📚 AP Physics C: Electricity and Magnetism Complete Concept Review | AP物理C电磁学知识点全面梳理
AP Physics C: Electricity and Magnetism covers the principles of electrostatics, electric circuits, magnetism, and electromagnetic induction using calculus. This article provides a streamlined concept review focused on key formulas, laws, and applications commonly tested on the exam.
AP物理C电磁学涵盖静电学、电路、磁学和电磁感应，并使用微积分方法。本文梳理核心概念、关键公式、定律和常见考试应用，帮助考生构建完整的知识框架。
1. Electric Charge and Coulomb’s Law | 电荷与库仑定律
Electric charge is a fundamental property. There are two types – positive and negative. Charge is conserved and quantized: the elementary charge is e = 1.602 × 10−19 C. Coulomb’s law describes the electrostatic force between two point charges: the magnitude is proportional to the product of the charges and inversely proportional to the square of the distance.
电荷是基本属性，分正、负两种。电荷守恒且量子化，元电荷 e = 1.602 × 10−19 C。库仑定律描述两点电荷间的静电力：力的大小与电荷量的乘积成正比，与距离的平方成反比。
F = k |q₁ q₂| / r² , k = 1/(4πε₀) ≈ 8.99 × 10⁹ N·m²/C²
The force acts along the line joining the charges. In vector form, F₁₂ = (k q₁ q₂ / r²) r̂₁₂, where r̂₁₂ is the unit vector from q₂ to q₁. Like charges repel, opposite charges attract.
力沿两电荷连线方向。矢量形式为 F₁₂ = (k q₁ q₂ / r²) r̂₁₂，其中 r̂₁₂ 是从 q₂ 指向 q₁ 的单位矢量。同号相斥，异号相吸。
For multiple point charges, the net force on a charge is the vector superposition of all individual Coulomb forces. The force obeys Newton’s third law.
对于多个点电荷，某电荷所受合力是各库仑力的矢量叠加。库仑力遵守牛顿第三定律。
2. Electric Field and Field Lines | 电场与电场线
The electric field E at a point is the force per unit test charge: E = F / q₀. It is a vector field with units N/C or V/m. For a point charge, the field is radial: E = k q / r² radially outward (positive q) or inward (negative q).
电场 E 定义为测试电荷每单位电荷所受的力：E = F / q₀。它是矢量场，单位 N/C 或 V/m。点电荷的电场呈辐射状：E = k q / r²，正电荷向外，负电荷向内。
Continuous charge distributions require integration: dE = k dq / r² r̂. The total field is the vector sum (integral) of contributions from all charge elements dq. Linear charge density λ, surface charge density σ, and volume charge density ρ are used to express dq.
连续电荷分布需用积分求解：dE = k dq / r² r̂。总场强为所有电荷元 dq 贡献的矢量积分。常用线电荷密度 λ、面电荷密度 σ 和体电荷密度 ρ 表示 dq。
Electric field lines start on positive charges and end on negative charges. Field strength is proportional to the density of lines. In uniform fields (like between parallel plates), E is constant.
电场线起于正电荷，止于负电荷。场强与电场线密度成正比。在平行板间的匀强电场中，E 为恒量。
3. Electric Potential and Potential Energy | 电势与电势能
Electric potential difference ΔV between two points is the negative work done by the electric field per unit charge: ΔV = Vb − Va = −∫ab E·dl. The potential at a point due to a point charge is V = k q / r (taking V=0 at infinity). Potential energy of a pair of charges is U = k q₁ q₂ / r.
两点间的电势差 ΔV 等于电场力对单位电荷做负功：ΔV = Vb − Va = −∫ab E·dl。点电荷的电势（无穷远为 0）为 V = k q / r。两个点电荷系统的电势能 U = k q₁ q₂ / r。
For a system of charges, the total potential is the algebraic sum of individual potentials (scalar superposition). Energy of a point charge in an external field is U = qV. Equipotential surfaces are always perpendicular to electric field lines.
电荷系的电势是各电荷电势的代数和（标量叠加）。点电荷在外电场中的电势能 U = qV。等势面始终垂直于电场线。
E = −∇V (or in one dimension, Eₓ = −dV/dx)
The electric field points in the direction of steepest potential decrease. Conductors in electrostatic equilibrium are equipotential volumes; the electric field inside is zero and the potential is constant.
电场指向电势下降最快的方向。静电平衡下的导体是等势体，内部电场为零，电势恒定。
4. Gauss’s Law | 高斯定律
Electric flux through a closed surface is Φₑ = ∮ E·dA. Gauss’s law states that the net flux is proportional to the enclosed charge: ∮ E·dA = Qenc / ε₀.
通过闭合曲面的电通量为 Φₑ = ∮ E·dA。高斯定律指出净电通量与面内净电荷成正比：∮ E·dA = Qenc / ε₀。
Gauss’s law is most useful for highly symmetric charge distributions (spherical, cylindrical, planar). Choose a Gaussian surface that matches the symmetry so that E is constant over the surface and parallel or perpendicular to dA.
高斯定律对高度对称的电荷分布（球、圆柱、平面）最有效。选取与对称性匹配的高斯面，使面上 E 大小恒定，且方向与 dA 平行或垂直。
| Symmetry | Common result | 对称性 | 典型结果 |
|---|---|---|---|
| Spherical (point/sphere) | E = (1/4πε₀) Q / r² (r ≥ R) | 球对称（点/球壳） | E = (1/4πε₀) Q / r² (r ≥ R) |
| Infinite line (cylindrical) | E = (1/2πε₀) λ / r | 无限长直线（柱对称） | E = (1/2πε₀) λ / r |
| Infinite plane (planar) | E = σ / (2ε₀) (uniform) | 无限大平面（面对称） | E = σ / (2ε₀) （匀强） |
For conductors, any net charge resides on the surface. The field just outside a conductor is E = σ/ε₀ perpendicular to the surface, and the field inside is zero.
导体中净电荷分布在表面。紧邻导体外侧的电场 E = σ/ε₀ 垂直于表面，内部场强为零。
5. Capacitance and Dielectrics | 电容与电介质
A capacitor stores charge and energy. Capacitance C = Q / V. For a parallel-plate capacitor with vacuum between plates, C₀ = ε₀ A / d. When the space is filled with a dielectric of constant κ, the capacitance becomes C = κ ε₀ A / d = κ C₀.
电容器储存电荷与能量。电容 C = Q / V。真空平行板电容器 C₀ = ε₀ A / d。当填充介电常数为 κ 的电介质时，电容 C = κ ε₀ A / d = κ C₀。
Dielectric reduces the effective electric field and potential difference for given charge, increasing capacitance. The dielectric strength is the maximum electric field a material can withstand without breakdown.
电介质减弱有效电场和电势差，从而增大电容。介电强度是材料不被击穿所能承受的最大电场。
Energy stored in a capacitor: U = ½ QV = ½ CV² = Q²/(2C). Energy density in an electric field (in vacuum) is u = ½ ε₀ E². With dielectric, u = ½ κ ε₀ E².
电容器储存的能量：U = ½ QV = ½ CV² = Q²/(2C)。真空电场能量密度 u = ½ ε₀ E²，介质中 u = ½ κ ε₀ E²。
Capacitors in parallel have the same voltage; equivalent capacitance Ceq = C₁ + C₂ + … In series, they carry the same charge; 1/Ceq = 1/C₁ + 1/C₂ + … .
并联电容器电压相同，等效电容 Ceq = C₁ + C₂ + …。串联电容器电荷相同，1/Ceq = 1/C₁ + 1/C₂ + …。
6. Current, Resistance, and DC Circuits | 电流、电阻与直流电路
Electric current I = dq/dt is the rate of flow of charge through a cross section. Current density J = I/A (conductor) and J = σ E (microscopic Ohm’s law), where σ is conductivity.
电流 I = dq/dt 是电荷通过截面的流动速率。电流密度 J = I/A（导体），微观欧姆定律 J = σ E，其中 σ 为电导率。
Resistance R = V/I for ohmic materials. For a uniform wire, R = ρ L/A, where ρ = 1/σ is resistivity. Resistivity varies with temperature: ρ(T) = ρ₀[1 + α(T − T₀)].
欧姆材料的电阻 R = V/I。均匀导线 R = ρ L/A，ρ = 1/σ 为电阻率。电阻率随温度变化：ρ(T) = ρ₀[1 + α(T − T₀)]。
Electric power dissipated in a resistor: P = IV = I² R = V²/R. The energy is converted to heat.
电阻消耗的电功率：P = IV = I² R = V²/R，电能转化为热能。
In a simple DC circuit, electromotive force (emf) ε of a battery provides energy. Terminal voltage V = ε − Ir, where r is internal resistance. Resistors in series: Req = R₁ + R₂ + … ; in parallel: 1/Req = 1/R₁ + 1/R₂ + … .
简单直流电路中，电池电动势 ε 提供能量。路端电压 V = ε − Ir，其中 r 为内阻。电阻串联：Req = R₁ + R₂ + …；并联：1/Req = 1/R₁ + 1/R₂ + …。
7. Kirchhoff’s Laws and RC Circuits | 基尔霍夫定律与 RC 电路
Kirchhoff’s junction rule (current law): At any node, the sum of currents entering equals the sum leaving (ΣIin = ΣIout) – a consequence of charge conservation. Kirchhoff’s loop rule (voltage law): Around any closed loop, the algebraic sum of potential differences equals zero (ΣΔV = 0) – a consequence of energy conservation.
基尔霍夫节点定律（电流定律）：进入任一节点的电流之和等于流出电流之和（ΣIin = ΣIout），是电荷守恒的体现。基尔霍夫回路定律（电压定律）：沿任一闭合回路，各部分电势差的代数和为零（ΣΔV = 0），是能量守恒的体现。
An RC circuit contains a resistor and a capacitor. Charging: q(t) = C ε (1 − e−t/RC), I(t) = (ε/R) e−t/RC. Discharging: q(t) = Q₀ e−t/RC, I(t) = (Q₀/RC) e−t/RC. The time constant τ = RC characterizes how fast the circuit charges or discharges.
RC 电路包含电阻和电容。充电过程：q(t) = C ε (1 − e−t/RC)，I(t) = (ε/R) e−t/RC。放电过程：q(t) = Q₀ e−t/RC，I(t) = (Q₀/RC) e−t/RC。时间常数 τ = RC 表征充放电的快慢。
After one time constant, the charge on a charging capacitor reaches ~63% of its final value; during discharge, it drops to ~37% of the initial charge.
经过一个时间常数，充电电容的电荷量达到最终值的约 63%；放电时电荷降至初始值的约 37%。
8. Magnetic Fields and Forces | 磁场与磁场力
Magnetic fields exert forces on moving charges and current-carrying wires. The magnetic force on a moving charge is F = q v × B. Magnitude: F = q v B sinθ, where θ is the angle between v and B. The direction is given by the right-hand rule.
磁场对运动电荷和载流导线施加力。运动
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