AP Physics C: Mechanics and Electromagnetism: Complete Knowledge Review | AP 物理C力学与电磁学：知识点全面梳理

📚 AP Physics C: Mechanics and Electromagnetism: Complete Knowledge Review | AP 物理C力学与电磁学：知识点全面梳理

AP Physics C: Mechanics and Electromagnetism is a calculus-based, college-level physics sequence that demands both conceptual understanding and mathematical fluency. This guide provides a thorough yet concise review of every essential topic, from kinematics to Maxwell’s equations, linking theory to typical problem-solving techniques.

AP 物理C力学与电磁学是一门基于微积分的大学水平物理课程，要求学生既具备概念理解又拥有数学运算的熟练度。本指南全面而精炼地梳理了从运动学到麦克斯韦方程组的每一个重要知识点，并将理论与典型解题技巧相结合。

1. Kinematics | 运动学

Kinematics describes motion in terms of position, velocity, and acceleration without reference to forces. Position vector r(t) = x(t) i + y(t) j; velocity v = dr/dt; acceleration a = dv/dt = d²r/dt². In one dimension with constant acceleration a, the standard equations apply: v = v₀ + a t, x = x₀ + v₀ t + ½ a t², v² = v₀² + 2a(x−x₀). For projectile motion, horizontal velocity is constant, vertical motion has a_y = −g; the trajectory is parabolic. Uniform circular motion involves centripetal acceleration a_c = v²/r toward the center; angular velocity ω = dθ/dt, and v = rω.

运动学在不涉及力的情况下描述位置、速度和加速度。位置矢量 r(t) = x(t) i + y(t) j；速度 v = dr/dt；加速度 a = dv/dt = d²r/dt²。在一维恒定加速度 a 下，标准方程适用：v = v₀ + a t, x = x₀ + v₀ t + ½ a t², v² = v₀² + 2a(x−x₀)。对于抛体运动，水平速度恒定，垂直方向加速度 a_y = −g，轨迹为抛物线。匀速圆周运动具有指向圆心的向心加速度 a_c = v²/r；角速度 ω = dθ/dt，且 v = rω。

---

2. Newton’s Laws of Motion | 牛顿运动定律

Newton’s First Law: an object at rest stays at rest, and an object in motion stays in motion with constant velocity unless acted upon by a net external force. Second Law: ΣF = ma (calculus form: ΣF = dp/dt). Third Law: forces come in action-reaction pairs, equal in magnitude and opposite in direction. Free-body diagrams are essential for isolating systems and identifying all forces: gravity (mg), normal force (N), tension (T), friction (static f_s ≤ μ_sN, kinetic f_k = μ_kN), spring force (Hooke’s law: F = −kx), and drag forces. Equilibrium occurs when ΣF = 0 and Στ = 0.

牛顿第一定律：若物体不受净外力作用，静止的物体保持静止，运动的物体以恒定速度运动。第二定律：ΣF = ma（微积分形式：ΣF = dp/dt）。第三定律：力以作用力与反作用力对出现，大小相等、方向相反。受力分析图对于隔离系统并识别所有力至关重要：重力 (mg)，法向力 (N)，拉力 (T)，摩擦力（静摩擦 f_s ≤ μ_sN，动摩擦 f_k = μ_kN），弹簧力（胡克定律：F = −kx）以及阻力。当 ΣF = 0 且 Στ = 0 时，物体处于平衡状态。

---

3. Work, Energy, and Power | 功、能量和功率

Work done by a force is W = ∫ F · dr. For a constant force along a straight line, W = F d cosθ. The work-energy theorem states W_net = ΔK = ½mv² − ½mv₀². Kinetic energy K = ½mv². Conservative forces (gravity, spring, electrostatic) have associated potential energy: gravitational U_g = mgy (near Earth), U_g = −G M m / r (universal); spring U_s = ½kx². Mechanical energy E = K + U is conserved when only conservative forces do work. Power P = dW/dt = F · v.

力所做的功为 W = ∫ F · dr。对于沿直线的恒力，W = F d cosθ。动能定理指出 W_net = ΔK = ½mv² − ½mv₀²。动能 K = ½mv²。保守力（重力、弹簧力、静电力）具有相应的势能：重力势能 U_g = mgy（近地表），U_g = −G M m / r（万有引力）；弹性势能 U_s = ½kx²。当只有保守力做功时，机械能 E = K + U 守恒。功率 P = dW/dt = F · v。

---

4. Systems of Particles and Linear Momentum | 质点系与线性动量

Linear momentum p = mv. For a system of particles, total momentum P = Σ m_i v_i = M_total v_cm. Impulse J = ∫ F dt = Δp. Conservation of momentum: if ΣF_ext = 0, then P is constant. Collisions can be elastic (kinetic energy conserved) or inelastic (some kinetic energy lost, perfectly inelastic means objects stick together). Center of mass: r_cm = (Σ m_i r_i) / M_total; velocity of cm v_cm = dr_cm/dt.

线性动量 p = mv。对于质点系，总动量 P = Σ m_i v_i = M_total v_cm。冲量 J = ∫ F dt = Δp。动量守恒：若 ΣF_ext = 0，则 P 恒定。碰撞分为弹性碰撞（动能守恒）和非弹性碰撞（部分动能损失；完全非弹性碰撞指物体粘在一起）。质心：r_cm = (Σ m_i r_i) / M_total；质心速度 v_cm = dr_cm/dt。

---

5. Rotation | 转动

Rotational kinematics parallels linear kinematics: θ = θ₀ + ω₀ t + ½α t², ω = ω₀ + α t, ω² = ω₀² + 2α(θ−θ₀). Torque τ = r × F, magnitude τ = r F sinθ. Moment of inertia I = Σ m_i r_i² = ∫ r² dm; parallel-axis theorem: I = I_cm + Md². Rotational dynamics: Στ = I α. Rotational kinetic energy K_rot = ½ I ω². Angular momentum L = Iω for rigid body about fixed axis; general L = r × p. Conservation of angular momentum: if Στ_ext = 0, L is constant. Rolling without slipping: v_cm = r ω.

转动运动学与直线运动学类似：θ = θ₀ + ω₀ t + ½α t²，ω = ω₀ + α t，ω² = ω₀² + 2α(θ−θ₀)。力矩 τ = r × F，大小 τ = r F sinθ。转动惯量 I = Σ m_i r_i² = ∫ r² dm；平行轴定理：I = I_cm + Md²。转动动力学：Στ = I α。转动动能 K_rot = ½ I ω²。绕固定轴刚体的角动量 L = Iω；一般形式 L = r × p。角动量守恒：若 Στ_ext = 0，则 L 恒定。纯滚动条件：v_cm = r ω。

---

6. Oscillations and Gravitation | 振动与万有引力

Simple harmonic motion (SHM) occurs when restoring force F = −kx. Equation of motion: d²x/dt² + (k/m) x = 0, yielding x(t) = A cos(ωt + φ) with ω = √(k/m). Period T = 2π/ω. Energy in SHM: total E = ½kA², constant. For a pendulum, for small angles, ω = √(g/L) for simple pendulum; physical pendulum ω = √(mgd/I). Universal gravitation: F = −(G M m / r²) r̂. Gravitational potential energy U = −G M m / r. Kepler’s laws: elliptical orbits, equal areas in equal times, T² ∝ a³. For circular orbits, v = √(GM/r).

当回复力 F = −kx 时，物体做简谐运动。运动方程：d²x/dt² + (k/m) x = 0，解为 x(t) = A cos(ωt + φ)，其中 ω = √(k/m)。周期 T = 2π/ω。简谐运动中的能量：总能量 E = ½kA²，保持恒定。对于单摆，在小角度下 ω = √(g/L)；物理摆 ω = √(mgd/I)。万有引力：F = −(G M m / r²) r̂。引力势能 U = −G M m / r。开普勒定律：椭圆轨道、相等时间扫过相等面积、T² ∝ a³。对于圆轨道，v = √(GM/r)。

---

7. Electrostatics | 静电场

Coulomb’s law: F = k q₁ q₂ / r² r̂, where k = 1/(4πε₀). Electric field E = F/q₀; for a point charge, E = k q / r² r̂. Gauss’s law: ∮ E · dA = Q_enc/ε₀. This is especially useful for symmetric charge distributions: spherical, cylindrical, planar. Electric potential V is defined such that ΔV = −∫ E · dl; for a point charge, V = k q / r. Potential energy of a system of charges: U = k q₁ q₂ / r. Relation: E = −∇V; in one dimension, E_x = −dV/dx.

库仑定律：F = k q₁ q₂ / r² r̂，其中 k = 1/(4πε₀)。电场 E = F/q₀；对于点电荷，E = k q / r² r̂。高斯定律：∮ E · dA = Q_enc/ε₀。这对于对称电荷分布尤为有用：球对称、柱对称、平面对称。电势 V 的定义为 ΔV = −∫ E · dl；对于点电荷，V = k q / r。电荷系的电势能：U = k q₁ q₂ / r。关系式：E = −∇V；在一维中，E_x = −dV/dx。

---

8. Conductors, Capacitors, and Dielectrics | 导体、电容器与电介质

In electrostatic equilibrium, the electric field inside a conductor is zero; any net charge resides on the surface; the electric field just outside a conductor is perpendicular to the surface with magnitude σ/ε₀. Capacitance C = Q / V. For a parallel plate capacitor, C = ε₀ A / d. Energy stored in a capacitor: U = ½ Q V = ½ C V² = Q²/(2C). When a dielectric with dielectric constant κ is inserted, capacitance becomes C = κ C₀; the electric field between plates is reduced by factor κ for fixed charge. Capacitors in series: 1/C_eq = Σ 1/C_i; in parallel: C_eq = Σ C_i.

在静电平衡下，导体内部电场为零；所有净电荷分布在表面；紧贴导体外表面的电场垂直于表面，大小为 σ/ε₀。电容 C = Q / V。对于平行板电容器，C = ε₀ A / d。电容器储存的能量：U = ½ Q V = ½ C V² = Q²/(2C)。当插入介电常数为 κ 的电介质时，电容变为 C = κ C₀；在电荷固定的情况下，板间电场减为原来的 1/κ。电容器串联：1/C_eq = Σ 1/C_i；并联：C_eq = Σ C_i。

---

9. Electric Circuits | 电路

Current I = dQ/dt. Ohm’s law: V = I R. Resistance R = ρ L / A. Power dissipated in a resistor: P = I V = I² R = V²/R. Kirchhoff’s rules: junction rule (Σ I_in = Σ I_out) based on charge conservation; loop rule (Σ V = 0 around any closed loop) based on energy conservation. RC circuits: charging q(t) = Q(1 − e^−t/τ), discharging q(t) = Q₀ e^−t/τ, where time constant τ = RC. Voltmeters and ammeters: ideal voltmeter has infinite resistance; ideal ammeter has zero resistance.

电流 I = dQ/dt。欧姆定律：V = I R。电阻 R = ρ L / A。电阻耗散的功率：P = I V = I² R = V²/R。基尔霍夫定律：节点电流定律 (Σ I_in = Σ I_out) 基于电荷守恒；回路电压定律（任意闭合回路 Σ V = 0）基于能量守恒。RC 电路：充电时 q(t) = Q(1 − e^−t/τ)，放电时 q(t) = Q₀ e^−t/τ，其中时间常数 τ = RC。电压表与电流表：理想电压表内阻无穷大；理想电流表内阻为零。

---

10. Magnetic Fields and Electromagnetism | 磁场与电磁学

Magnetic force on a moving charge: F = q v × B. Magnitude F = q v B sinθ, direction by right-hand rule. Force on a current-carrying wire: F = I L × B. The Biot-Savart law gives the magnetic field of a current element: dB = (μ₀/4π) (I dl × r̂) / r². Ampere’s law: ∮ B · dl = μ₀ I_enc. For a long straight wire, B = μ₀ I / (2πr); for a solenoid, B = μ₀ n I inside. Force between parallel currents: parallel currents attract, antiparallel repel. Motion of charged particles in uniform B: circular path with radius r = m v / (q B), cyclotron frequency ω = q B / m.

运动电荷在磁场中所受洛伦兹力：F = q v × B。大小 F = q v B sinθ，方向由右手定则确定。载流导线所受磁力：F = I L × B。毕奥-萨伐尔定律给出电流元产生的磁场：dB = (μ₀/4π) (I dl × r̂) / r²。安培环路定理：∮ B · dl = μ₀ I_enc。无限长直导线周围的磁场 B = μ₀ I / (2πr)；长直螺线管内部 B = μ₀ n I。平行电流之间的力：同向吸引，异向排斥。带电粒子在均匀磁场中的运动：做匀速圆周运动，半径 r = m v / (q B)，回旋频率 ω = q B / m。

---

11. Electromagnetic Induction | 电磁感应

Faraday’s law: induced emf ε = −dΦ_B/dt, where magnetic flux Φ_B = ∫ B · dA. Lenz’s law: the induced current opposes the change in flux. Motional emf: ε = B L v for a conductor moving perpendicularly in a magnetic field. Inductance L: ε_L = −L dI/dt. Self-inductance of a solenoid: L = μ₀ n² A l. Energy stored in an inductor: U = ½ L I². LR circuits: current growth I(t) = (ε/R)(1 − e^−t/τ) with τ = L/R; decay I(t) = I₀ e^−t/τ. Mutual inductance: ε₂ = −M dI₁/dt.

法拉第电磁感应定律：感应电动势 ε = −dΦ_B/dt，其中磁通量 Φ_B = ∫ B · dA。楞次定律：感应电流的磁场阻碍磁通量的变化。动生电动势：对于在磁场中垂直运动的导体，ε = B L v。电感系数 L：ε_L = −L dI/dt。长直螺线管的自感 L = μ₀ n² A l。电感器中储存的能量：U = ½ L I²。LR 电路：电流增长 I(t) = (ε/R)(1 − e^−t/τ)，时间常数 τ = L/R；衰减 I(t) = I₀ e^−t/τ。互感：ε₂ = −M dI₁/dt。

---

12. Maxwell’s Equations and Review | 麦克斯韦方程组与总结

Maxwell’s equations unify electricity and magnetism. In integral form: (1) Gauss’s law for electricity: ∮ E · dA = Q_enc/ε₀. (2) Gauss’s law for magnetism: ∮ B · dA = 0. (3) Faraday’s law: ∮ E · dl = −dΦ_B/dt. (4) Ampere-Maxwell law: ∮ B · dl = μ₀ (I_enc + ε₀ dΦ_E/dt). The displacement current term ε₀ dΦ_E/dt allows for the existence of electromagnetic waves. In a vacuum, the wave equation emerges with speed c = 1/√(μ₀ ε₀). Poynting vector S = (1/μ₀) E × B represents energy flux.

麦克斯韦方程组将电与磁统一起来。积分形式：(1) 电场高斯定律：∮ E · dA = Q_enc/ε₀。(2) 磁场高斯定律：∮ B · dA = 0。(3) 法拉第定律：∮ E · dl = −dΦ_B/dt。(4) 安培-麦克斯韦定律：∮ B · dl = μ₀ (I_enc + ε₀ dΦ_E/dt)。位移电流项 ε₀ dΦ_E/dt 使得电磁波的存在成为可能。在真空中，由此导出的波动方程给出波速 c = 1/√(μ₀ ε₀)。坡印廷矢量 S = (1/μ₀) E × B 代表能流密度。

Published by TutorHao | AP Physics C Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）