量都已掌握且单位正确。
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