📚 AP Physics C: Mechanics and Electricity & Magnetism Knowledge Review | AP 物理C：力学与电磁学知识梳理
AP Physics C is a calculus-based, college-level physics course divided into two separate exams: Mechanics and Electricity & Magnetism (E&M). Success in both requires a deep conceptual understanding paired with the ability to set up and solve differential and integral calculus problems. This review outlines the essential knowledge for each major topic, highlighting the equations, graphs, and physical reasoning you must master.
AP 物理C是一门基于微积分的大学水平物理课程，分为力学和电磁学两门独立考试。要在这两门考试中取得成功，不仅要深刻理解物理概念，还必须能够建立并求解涉及微分和积分的计算问题。本文梳理了每个主要模块的核心知识点，重点介绍必须掌握的方程、图像以及物理推理方法。
1. Kinematics | 运动学
Kinematics describes motion without regard to its causes. In AP Physics C, you must be comfortable with position, velocity, and acceleration as functions of time, and their relationships through differentiation and integration. The key vectors are position r(t), velocity v(t) = dr/dt, and acceleration a(t) = dv/dt. For constant acceleration, the standard equations apply: v = v₀ + at, x = x₀ + v₀t + ½at², and v² = v₀² + 2a(x − x₀). Be prepared to interpret motion graphs: slope of x-t gives velocity, area under v-t gives displacement, and slope of v-t gives acceleration.
运动学研究物体的运动而不考虑引起运动的原因。在AP 物理C中，你必须熟练地将位置、速度和加速度表示为时间的函数，并掌握它们之间通过求导和积分建立的关系。关键矢量包括位置 r(t)、速度 v(t) = dr/dt 和加速度 a(t) = dv/dt。对于匀加速运动，标准公式适用：v = v₀ + at，x = x₀ + v₀t + ½at²，以及 v² = v₀² + 2a(x − x₀)。要能够解读运动图像：x-t 图的斜率代表速度，v-t 图下的面积代表位移，v-t 图的斜率代表加速度。
For motion in two dimensions, treat the x- and y-components independently. Projectile motion is the classic example: aₓ = 0, aᵧ = −g. The range equation is often useful but only applies when launch and landing heights are equal. Use parametric equations and calculus to find velocity components, maximum height, and time of flight.
对于二维运动，需要将 x 和 y 方向独立处理。抛体运动是最经典的例子：aₓ = 0，aᵧ = −g。射程公式通常很有用，但只在发射高度与落地高度相同时才适用。要学会使用参数方程和微积分来求解速度分量、最大高度和飞行时间。
2. Newton’s Laws of Motion | 牛顿运动定律
Newton’s laws connect force and motion. The first law introduces inertia; the second law, ΣF = ma, is the core mathematical model; the third law addresses action-reaction pairs. Free-body diagrams are essential for identifying all forces acting on a body: weight (mg), normal force (N), tension (T), friction (fₖ or fₛ), springs (F = −kx), and any applied forces. Remember that static friction fₛ ≤ μₛN adjusts to prevent slipping, while kinetic friction fₖ = μₖN acts opposite to relative motion.
牛顿定律将力与运动联系起来。第一定律引入了惯性；第二定律 ΣF = ma 是核心数学模型；第三定律指出作用力与反作用力关系。受力图对于确定作用在物体上的所有力至关重要：重力 (mg)、法向力 (N)、张力 (T)、摩擦力 (fₖ 或 fₛ)、弹力 (F = −kx) 以及任何施加的外力。需要牢记：静摩擦力 fₛ ≤ μₛN 会自行调整以防止滑动，而动摩擦力 fₖ = μₖN 的方向与相对运动方向相反。
When solving dynamics problems, write the sum of forces in each direction, equate to mass times acceleration, and solve the resulting differential equations if acceleration is not constant. In uniform circular motion, the net force toward the center is m v²/r. Always define a coordinate system that simplifies the problem, such as aligning one axis along the acceleration direction.
在求解动力学问题时，要先写出各个方向上的合力，令其等于质量乘以加速度，并在加速度不恒定时求解相应的微分方程。在匀速圆周运动中，指向圆心的合力大小为 m v²/r。务必建立一个能简化问题的坐标系，例如将某一坐标轴与加速度方向对齐。
3. Work, Energy, and Power | 功、能和功率
The work done by a force is W = ∫ F · dr. For a constant force, this simplifies to W = F d cos θ. The work-energy theorem states that the net work equals the change in kinetic energy: W_net = ΔK = ½ m v² − ½ m v₀². Conservative forces (gravity, spring force, electrostatic force) have an associated potential energy function: ΔU = −W_conservative. Gravitational potential near Earth: U_g = mgy; universal: U_g = −G M m / r. Spring potential: Uₛ = ½ k x².
力所做的功定义为 W = ∫ F · dr。对于恒力，简化为 W = F d cos θ。功能定理指出，合力所做的净功等于动能的变化量：W_net = ΔK = ½ m v² − ½ m v₀²。保守力（重力、弹力、静电力）都具有与之对应的势能函数：ΔU = −W_conservative。近地重力势能：U_g = mgy；万有引力势能：U_g = −G M m / r。弹簧势能：Uₛ = ½ k x²。
Mechanical energy E = K + U is conserved when only conservative forces do work. If non-conservative forces act, W_nc = ΔE. Power is the rate at which work is done: P = dW/dt. For a constant force, instantaneous power can be written as P = F · v. Be comfortable deriving potential energy functions from a given force function using integration.
当只有保守力做功时，机械能 E = K + U 守恒。若存在非保守力做功，则 W_nc = ΔE。功率是做功的速率：P = dW/dt。对于恒力，瞬时功率可写为 P = F · v。应熟练掌握从一个给定的力函数通过积分推导出势能函数的方法。
4. Systems of Particles and Linear Momentum | 质点系与动量
The linear momentum of a particle is p = mv. For a system of particles, the total momentum is the vector sum of individual momenta. Newton’s second law in momentum form is ΣF = dp/dt. If the net external force is zero, total momentum is conserved. This principle is crucial for analyzing collisions and explosions. The impulse delivered by a force is J = ∫ F dt = Δp, which equals the area under a force-time graph.
质点的线动量定义为 p = mv。对于质点系，总动量等于各质点动量的矢量和。牛顿第二定律的动量形式为 ΣF = dp/dt。若系统所受合外力为零，则总动量守恒。这一原理对于分析碰撞和爆炸至关重要。力产生的冲量为 J = ∫ F dt = Δp，其在数值上等于力-时间图像下的面积。
Collisions are classified as elastic (kinetic energy conserved) or inelastic (some KE lost). In perfectly inelastic collisions, objects stick together and move with a common final velocity. The center of mass of a system moves as if all mass were concentrated there and all external forces acted at that point. Its position is r_cm = (Σ mᵢ rᵢ) / Σ mᵢ, and for continuous bodies, this becomes an integral.
碰撞分为弹性碰撞（动能守恒）和非弹性碰撞（部分动能损失）。在完全非弹性碰撞中，物体粘在一起并以共同的速度运动。系统的质心就像所有质量集中于该点、所有外力作用在该点一样运动。质心位置为 r_cm = (Σ mᵢ rᵢ) / Σ mᵢ，对于连续体，该式转化为积分形式。
5. Rotation | 转动
Rotational kinematics mirrors linear kinematics with angular displacement θ, angular velocity ω = dθ/dt, and angular acceleration α = dω/dt. For constant α: ω = ω₀ + αt, θ = θ₀ + ω₀t + ½ α t², ω² = ω₀² + 2α(θ − θ₀). The vector direction of ω and α follows the right-hand rule. The rotational inertia (moment of inertia) I = Σ mᵢ rᵢ² or ∫ r² dm depends on both mass distribution and axis of rotation. The parallel-axis theorem states I = I_cm + M d².
转动运动学与直线运动学一一对应，包括角位移 θ、角速度 ω = dθ/dt 和角加速度 α = dω/dt。对于恒定 α：ω = ω₀ + αt，θ = θ₀ + ω₀t + ½ α t²，ω² = ω₀² + 2α(θ − θ₀)。ω 和 α 的矢量方向由右手定则确定。转动惯量 I = Σ mᵢ rᵢ² 或 ∫ r² dm 既取决于质量分布，也取决于转轴。平行轴定理为 I = I_cm + M d²。
Torque τ = r × F, magnitude r F sin θ. Newton’s second law for rotation is Στ = I α. Rolling without slipping is a key condition: v_cm = R ω and a_cm = R α. Rotational kinetic energy is K_rot = ½ I ω². Total kinetic energy of a rolling object is K = ½ M v_cm² + ½ I_cm ω². Angular momentum L = r × p for a particle, and L = I ω for a rigid body about a fixed axis. If net external torque is zero, angular momentum is conserved.
力矩 τ = r × F，大小为 r F sin θ。转动情况下的牛顿第二定律为 Στ = I α。无滑滚动是一个关键条件：v_cm = R ω 且 a_cm = R α。转动动能为 K_rot = ½ I ω²。滚动体的总动能为 K = ½ M v_cm² + ½ I_cm ω²。质点角动量 L = r × p，定轴刚体的角动量则为 L = I ω。若合外力矩为零，则角动量守恒。
6. Oscillations and Gravitation | 振动与万有引力
Simple harmonic motion (SHM) occurs when the restoring force is proportional to displacement: F = −kx. This leads to a = −(k/m)x = −ω² x, so ω = √(k/m). The solution is x(t) = A cos(ωt + φ), with amplitude A and phase constant φ. Velocity and acceleration as functions of time are derived by differentiation. Energy in SHM continuously transforms between kinetic and potential: E_total = ½ k A² = ½ m v² + ½ k x². For a simple pendulum, ω = √(g/L); for a physical pendulum, ω = √(mgd/I).
简谐运动 (SHM) 发生在回复力与位移成正比时：F = −kx。由此可得 a = −(k/m)x = −ω² x，因此 ω = √(k/m)。位移的解为 x(t) = A cos(ωt + φ)，其中 A 为振幅，φ 为初相。速度和加速度作为时间的函数可通过求导得到。简谐运动中的能量在动能和势能之间持续转化：E_total = ½ k A² = ½ m v² + ½ k x²。对于单摆，ω = √(g/L)；对于物理摆，ω = √(mgd/I)。
Newton’s law of gravitation gives the force between two point masses: F = G M m / r². Gravitational potential energy for two-body systems is U = −G M m / r. The total mechanical energy in an elliptical orbit is E = −G M m / (2a), where a is the semi-major axis. Kepler’s laws describe planetary motion: elliptical orbits with the Sun at a focus, equal areas in equal times, and T² ∝ a³. Escape velocity is v_esc = √(2GM/R).
牛顿万有引力定律给出了两质点间的引力：F = G M m / r²。双体系统的引力势能为 U = −G M m / r。椭圆轨道中，总机械能为 E = −G M m / (2a)，其中 a 为半长轴。开普勒定律描述了行星运动：行星沿椭圆轨道运动，太阳位于焦点；行星与太阳的连线在相等时间内扫过相等面积；以及 T² ∝ a³。逃逸速度为 v_esc = √(2GM/R)。
7. Electrostatics | 静电学
Electric charge is quantized and conserved. Coulomb’s law describes the force between two point charges: F = (1/(4πϵ₀)) q₁ q₂ / r², with direction along the line joining them. The electric field E at a point is the force per unit test charge: E = F/q. For a point charge, E = (1/(4πϵ₀)) q / r². By the superposition principle, the net field is the vector sum of individual fields. Electric field lines start on positive charges and end on negative charges.
电荷是量子化的且守恒。库仑定律描述两点电荷间的作用力：F = (1/(4πϵ₀)) q₁ q₂ / r²，方向沿二者连线。电场强度 E 定义为单位试探电荷所受的力：E = F/q。对于点电荷，E = (1/(4πϵ₀)) q / r²。根据叠加原理，合场强为各个场强的矢量和。电场线始于正电荷、终于负电荷。
Electric flux Φ_E = ∫ E · dA. Gauss’s law states Φ_E = Q_enclosed / ϵ₀, and it is immensely useful for finding E-fields of symmetric charge distributions (spherical, cylindrical, planar). You must be able to choose an appropriate Gaussian surface and argue why E is constant and perpendicular or parallel on its faces. Common results: for a point charge, E falls as 1/r²; for an infinite line of charge, E ∝ 1/r; for an infinite plane, E = σ/(2ϵ₀), constant.
电通量 Φ_E = ∫ E · dA。高斯定理指出 Φ_E = Q_enclosed / ϵ₀，这对于求解具有对称性（球对称、柱对称、平面对称）的电荷分布的电场极为有用。你必须能够选取合适的高斯面，并论证为何在其表面上 E 的大小恒定且方向垂直或平行。常见的结果有：点电荷的电场按 1/r² 衰减；无限长线电荷的电场 ∝ 1/r；无限大带电平面的电场 E = σ/(2ϵ₀)，为恒量。
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, excess charge resides on the surface, and the external electric field just outside is perpendicular to the surface with magnitude E = σ/ϵ₀. The potential difference between two points is ΔV = −∫ E · dl. For a point charge, V = (1/(4πϵ₀)) q/r. Equipotential surfaces are perpendicular to field lines; a conductor’s surface is an equipotential.
在静电平衡条件下，导体内部的电场为零，过剩电荷分布在表面，导体外表面附近的电场方向垂直于表面且大小为 E = σ/ϵ₀。两点之间的电势差定义为 ΔV = −∫ E · dl。点电荷的电势为 V = (1/(4πϵ₀)) q/r。等势面处处与电场线垂直；导体的表面是一个等势面。
A capacitor consists of two conductors carrying equal and opposite charges ±Q, with capacitance C = Q/ΔV. For a parallel-plate capacitor, C = κϵ₀ A/d, where κ is the dielectric constant. Energy stored: U_C = ½ Q ΔV = ½ C (ΔV)² = Q²/(2C). When a dielectric is inserted, capacitance increases by a factor κ, and the electric field between plates decreases. Understand how inserting a dielectric with or without a connected battery changes Q, V, E, and stored energy.
电容器由两个带等量异号电荷 ±Q 的导体组成，电容定义为 C = Q/ΔV。对于平行板电容器，C = κϵ₀ A/d，其中 κ 为介电常数。储存的能量为：U_C = ½ Q ΔV = ½ C (ΔV)² = Q²/(2C)。当插入电介质时，电容增大为原来的 κ 倍，板间电场减弱。需理解在保持与电池连接或断开电池两种情况下，插入电介质将如何改变 Q、V、E 和储存的能量。
9. Electric Circuits | 电路
Current is I = dQ/dt. Resistance R = ρ L / A, where ρ is resistivity. Ohm’s law for a resistor: V = IR. Power dissipated is P = IV = I²R = V²/R. The electromotive force (emf) ε of a source is the work per unit charge done by a non-electrostatic field; terminal voltage is V = ε − Ir, with internal resistance r. Kirchhoff’s junction rule (ΣI_in = ΣI_out) and loop rule (ΣV = 0 around any closed loop) are essential for analyzing multiloop circuits.
电流 I = dQ/dt。电阻 R = ρ L / A，ρ 是电阻率。电阻的欧姆定律：V = IR。消耗的功率 P = IV = I²R = V²/R。电源的电动势 ε 是非静电力对单位电荷所做的功；路端电压为 V = ε − Ir，其中 r 为内阻。基尔霍夫节点定律（ΣI_in = ΣI_out）和回路定律（ΣV = 0 沿任一闭合回路）是分析多回路电路的基础。
Equivalent resistance: series R_eq = Σ Rᵢ, parallel 1/R_eq = Σ 1/Rᵢ. Capacitors store charge; equivalent capacitance: parallel C_eq = Σ Cᵢ, series 1/C_eq = Σ 1/Cᵢ. In an RC circuit, the charge on a capacitor follows q(t) = Q_final (1 − e^(−t/τ)) during charging, and q(t) = Q₀ e^(−t/τ) during discharging, where time constant τ = RC. Be able to derive these from the differential equation using separation of variables.
等效电阻：串联 R_eq = Σ Rᵢ，并联 1/R_eq = Σ 1/Rᵢ。电容器储存电荷；等效电容：并联 C_eq = Σ Cᵢ，串联 1/C_eq = Σ 1/Cᵢ。在 RC 电路中，充电过程电容器所带电荷满足 q(t) = Q_final (1 − e^(−t/τ))，放电过程满足 q(t) = Q₀ e^(−t/τ)，其中时间常数 τ = RC。要能通过分离变量法从微分方程推导出这些表达式。
10. Magnetic Fields and Forces | 磁场与磁力
The magnetic force on a moving charge is F_B = q v × B, with magnitude |q| v B sin θ. The force on a current-carrying wire segment in a uniform field is F_B = I L × B. Charged particles moving perpendicular to a uniform magnetic field undergo uniform circular motion with radius r = m v / (|q| B). Magnetic fields do no work because the force is always perpendicular to velocity.
运动电荷在磁场中所受的力为 F_B = q v × B，大小为 |q| v B sin θ。一段载流直导线在匀强磁场中所受的力为 F_B = I L × B。带电粒子垂直于匀强磁场运动时，将做匀速圆周运动，轨道半径 r = m v / (|q| B)。由于磁场力始终垂直于速度方向，磁场对运动电荷不做功。
The Biot-Savart law gives the magnetic field created by a current element: dB = (μ₀/(4π)) (I dl × r̂) / r². Ampere’s law, ∮ B · dl = μ₀ I_enclosed, is the magnetic analog of Gauss’s law and is crucial for fields with high symmetry. Standard results: straight wire B = μ₀ I / (2π r); center of a circular loop B = μ₀ I / (2R); interior of an ideal solenoid B = μ₀ n I, where n is turns per unit length.
毕奥-萨伐尔定律给出了电流元产生的磁场：dB = (μ₀/(4π)) (I dl × r̂) / r²。安培环路定理 ∮ B · dl = μ₀ I_enclosed 是磁学中的“高斯定律”，对于求解具有高对称性的磁场至关重要。标准结论：无限长直载流导线的磁场 B = μ₀ I / (2π r)；圆形载流线圈圆心处的磁场 B = μ₀ I / (2R)；理想长直螺线管内部的磁场 B = μ₀ n I，其中 n 为单位长度匝数。
11. Electromagnetic Induction | 电磁感应
Faraday’s law states that a changing magnetic flux induces an emf: ε = −dΦ_B / dt. For a coil of N turns, ε = −N dΦ_B/dt. Lenz’s law gives the direction of induced current: it opposes the change in flux. Motional emf for a conductor moving through a field: ε = −dΦ_B/dt = B L v when the conductor, its velocity, and the field are mutually perpendicular. Be prepared to calculate induced emf in rods, loops, and sliding bars.
法拉第定律指出，变化的磁通量会感应出电动势：ε = −dΦ_B / dt。对于 N 匝线圈，ε = −N dΦ_B/dt。楞次定律给出了感应电流的方向：它总是阻碍引起感应的磁通量变化。当导体在磁场中运动时，若导体、速度和磁场三者相互垂直，则动生电动势 ε = B L v。要求能够计算杆、线圈和滑轨中的感应电动势。
Inductance L is defined by ε_L = −L dI/dt. The inductance of an ideal solenoid: L = μ₀ n² A l. Energy stored in an inductor: U_L = ½ L I². In an RL circuit, current buildup follows I(t) = (ε/R) (1 − e^(−t/τ_L)) and decay I(t) = I₀ e^(−t/τ_L), where inductive time constant τ_L = L/R. The LC circuit exhibits simple harmonic oscillation of charge with angular frequency ω = 1/√(LC).
电感 L 由关系式 ε_L = −L dI/dt 定义。理想长直螺线管的自感系数为 L = μ₀ n² A l。电感器中储存的能量为 U_L = ½ L I²。在 RL 电路中，电流在增长过程中遵循 I(t) = (ε/R) (1 − e^(−t/τ_L))，衰减过程遵循 I(t) = I₀ e^(−t/τ_L)，其中电感时间常数 τ_L = L/R。LC 回路中电荷呈现简谐振动，角频率 ω = 1/√(LC)。
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