📚 AP Physics C: Mechanics & E&M Key Concepts Review | AP物理C电磁学与力学重点知识梳理
Welcome to this focused review of the essential topics in AP Physics C: Mechanics and Electricity & Magnetism. This guide consolidates the core principles, equations, and problem-solving strategies you need to master for the exams. We cover kinematics, Newton’s laws, energy, momentum, rotation, oscillations, gravitation, electrostatics, circuits, magnetism, induction, and Maxwell’s equations, all presented with paired English-Chinese explanations to reinforce understanding.
欢迎阅读 AP 物理 C 力学与电磁学重点知识梳理。本指南整合了考试必须掌握的核心原理、方程与解题策略，涵盖运动学、牛顿定律、能量、动量、转动、振动与引力、静电学、电路、磁学、电磁感应以及麦克斯韦方程组，每部分均以中英对照讲解，帮助加深理解。
1. Kinematics | 运动学
Position, velocity, and acceleration are related by derivatives: v = dx/dt, a = dv/dt. For constant acceleration, the four kinematic equations describe motion in one dimension.
位置、速度和加速度通过导数联系：v = dx/dt，a = dv/dt。对于匀加速度，四个运动学方程描述一维运动。
v = v₀ + at x = x₀ + v₀t + ½at² v² = v₀² + 2aΔx x = x₀ + ½(v₀ + v)t
Projectile motion is treated as independent horizontal and vertical components; horizontal velocity is constant, vertical acceleration is -g.
抛体运动分解为独立的水平和竖直分量；水平速度恒定，竖直加速度为 -g。
Angular kinematics mirrors linear kinematics with θ, ω, α replacing x, v, a: ω = dθ/dt, α = dω/dt, and analogous equations for constant α.
角运动学与线运动学对应，用 θ、ω、α 代替 x、v、a： ω = dθ/dt, α = dω/dt，且匀角加速度下有类似方程。
2. Newton’s Laws & Applications | 牛顿定律及其应用
Newton’s Second Law in its differential form: F_net = dp/dt = ma (for constant mass). Free-body diagrams are essential for identifying forces.
牛顿第二定律的微分形式： F_net = dp/dt = ma（质量恒定）。受力分析图对识别各力至关重要。
Common forces: gravity (F_g = mg), normal force, tension, friction (static f_s ≤ μ_s N, kinetic f_k = μ_k N), and spring force (Hooke’s law: F_s = -kx).
常见力：重力 (F_g = mg)、法向力、张力、摩擦力（静摩擦 f_s ≤ μ_s N，动摩擦 f_k = μ_k N）和弹簧力（胡克定律：F_s = -kx）。
Uniform circular motion requires a centripetal acceleration a_c = v²/r, directed toward the center, provided by net radial force F_net = mv²/r.
匀速圆周运动需要向心加速度 a_c = v²/r，指向圆心，由径向合力提供 F_net = mv²/r。
Resistive forces: drag force often proportional to v or v²; terminal velocity occurs when drag equals weight.
阻力：阻力常正比于 v 或 v²；当阻力等于重力时达到终极速度。
3. Work, Energy & Power | 功、能量与功率
Work done by a force: W = ∫ F · dr. For a constant force, W = Fd cosθ. The work-energy theorem: W_net = ΔK = ½mv² – ½mv₀².
力做的功：W = ∫ F · dr。恒力做功 W = Fd cosθ。功能定理：W_net = ΔK = ½mv² – ½mv₀²。
Conservative forces (gravity, spring, electrostatic) have associated potential energy: U_g = mgy, U_s = ½kx². The negative gradient gives force: F = -dU/dx.
保守力（重力、弹力、静电力）对应势能：U_g = mgy，U_s = ½kx²。势能负梯度等于力：F = -dU/dx。
Mechanical energy conservation: E = K + U = constant if only conservative forces do work. Power is the rate of doing work: P = dW/dt = F·v.
机械能守恒：若仅保守力做功，E = K + U 恒定。功率是做功速率：P = dW/dt = F·v。
4. Linear Momentum & Collisions | 线动量与碰撞
Momentum p = mv. Impulse J = ∫ F dt = Δp. Momentum conservation: if net external force is zero, total momentum of a system is constant.
动量 p = mv。冲量 J = ∫ F dt = Δp。动量守恒：若合外力为零，系统总动量守恒。
Elastic collisions conserve kinetic energy and momentum; in one dimension for equal masses, velocities are exchanged. Inelastic collisions conserve momentum only, with maximum loss in perfectly inelastic collisions where objects stick together.
弹性碰撞动能和动量均守恒；一维等质量碰撞交换速度。非弹性碰撞仅动量守恒，完全非弹性碰撞中物体粘合，动能损失最大。
The center of mass position: r_cm = (Σ m_i r_i)/M. Its velocity and acceleration relate to total momentum and net external force: F_net_ext = M a_cm.
质心位置：r_cm = (Σ m_i r_i)/M。其速度和加速度与总动量和合外力相关：F_net_ext = M a_cm。
5. Rotational Motion | 转动
Torque τ = r × F, magnitude τ = rF sinθ. Rotational analog of Newton’s Second Law: Στ = Iα, where moment of inertia I = Σ m_i r_i² or ∫ r² dm.
力矩 τ = r × F，大小 τ = rF sinθ。转动中的牛顿第二定律：Στ = Iα，其中转动惯量 I = Σ m_i r_i² 或 ∫ r² dm。
Parallel-axis theorem: I = I_cm + Md². Rolling without slipping: v_cm = ωR, a_cm = αR. Kinetic energy of a rolling body: K = ½I_cm ω² + ½Mv_cm².
平行轴定理：I = I_cm + Md²。无滑滚动：v_cm = ωR, a_cm = αR。滚动体的动能：K = ½I_cm ω² + ½Mv_cm²。
Angular momentum L = r × p = Iω. For a rigid body, Στ_ext = dL/dt. Conservation of angular momentum: if net external torque is zero, L is constant.
角动量 L = r × p = Iω。刚体满足 Στ_ext = dL/dt。角动量守恒：若合外力矩为零，L 恒定。
6. Oscillations & Gravitation | 振动与引力
Simple harmonic motion (SHM) occurs when restoring force F = -kx. Angular frequency ω = √(k/m) for mass-spring, ω = √(g/L) for a simple pendulum (small angles).
简谐运动发生于回复力 F = -kx 时。弹簧振子角频率 ω = √(k/m)，单摆（小角度）ω = √(g/L)。
General solution: x(t) = A cos(ωt + φ). Energy in SHM: E = ½kA², and it continuously interchanges between kinetic and potential.
通解：x(t) = A cos(ωt + φ)。简谐运动的能量：E = ½kA²，在动能与势能之间不断转化。
Newton’s law of gravitation: F = GMm/r². Gravitational potential energy U = -GMm/r. For orbits, Kepler’s laws apply, and for circular orbits: v = √(GM/r).
牛顿引力定律：F = GMm/r²。引力势能 U = -GMm/r。轨道运动遵循开普勒定律，圆轨道速度：v = √(GM/r)。
7. Electrostatics & Gauss’s Law | 静电学与高斯定律
Coulomb’s law: F = k q₁ q₂ / r², with k = 1/(4πε₀). Electric field E = F/q, and for a point charge: E = k q/r² (radial direction).
库仑定律：F = k q₁ q₂ / r²，k = 1/(4πε₀)。电场 E = F/q，点电荷电场：E = k q/r²（径向）。
Gauss’s law: ∮_S E·dA = q_enc / ε₀. Apply symmetry to find fields for spheres, cylinders, and planes.
高斯定律：∮_S E·dA = q_enc / ε₀。利用对称性可求球、柱和平面电荷分布的电场。
Conductors in electrostatic equilibrium: E inside is zero, excess charge resides on surface, and E just outside is perpendicular with magnitude σ/ε₀.
静电平衡的导体：内部电场为零，净电荷分布于表面，外表面附近电场垂直于表面且大小为 σ/ε₀。
8. Electric Potential & Capacitance | 电势与电容
Electric potential difference ΔV = -∫ E·dr. For a point charge, V = k q/r (with V=0 at infinity). Equipotential surfaces are perpendicular to E.
电势差 ΔV = -∫ E·dr。点电荷电势 V = k q/r（无穷远为 V=0）。等势面与电场线垂直。
Capacitance C = Q/V; for parallel plates C = ε₀ A/d. Energy stored in a capacitor: U = ½QV = ½CV². With a dielectric, C’ = κC.
电容 C = Q/V；平行板电容 C = ε₀ A/d。电容器储能：U = ½QV = ½CV²。加入电介质后 C’ = κC。
9. Current & DC Circuits | 电流与直流电路
Current I = dQ/dt, current density J = I/A. Ohm’s law: V = IR, resistance R = ρL/A. Power dissipated: P = IV = I²R = V²/R.
电流 I = dQ/dt，电流密度 J = I/A。欧姆定律：V = IR，电阻 R = ρL/A。功率耗散：P = IV = I²R = V²/R。
Kirchhoff’s rules: junction rule (ΣI_in = ΣI_out) and loop rule (ΣΔV = 0). Used to analyze complex circuits with resistors and batteries.
基尔霍夫定律：节点电流定律 (ΣI_in = ΣI_out) 与回路电压定律 (ΣΔV = 0)，用于分析电阻与电池组成的复杂电路。
RC circuits: charging q(t) = Cε(1 – e⁻ᵗ/ʳᶜ), discharging q(t) = Q₀e⁻ᵗ/ʳᶜ, time constant τ = RC.
RC 电路：充电 q(t) = Cε(1 – e⁻ᵗ/ʳᶜ)，放电 q(t) = Q₀e⁻ᵗ/ʳᶜ，时间常数 τ = RC。
10. Magnetic Fields & Forces | 磁场与磁力
Magnetic force on a moving charge: F = qv × B, magnitude F = |q|vB sinθ. Force on a current-carrying wire: F = I L × B.
运动电荷在磁场中的受力：F = qv × B，大小 F = |q|vB sinθ。载流导线受力：F = I L × B。
A charged particle moves in a uniform B field with circular path radius r = mv/(|q|B), cyclotron frequency ω = |q|B/m.
带电粒子在匀强磁场中做圆周运动，半径 r = mv/(|q|B)，回旋频率 ω = |q|B/m。
Biot-Savart law: dB = (μ₀/4π) (I dL × r̂)/r². Ampere’s law: ∮ B·dL = μ₀ I_enc. For a long straight wire, B = μ₀I/(2πr); solenoid, B = μ₀nI.
比奥-萨伐尔定律：dB = (μ₀/4π) (I dL × r̂)/r²。安培定律：∮ B·dL = μ₀ I_enc。长直导线 B = μ₀I/(2πr)；螺线管内部 B = μ₀nI。
11. Electromagnetic Induction | 电磁感应
Magnetic flux Φ_B = ∫ B·dA. Faraday’s law: ε = -dΦ_B/dt. Lenz’s law: induced current opposes the change in flux.
磁通量 Φ_B = ∫ B·dA。法拉第定律：ε = -dΦ_B/dt。楞次定律：感应电流阻碍磁通量的变化。
Motional EMF: ε = BLv for a rod moving perpendicular to a uniform B. Generators produce alternating EMF by rotating a coil in a magnetic field.
动生电动势：垂直于匀强磁场运动的棒产生 ε = BLv。发电机通过线圈在磁场中转动产生交变电动势。
Induced electric fields are non-conservative and circulate around changing magnetic flux, even without a conductor.
感应电场是非保守场，环绕变化的磁通量存在，即使没有导体。
12. Inductance & Maxwell’s Equations | 电感与麦克斯韦方程组
Self-inductance L = Φ_B/I. Induced EMF ε_L = -L dI/dt. Energy stored in an inductor: U = ½LI². Mutual inductance M describes flux linkage between two coils.
自感 L = Φ_B/I。自感电动势 ε_L = -L dI/dt。电感储能：U = ½LI²。互感 M 描述两线圈间的磁通耦合。
LR circuits: current growth I(t) = (ε/R)(1 – e^{-Rt/L}) and decay I(t) = I₀ e^{-Rt/L}, with time constant τ_L = L/R.
LR 电路：电流增长 I(t) = (ε/R)(1 – e^{-Rt/L})，衰减 I(t) = I₀ e^{-Rt/L}，时间常数 τ_L = L/R。
Maxwell’s equations unify electricity and magnetism, adding displacement current (I_d = ε₀ dΦ_E/dt) to Ampere’s law, predicting electromagnetic waves with speed c = 1/√(ε₀μ₀).
麦克斯韦方程组统一了电与磁，在安培定律中加入位移电流 (I_d = ε₀ dΦ_E/dt)，预言电磁波以 c = 1/√(ε₀μ₀) 传播。
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