Pre-U CCEA 物理：公式定理速查手册

Pre-U CCEA 物理：公式定理速查手册

Pre-U CCEA Physics: Formula & Theorem Quick Reference Handbook

Pre-U 物理是一门要求很高、内容深入的课程，涵盖了从经典力学到现代物理学的广泛领域。对于 CCEA 考试局的学生来说，熟练掌握公式和定理不仅是应对考试的关键，更是建立扎实物理直觉的基础。本手册按照主题模块系统整理了 Pre-U CCEA 物理课程中的核心公式与定理，帮助你在复习时快速查阅、巩固记忆。

Pre-U Physics is a demanding and in-depth course that spans a broad range of topics from classical mechanics to modern physics. For CCEA examination board students, mastering formulas and theorems is not only essential for tackling exams but also fundamental to building a solid physical intuition. This handbook systematically organises the core formulas and theorems from the Pre-U CCEA Physics syllabus by topic module, helping you quickly reference and reinforce your memory during revision.

一、力学（Mechanics）

1. Mechanics

1.1 运动学（Kinematics）

1.1 Kinematics

运动学是力学的基础，描述物体的运动而不考虑引起运动的力。在 Pre-U 阶段，你需要掌握匀速直线运动、匀加速直线运动和抛体运动的定量描述方法。

Kinematics is the foundation of mechanics, describing the motion of objects without considering the forces that cause that motion. At the Pre-U level, you need to master the quantitative description of uniform motion, uniformly accelerated motion, and projectile motion.

匀速直线运动公式：v = s / t，其中 v 为速度，s 为位移，t 为时间。这是最基本的运动学关系，适用于速度恒定的情况。

Uniform motion formula: v = s / t, where v is velocity, s is displacement, and t is time. This is the most basic kinematic relationship, applicable when velocity is constant.

匀加速直线运动公式（SUVAT 方程）：这些方程是 Pre-U 物理的核心，在所有运动学问题中反复出现。五个变量为：s（位移）、u（初速度）、v（末速度）、a（加速度）、t（时间）。

Uniformly accelerated linear motion formulas (SUVAT equations): These equations are central to Pre-U Physics and appear repeatedly in kinematic problems. The five variables are: s (displacement), u (initial velocity), v (final velocity), a (acceleration), and t (time).

核心方程一：v = u + at —— 描述速度随时间的变化。
核心方程二：s = ut + ½at² —— 在已知初速度和加速度时计算位移。
核心方程三：v² = u² + 2as —— 当时间未知时非常有用。
核心方程四：s = ½(u + v)t —— 利用平均速度计算位移。

Equation 1: v = u + at — describes the change in velocity over time.
Equation 2: s = ut + ½at² — calculates displacement when initial velocity and acceleration are known.
Equation 3: v² = u² + 2as — particularly useful when time is unknown.
Equation 4: s = ½(u + v)t — calculates displacement using average velocity.

自由落体运动：在重力场中，加速度 a = g ≈ 9.81 m/s²（方向向下）。注意在建立坐标系时选择正方向，这将影响 u、v、a 和 s 的符号。

Free fall motion: In a gravitational field, acceleration a = g ≈ 9.81 m/s² (directed downwards). Pay attention to your choice of positive direction when setting up the coordinate system, as this will affect the signs of u, v, a, and s.

抛体运动：将运动分解为水平方向和竖直方向两个独立的分量。水平方向：匀速运动，vₓ = uₓ = u·cosθ，sₓ = uₓ·t。竖直方向：匀加速运动，使用 SUVAT 方程，a = -g（取向上为正）。总飞行时间由竖直运动决定，水平射程由水平速度乘以总时间得到。

Projectile motion: Resolve the motion into independent horizontal and vertical components. Horizontal: uniform motion, vₓ = uₓ = u·cosθ, sₓ = uₓ·t. Vertical: uniformly accelerated motion, applying the SUVAT equations with a = -g (taking upward as positive). The total time of flight is determined by the vertical motion, and the horizontal range is the horizontal velocity multiplied by the total time.

1.2 动力学（Dynamics）

1.2 Dynamics

动力学研究力与运动之间的关系。牛顿三大定律是整个经典力学的基石，Pre-U 考试中几乎所有力学问题都可以追溯到这三个定律。

Dynamics studies the relationship between forces and motion. Newton’s three laws are the cornerstone of all classical mechanics, and virtually every mechanics problem in Pre-U exams can be traced back to these three laws.

牛顿第一定律（惯性定律）：在没有净外力作用的情况下，物体将保持静止或匀速直线运动状态。这一定律引入了惯性的概念——质量是惯性的量度。

Newton’s First Law (Law of Inertia): An object will remain at rest or in uniform motion in a straight line unless acted upon by a net external force. This law introduces the concept of inertia — mass is a measure of inertia.

牛顿第二定律：F = ma，物体加速度的大小与所受合外力成正比，与物体质量成反比，加速度方向与合外力方向相同。这是 Pre-U 物理中最重要的方程之一。注意 F 是合外力（net force），而非单个力。

Newton’s Second Law: F = ma, the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass, with the acceleration in the same direction as the net force. This is one of the most important equations in Pre-U Physics. Note that F is the net force, not a single force.

牛顿第三定律（作用力与反作用力）：当物体 A 对物体 B 施加力时，物体 B 同时对物体 A 施加大小相等、方向相反的力。这两个力作用在不同物体上，因此不能相互抵消。

Newton’s Third Law (Action-Reaction): When object A exerts a force on object B, object B simultaneously exerts an equal and opposite force on object A. These two forces act on different objects and therefore cannot cancel each other out.

动量：p = mv，动量是矢量，方向与速度相同。动量守恒定律：在没有外力作用的系统中，总动量守恒。碰撞问题几乎总是用动量守恒来解决。

Momentum: p = mv, momentum is a vector quantity with the same direction as velocity. Law of Conservation of Momentum: in a system with no external forces, total momentum is conserved. Collision problems are almost always solved using momentum conservation.

冲量：I = F·Δt = Δp，冲量等于动量的变化量。在力-时间图像中，曲线下的面积等于冲量。

Impulse: I = F·Δt = Δp, impulse equals the change in momentum. In a force-time graph, the area under the curve equals the impulse.

1.3 功、能与功率（Work, Energy and Power）

1.3 Work, Energy and Power

能量是物理学中最核心的概念之一，在 Pre-U CCEA 课程中你需要熟练运用功-能原理和能量守恒来解决各类问题。

Energy is one of the most central concepts in physics, and in the Pre-U CCEA course you need to skilfully apply the work-energy principle and energy conservation to solve a variety of problems.

功：W = F·s·cosθ，其中 θ 是力与位移之间的夹角。功的单位是焦耳（J），1 J = 1 N·m。当力与位移方向相同时，W = Fs。当力与位移垂直时（如向心力），做功为零。

Work: W = F·s·cosθ, where θ is the angle between the force and displacement. The unit of work is the joule (J), 1 J = 1 N·m. When force and displacement are in the same direction, W = Fs. When force is perpendicular to displacement (e.g. centripetal force), the work done is zero.

动能：KE = ½mv²。动能定理：合外力对物体所做的总功等于物体动能的变化量，即 W_net = ΔKE。

Kinetic energy: KE = ½mv². Work-Kinetic Energy Theorem: the total work done by the net force on an object equals the change in its kinetic energy, i.e. W_net = ΔKE.

重力势能：PE_grav = mgh，其中 h 是相对于参考零点的高度。弹性势能：PE_elas = ½kx²，其中 k 是弹簧的劲度系数，x 是相对于平衡位置的位移。

Gravitational potential energy: PE_grav = mgh, where h is the height relative to a reference zero point. Elastic potential energy: PE_elas = ½kx², where k is the spring constant and x is the displacement from the equilibrium position.

能量守恒：在一个封闭系统中，总能量保持不变。在忽略摩擦和空气阻力的理想情况下，机械能守恒：KE + PE = 常数。当存在非保守力时，能量会转化为热能等其他形式。

Conservation of energy: In a closed system, total energy remains constant. In ideal situations neglecting friction and air resistance, mechanical energy is conserved: KE + PE = constant. When non-conservative forces are present, energy is converted into other forms such as thermal energy.

功率：P = W / t = F·v（当力和速度同向时）。功率的单位是瓦特（W），1 W = 1 J/s。效率 = 有用输出功率 / 总输入功率 × 100%。

Power: P = W / t = F·v (when force and velocity are in the same direction). The unit of power is the watt (W), 1 W = 1 J/s. Efficiency = useful output power / total input power × 100%.

1.4 圆周运动（Circular Motion）

1.4 Circular Motion

匀速圆周运动是 Pre-U 物理的重要组成部分，虽然速度大小不变，但方向时刻改变，因此是一种加速运动。

Uniform circular motion is an important component of Pre-U Physics. Although the speed magnitude remains constant, the direction changes continuously, making it an accelerated motion.

角速度：ω = Δθ / Δt = 2π / T = 2πf，其中 T 是周期，f 是频率。角速度的单位是 rad/s。

Angular velocity: ω = Δθ / Δt = 2π / T = 2πf, where T is the period and f is the frequency. The unit of angular velocity is rad/s.

线速度与角速度的关系：v = ωr。向心加速度：a = v² / r = ω²r，方向始终指向圆心。

Relationship between linear and angular velocity: v = ωr. Centripetal acceleration: a = v² / r = ω²r, always directed towards the centre of the circle.

向心力：F = ma = mv² / r = mω²r。向心力不是一个独立的”新力”，而是由其他力提供的，例如重力（行星轨道）、张力（绳系物体旋转）、摩擦力（车辆转弯）或法向反作用力。

Centripetal force: F = ma = mv² / r = mω²r. Centripetal force is not an independent “new force” but is provided by other forces, such as gravity (planetary orbits), tension (object swung on a string), friction (vehicles turning), or normal reaction force.

二、波与光学（Waves and Optics）

2. Waves and Optics

2.1 波的基本性质（Basic Wave Properties）

2.1 Basic Wave Properties

波是 Pre-U 物理中贯穿多个领域的核心概念——从声学到光学再到量子力学。理解波的基本性质对于后续学习至关重要。

Waves are a core concept in Pre-U Physics that cuts across multiple domains — from acoustics to optics to quantum mechanics. Understanding basic wave properties is crucial for subsequent learning.

波速方程：v = fλ，这是连接波速、频率和波长三个最基本参数的公式。波速由介质决定，频率由波源决定，波长则随之调整。

Wave speed equation: v = fλ, the formula connecting the three most fundamental parameters: wave speed, frequency, and wavelength. Wave speed is determined by the medium, frequency is determined by the source, and wavelength adjusts accordingly.

相位与相位差：相位 φ = 2π(x/λ) + φ₀。两个点之间的相位差 Δφ = 2π(Δx/λ)。相位差为 2π 的整数倍时两波同相，为 π 的奇数倍时反相。

Phase and phase difference: Phase φ = 2π(x/λ) + φ₀. The phase difference between two points is Δφ = 2π(Δx/λ). When the phase difference is an integer multiple of 2π, waves are in phase; when it is an odd multiple of π, they are in anti-phase.

波的强度：I ∝ A²，波的强度与振幅的平方成正比。对于点源发出的球面波，I ∝ 1/r²（平方反比定律）。

Wave intensity: I ∝ A², the intensity of a wave is proportional to the square of its amplitude. For spherical waves from a point source, I ∝ 1/r² (inverse square law).

2.2 叠加与干涉（Superposition and Interference）

2.2 Superposition and Interference

叠加原理：当两列或以上波在同一介质中相遇时，介质中任一点的合位移等于各列波单独在该点产生的位移的矢量和。

Principle of Superposition: When two or more waves meet in the same medium, the resultant displacement at any point is the vector sum of the displacements produced by each individual wave at that point.

双缝干涉（杨氏实验）：条纹间距 Δy = λD / d，其中 D 是缝到屏幕的距离，d 是双缝间距。亮纹条件：d·sinθ = nλ（n = 0, 1, 2, …）；暗纹条件：d·sinθ = (n + ½)λ。

Double-slit interference (Young’s experiment): Fringe spacing Δy = λD / d, where D is the distance from slits to screen and d is the slit separation. Bright fringe condition: d·sinθ = nλ (n = 0, 1, 2, …); dark fringe condition: d·sinθ = (n + ½)λ.

衍射光栅：d·sinθ = nλ，这与双缝干涉的亮纹条件形式相同。光栅能产生更锐利、更明亮的干涉条纹，因此广泛用于光谱分析。光栅常数 d 是相邻狭缝之间的距离。

Diffraction grating: d·sinθ = nλ, the same form as the double-slit bright fringe condition. Gratings produce sharper and brighter interference fringes and are therefore widely used in spectroscopy. The grating constant d is the distance between adjacent slits.

2.3 驻波（Stationary/Standing Waves）

2.3 Stationary/Standing Waves

驻波是由两列频率相同、振幅相同、传播方向相反的波叠加而成的特殊波形。在 Pre-U 考试中，弦上的驻波和空气柱中的驻波是最常出现的两种情形。

Stationary waves are a special wave pattern formed by the superposition of two waves of the same frequency and amplitude travelling in opposite directions. In Pre-U exams, stationary waves on strings and in air columns are the two most commonly encountered scenarios.

弦上的驻波：基频 f₁ = (1/2L)·√(T/μ)，其中 L 是弦长，T 是张力，μ 是线密度（单位长度的质量）。第 n 次谐波频率 f_n = n·f₁。两端固定的弦，节点在两端，波腹在中间。

Stationary waves on a string: Fundamental frequency f₁ = (1/2L)·√(T/μ), where L is string length, T is tension and μ is linear density (mass per unit length). The nth harmonic frequency is f_n = n·f₁. For a string fixed at both ends, nodes are at the ends and antinodes are in between.

空气柱中的驻波：一端闭口的空气柱，基频 f₁ = v/(4L)，只有奇数次谐波（f₁, 3f₁, 5f₁, …）。两端开口的空气柱，基频 f₁ = v/(2L)，所有谐波都存在。

Stationary waves in air columns: For a closed-at-one-end pipe, fundamental frequency f₁ = v/(4L), and only odd harmonics exist (f₁, 3f₁, 5f₁, …). For an open-at-both-ends pipe, f₁ = v/(2L), and all harmonics are present.

三、电学（Electricity）

3. Electricity

3.1 电路基础（Circuit Fundamentals）

3.1 Circuit Fundamentals

电学是 Pre-U CCEA 物理的重中之重，从基本电路分析到电磁感应，需要你建立清晰的物理图像和严谨的数学推理能力。

Electricity is one of the most heavily weighted topics in Pre-U CCEA Physics. From basic circuit analysis to electromagnetic induction, you need to develop a clear physical picture and rigorous mathematical reasoning.

电流：I = ΔQ / Δt，电流定义为单位时间内通过导体横截面的电荷量。单位是安培（A），1 A = 1 C/s。在金属导体中，I = nAvq，其中 n 是载流子密度，A 是横截面积，v 是漂移速度，q 是每个载流子的电荷量。

Current: I = ΔQ / Δt, current is defined as the charge passing through a cross-section of a conductor per unit time. The unit is the ampere (A), 1 A = 1 C/s. In metallic conductors, I = nAvq, where n is carrier density, A is cross-sectional area, v is drift velocity, and q is the charge per carrier.

电压（电位差）：V = W / Q，单位电荷通过元件时电场力所做的功。单位是伏特（V），1 V = 1 J/C。电动势（EMF）ε = W / Q，是电源将其他形式的能量转换为电能的能力。

Voltage (potential difference): V = W / Q, the work done by the electric field per unit charge passing through a component. The unit is the volt (V), 1 V = 1 J/C. Electromotive force (EMF) ε = W / Q, the ability of a source to convert other forms of energy into electrical energy.

欧姆定律：V = IR，适用于欧姆导体（电阻恒定）。电阻 R = ρL / A，其中 ρ 是电阻率，L 是长度，A 是横截面积。电阻随温度变化：R_T = R₀[1 + α(T – T₀)]，其中 α 是温度系数。

Ohm’s Law: V = IR, applicable to ohmic conductors (constant resistance). Resistance R = ρL / A, where ρ is resistivity, L is length and A is cross-sectional area. Resistance varies with temperature: R_T = R₀[1 + α(T – T₀)], where α is the temperature coefficient.

电功率：P = VI = I²R = V²/R。这些公式适用于任何电路元件。对于纯电阻电路，所有电能都转化为热能。

Electrical power: P = VI = I²R = V²/R. These formulas apply to any circuit component. For purely resistive circuits, all electrical energy is converted into thermal energy.

3.2 电路分析（Circuit Analysis）

3.2 Circuit Analysis

基尔霍夫定律：电流定律（KCL）：进入任一节点的电流之和等于离开该节点的电流之和，ΣI_in = ΣI_out。电压定律（KVL）：沿任何闭合回路的电位差代数和为零，ΣV = 0。

Kirchhoff’s Laws: Current Law (KCL): the sum of currents entering any junction equals the sum of currents leaving it, ΣI_in = ΣI_out. Voltage Law (KVL): the algebraic sum of potential differences around any closed loop is zero, ΣV = 0.

串联电路：R_total = R₁ + R₂ + R₃ + …，总电阻等于各电阻之和。电流处处相等，电压按电阻比例分配：V₁/V₂ = R₁/R₂。

Series circuits: R_total = R₁ + R₂ + R₃ + …, the total resistance equals the sum of individual resistances. Current is the same throughout, and voltage divides in proportion to resistance: V₁/V₂ = R₁/R₂.

并联电路：1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + …，总电阻的倒数等于各支路电阻倒数之和。各支路电压相等，电流按电阻的反比分配：I₁/I₂ = R₂/R₁。

Parallel circuits: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + …, the reciprocal of total resistance equals the sum of reciprocals of branch resistances. Each branch has the same voltage, and current divides inversely proportional to resistance: I₁/I₂ = R₂/R₁.

分压器：V_out = V_in × R₂/(R₁ + R₂)。这是电位器（可变电阻分压器）的工作原理，在传感器电路中非常常见。

Potential divider: V_out = V_in × R₂/(R₁ + R₂). This is the working principle of the potentiometer (variable potential divider), very common in sensor circuits.

内阻：对于一个有内阻 r 的电源，端电压 V = ε – Ir。当电路断开时（I = 0），端电压等于电动势。输出到外电阻 R 的功率 P = I²R，当 R = r 时功率最大（最大功率传递定理）。

Internal resistance: For a source with internal resistance r, terminal voltage V = ε – Ir. When the circuit is open (I = 0), terminal voltage equals the EMF. Power delivered to external resistance R is P = I²R, and maximum power transfer occurs when R = r (Maximum Power Transfer Theorem).

四、材料物理学（Materials Physics）

4. Materials Physics

4.1 应力与应变（Stress and Strain）

4.1 Stress and Strain

材料物理学研究物体在受力时的行为，是连接力学与工程应用的桥梁。Pre-U CCEA 课程要求你能够定量分析材料的弹性、塑性和断裂行为。

Materials physics studies the behaviour of objects under applied forces, serving as a bridge between mechanics and engineering applications. The Pre-U CCEA course requires you to be able to quantitatively analyse the elastic, plastic, and fracture behaviour of materials.

应力：σ = F / A，单位是帕斯卡（Pa），1 Pa = 1 N/m²。应力描述材料内部单位面积上承受的内力。

Stress: σ = F / A, with units of pascals (Pa), 1 Pa = 1 N/m². Stress describes the internal force per unit area within a material.

应变：ε = ΔL / L₀，无量纲量，通常以百分比表示。应变描述材料的相对变形程度。

Strain: ε = ΔL / L₀, a dimensionless quantity, often expressed as a percentage. Strain describes the relative degree of deformation of a material.

杨氏模量：E = σ / ε（在弹性范围内）。杨氏模量是材料刚度的量度，单位与应力相同（Pa）。对于给定材料，杨氏模量是常数，与物体的形状和尺寸无关。

Young’s modulus: E = σ / ε (within the elastic limit). Young’s modulus is a measure of material stiffness with the same units as stress (Pa). For a given material, Young’s modulus is a constant independent of the object’s shape and dimensions.

弹性极限与塑性变形：弹性极限是材料能够完全恢复原始形状的最大应力。超过弹性极限后，材料发生塑性变形（永久变形）。极限抗拉强度（UTS）是材料在断裂前能承受的最大应力。

Elastic limit and plastic deformation: The elastic limit is the maximum stress a material can sustain while still returning to its original shape. Beyond the elastic limit, the material undergoes plastic deformation (permanent deformation). Ultimate tensile strength (UTS) is the maximum stress a material can withstand before fracture.

力-伸长图像中的能量：弹性应变能 = ½FΔL（即力-伸长曲线下的面积，对弹性变形为三角形面积）。对于单位体积：应变能密度 = ½σε = ½Eε²。

Energy in force-extension graphs: Elastic strain energy = ½FΔL (the area under the force-extension curve, which is a triangle for elastic deformation). Per unit volume: strain energy density = ½σε = ½Eε².

五、场（Fields）

5. Fields

5.1 引力场（Gravitational Fields）

5.1 Gravitational Fields

引力场是 Pre-U 物理中场概念的第一个接触点。与 A-Level 不同，Pre-U 课程要求你使用微积分概念来理解引力场，并在径向场和均匀场之间建立清晰的区分。

Gravitational fields are the first contact point with the field concept in Pre-U Physics. Unlike A-Level, the Pre-U course requires you to use calculus concepts to understand gravitational fields and to clearly distinguish between radial and uniform fields.

牛顿万有引力定律：F = Gm₁m₂ / r²，其中 G = 6.67 × 10⁻¹¹ N·m²/kg²。引力遵循平方反比定律。

Newton’s Law of Universal Gravitation: F = Gm₁m₂ / r², where G = 6.67 × 10⁻¹¹ N·m²/kg². Gravitational force follows an inverse square law.

引力场强度：g = F / m，单位质量的物体在引力场中受到的力。单位是 N/kg，等价于 m/s²。点质量的径向场：g = GM / r²。在地球表面附近近似为均匀场，g ≈ 9.81 N/kg。

Gravitational field strength: g = F / m, the force per unit mass experienced by an object in a gravitational field. The unit is N/kg, equivalent to m/s². Radial field of a point mass: g = GM / r². Near the Earth’s surface, this is approximated as a uniform field with g ≈ 9.81 N/kg.

引力势：V_g = -GM / r，单位质量的引力势能。引力势是标量，在无穷远处取为零。两个质量之间的引力势能：U = -Gm₁m₂ / r。

Gravitational potential: V_g = -GM / r, the gravitational potential energy per unit mass. Gravitational potential is a scalar quantity, taken as zero at infinity. Gravitational potential energy between two masses: U = -Gm₁m₂ / r.

逃逸速度：v_esc = √(2GM / r)。当物体的动能足以克服引力势能时，它可以逃离天体的引力场。

Escape velocity: v_esc = √(2GM / r). When an object’s kinetic energy is sufficient to overcome the gravitational potential energy, it can escape the gravitational field of a celestial body.

5.2 电场（Electric Fields）

5.2 Electric Fields

电场与引力场有许多数学上的相似之处，但存在一个关键区别：电荷有正负两种，因此电场力可以是吸引力或排斥力。

Electric fields share many mathematical similarities with gravitational fields but with one key difference: charge comes in two types, positive and negative, so electric forces can be either attractive or repulsive.

库仑定律：F = kQ₁Q₂ / r²，其中 k = 1/(4πε₀) = 8.99 × 10⁹ N·m²/C²。ε₀ = 8.85 × 10⁻¹² F/m 是真空介电常数。

Coulomb’s Law: F = kQ₁Q₂ / r², where k = 1/(4πε₀) = 8.99 × 10⁹ N·m²/C². ε₀ = 8.85 × 10⁻¹² F/m is the permittivity of free space.

电场强度：E = F / q，定义为单位正电荷在电场中所受的力。单位是 N/C 或 V/m。点电荷的径向场：E = kQ / r²。匀强电场（平行板之间）：E = V / d。

Electric field strength: E = F / q, defined as the force per unit positive charge in an electric field. Units are N/C or V/m. Radial field of a point charge: E = kQ / r². Uniform field (between parallel plates): E = V / d.

电势：V_e = kQ / r，单位正电荷在电场中某点的电势能。电势是标量。电势差（电压）ΔV = ΔU / q。

Electric potential: V_e = kQ / r, the potential energy per unit positive charge at a point in an electric field. Electric potential is a scalar. Potential difference (voltage) ΔV = ΔU / q.

电场力做功：W = qΔV。对于在匀强电场中运动的电荷：F = qE，W = qEd。电子伏特：1 eV = 1.6 × 10⁻¹⁹ J，是一个能量单位。

Work done by electric field: W = qΔV. For a charge moving in a uniform electric field: F = qE, W = qEd. Electron-volt: 1 eV = 1.6 × 10⁻¹⁹ J, a unit of energy.

六、电磁感应（Electromagnetic Induction）

6. Electromagnetic Induction

6.1 磁通量与法拉第定律（Magnetic Flux and Faraday’s Law）

6.1 Magnetic Flux and Faraday’s Law

电磁感应是 Pre-U 物理中最具挑战性也最令人着迷的章节之一，它揭示了电与磁之间的深刻联系，是整个现代电力工业的理论基础。

Electromagnetic induction is one of the most challenging and fascinating chapters in Pre-U Physics. It reveals the profound connection between electricity and magnetism and serves as the theoretical foundation for the entire modern electrical power industry.

磁通量：Φ = BA·cosθ，其中 θ 是磁场方向与面积法线之间的夹角。单位是韦伯（Wb），1 Wb = 1 T·m²。当磁力线垂直于线圈平面时，Φ = BA（最大）；当磁力线平行于线圈平面时，Φ = 0。

Magnetic flux: Φ = BA·cosθ, where θ is the angle between the magnetic field direction and the area normal. The unit is the weber (Wb), 1 Wb = 1 T·m². When field lines are perpendicular to the coil plane, Φ = BA (maximum); when parallel, Φ = 0.

磁通链：对于 N 匝线圈，磁通链 = NΦ。法拉第电磁感应定律：感应电动势的大小等于磁通链随时间的变化率，ε = -N·dΦ/dt。负号表示楞次定律——感应电流产生的磁通量总是反抗引起感应的磁通量变化。

Flux linkage: For a coil of N turns, flux linkage = NΦ. Faraday’s Law of Electromagnetic Induction: the magnitude of the induced EMF equals the rate of change of flux linkage, ε = -N·dΦ/dt. The negative sign represents Lenz’s Law — the induced current produces a flux that always opposes the change in flux that caused it.

楞次定律：感应电流的方向总是使得它所产生的磁通量反抗引起感应的磁通量变化。这是能量守恒在电磁学中的体现——如果感应电流加强了原变化，系统就会获得无限能量。

Lenz’s Law: The direction of the induced current is always such that the flux it produces opposes the change in flux that caused the induction. This is the manifestation of energy conservation in electromagnetism — if the induced current reinforced the original change, the system would gain unlimited energy.

6.2 交流电与变压器（AC and Transformers）

6.2 AC and Transformers

交流发电机：当线圈在均匀磁场中以角速度 ω 旋转时，产生的正弦交变电动势 ε = NBAω·sin(ωt)。峰值电动势 ε₀ = NBAω。

AC generator: When a coil rotates with angular velocity ω in a uniform magnetic field, the sinusoidal alternating EMF produced is ε = NBAω·sin(ωt). Peak EMF ε₀ = NBAω.

RMS 值：对于正弦交流电，V_rms = V₀ / √2，I_rms = I₀ / √2。RMS 值是与等效直流电产生相同热效应的交流电值。

RMS values: For sinusoidal AC, V_rms = V₀ / √2, I_rms = I₀ / √2. The RMS value is the AC value that produces the same heating effect as an equivalent DC current.

理想变压器：V_s / V_p = N_s / N_p，I_s / I_p = N_p / N_s。输入功率等于输出功率：V_p·I_p = V_s·I_s（假设 100% 效率）。变压器只能工作在交流电下，因为需要变化的磁通量来感应电动势。

Ideal transformer: V_s / V_p = N_s / N_p, I_s / I_p = N_p / N_s. Input power equals output power: V_p·I_p = V_s·I_s (assuming 100% efficiency). Transformers only work with AC because a changing magnetic flux is required to induce EMF.

七、原子与核物理（Atomic and Nuclear Physics）

7. Atomic and Nuclear Physics

7.1 光量子与光电效应（Photons and Photoelectric Effect）

7.1 Photons and Photoelectric Effect

光电效应是建立量子力学的重要实验基础，也是 Pre-U 考试的高频考点。它揭示了光的粒子性，不能用经典波动理论解释。

The photoelectric effect is an important experimental foundation for quantum mechanics and a high-frequency topic in Pre-U exams. It reveals the particle nature of light and cannot be explained by classical wave theory.

光子能量：E = hf = hc/λ，其中 h = 6.63 × 10⁻³⁴ J·s 是普朗克常数，c = 3.00 × 10⁸ m/s 是真空中光速。光电效应方程：hf = φ + KE_max，其中 φ 是功函数（从金属表面释放一个电子所需的最小能量），KE_max 是逸出光电子的最大动能。

Photon energy: E = hf = hc/λ, where h = 6.63 × 10⁻³⁴ J·s is Planck’s constant and c = 3.00 × 10⁸ m/s is the speed of light in a vacuum. Photoelectric equation: hf = φ + KE_max, where φ is the work function (minimum energy to release an electron from a metal surface) and KE_max is the maximum kinetic energy of the emitted photoelectron.

阈频率与截止电压：f₀ = φ / h，低于阈频率的光无论强度多大都不会产生光电子。阻止电压 V_s 使得 KE_max = eV_s，因此 V_s = (hf – φ)/e。

Threshold frequency and stopping voltage: f₀ = φ / h, light below the threshold frequency will not produce photoelectrons regardless of intensity. The stopping voltage V_s satisfies KE_max = eV_s, therefore V_s = (hf – φ)/e.

7.2 原子能级与光谱（Atomic Energy Levels and Spectra）

7.2 Atomic Energy Levels and Spectra

能级跃迁：当电子从高能级 E₂ 跃迁到低能级 E₁ 时，发射光子：hf = ΔE = E₂ – E₁。当电子吸收光子从低能级跃迁到高能级时，需要光子能量精确匹配能级差。

Energy level transitions: When an electron transitions from a higher energy level E₂ to a lower level E₁, a photon is emitted: hf = ΔE = E₂ – E₁. For an electron to absorb a photon and transition to a higher level, the photon energy must precisely match the energy difference.

氢原子能级：E_n = -13.6 / n² eV（n = 1, 2, 3, …）。这是从玻尔模型推导出的公式，能够准确预测氢原子的发射和吸收光谱。

Hydrogen energy levels: E_n = -13.6 / n² eV (n = 1, 2, 3, …). This formula, derived from the Bohr model, accurately predicts the emission and absorption spectra of hydrogen.

线状光谱：每种元素都有独特的光谱特征线，这是原子能级量子化的直接证据。莱曼系（n→1，紫外）、巴尔末系（n→2，可见光）和帕邢系（n→3，红外）是氢原子的三大谱系。

Line spectra: Each element has its own unique set of spectral lines, providing direct evidence for quantised atomic energy levels. The Lyman series (n→1, ultraviolet), Balmer series (n→2, visible), and Paschen series (n→3, infrared) are the three major series for hydrogen.

7.3 核物理（Nuclear Physics）

7.3 Nuclear Physics

核物理研究原子核的结构、性质和变化，在 Pre-U 课程中涵盖放射性衰变、核反应和质能关系等核心内容。

Nuclear physics studies the structure, properties, and transformations of atomic nuclei. In the Pre-U course, it covers core topics including radioactive decay, nuclear reactions, and mass-energy relationships.

质量亏损与结合能：原子核的质量总是小于其组成核子（质子和中子）的质量之和，这个差值就是质量亏损 Δm。结合能 E_binding = Δm·c²，即将原子核完全分解为单个核子所需的能量。每个核子的结合能越大，原子核越稳定。

Mass defect and binding energy: The mass of a nucleus is always less than the sum of the masses of its constituent nucleons (protons and neutrons); this difference is the mass defect Δm. Binding energy E_binding = Δm·c², the energy required to completely separate a nucleus into individual nucleons. The greater the binding energy per nucleon, the more stable the nucleus.

放射性衰变定律：N = N₀e^{-λt}，其中 N 是 t 时刻剩余的原子核数，N₀ 是初始数量，λ 是衰变常数。半衰期 T_{½} = ln2/λ = 0.693/λ。活度 A = λN = A₀e^{-λt}，单位是贝克勒尔（Bq），1 Bq = 1 次衰变/秒。

Radioactive decay law: N = N₀e^{-λt}, where N is the number of nuclei remaining at time t, N₀ is the initial number, and λ is the decay constant. Half-life T_{½} = ln2/λ = 0.693/λ. Activity A = λN = A₀e^{-λt}, with units of becquerels (Bq), 1 Bq = 1 decay per second.

八、备考策略与常见误区

8. Exam Strategies and Common Pitfalls

掌握了以上所有公式和定理后，如何在考试中有效运用它们才是成功的关键。以下是 CCEA Pre-U 物理考试中需要特别注意的事项。

After mastering all the formulas and theorems above, the key to success lies in how effectively you apply them in the exam. Here are some specific points to watch out for in CCEA Pre-U Physics exams.

单位与量纲检查：在代入数值计算前，始终检查所有物理量是否使用了 SI 单位。一个快速验证方法是检查方程两边的量纲是否一致——如果不一致，你的推导必然有误。例如，½mv² 的量纲是 kg·(m/s)² = kg·m²/s² = J，与能量的量纲一致。

Unit and dimensional checks: Before substituting numerical values, always check that all quantities are in SI units. A quick verification method is to check that the dimensions on both sides of an equation are consistent — if they are not, your derivation must contain an error. For example, the dimensions of ½mv² are kg·(m/s)² = kg·m²/s² = J, consistent with energy.

矢量方向的符号约定：在力学和场问题中，建立一个清晰的符号约定并在整个计算过程中保持一致。对于抛体运动，通常取向上为正，此时加速度 a = -g。对于电场，电势沿电场线方向降低。

Sign conventions for vectors: In mechanics and field problems, establish a clear sign convention and maintain it consistently throughout the calculation. For projectile motion, it is common to take upward as positive, giving acceleration a = -g. For electric fields, potential decreases in the direction of electric field lines.

图像分析：Pre-U CCEA 考试非常重视图像分析能力。你需要能够从力-位移图像中求功（面积），从速度-时间图像中求加速度（斜率）和位移（面积），从应力-应变图像中求杨氏模量（初始线性区域的斜率），以及从力-伸长图像中求应变能（面积）。

Graph analysis: CCEA Pre-U exams place a strong emphasis on graph analysis skills. You need to be able to find work (area) from force-displacement graphs, acceleration (slope) and displacement (area) from velocity-time graphs, Young’s modulus (gradient of the initial linear region) from stress-strain graphs, and strain energy (area) from force-extension graphs.

物理推理的完整性：在解释题中，仅仅给出最终陈述是不够的。你需要展示完整的推理链条：从基本定律出发，说明各物理量之间的关系，最后得出结论。使用”因为…所以…因此…”的逻辑结构可以确保你的答案获得满分。

Completeness of physical reasoning: In explanation questions, simply stating the final conclusion is insufficient. You need to demonstrate a complete chain of reasoning: starting from fundamental laws, explaining the relationships between quantities, and finally arriving at the conclusion. Using a “because… therefore… hence…” logical structure ensures your answer scores full marks.

希望这本公式定理速查手册能够成为你复习 Pre-U CCEA 物理的得力助手。记住，公式是工具，理解是根本。在记忆公式的同时，深入理解每个公式背后的物理意义和适用条件，你将在考试中游刃有余。

We hope this Formula and Theorem Quick Reference Handbook serves as a valuable companion in your Pre-U CCEA Physics revision. Remember, formulas are tools, but understanding is fundamental. While memorising the formulas, deeply understand the physical meaning and conditions of applicability behind each one, and you will navigate your exams with confidence and ease.