📚 Year 13 SQA Physics: Formula and Theorem Quick-Reference Handbook | SQA 物理（Year 13）公式定理速查手册
This handbook distils the essential formulas, laws, and theorems required for the SQA Advanced Higher Physics course (Year 13). It is designed as a fast revision tool, linking concepts across mechanics, electromagnetism, waves, and quantum phenomena. Each entry states the formula, explains the symbols, and gives a brief comment on its application or derivation.
本手册浓缩了 SQA 高级物理（Year 13）课程必需的核心公式、定律与定理，适合作为快速回顾工具。内容覆盖力学、电磁学、波动和量子现象，每个条目均列出公式、解释符号含义，并简要说明其应用或推导背景。
1. Equations of Motion with Constant Acceleration | 匀加速运动方程
These four equations describe linear motion with uniform acceleration a. They are the foundation of kinematics and are used extensively in projectile and rectilinear motion problems.
这组四个方程描述了加速度 a 恒定的直线运动，是运动学的基础，广泛用于抛物线与直线运动问题。
v = u + at
v: final velocity, u: initial velocity, a: acceleration, t: time
v：末速度，u：初速度，a：加速度，t：时间
s = ut + ½ at²
s: displacement, signs of u and a depend on direction chosen as positive.
s：位移，u 和 a 的符号依赖于所选的正方向。
s = ½ (u + v) t
Useful when acceleration is constant but not directly given.
当加速度恒定但未直接给定时非常有用。
v² = u² + 2as
Time-independent form; often used in energy-based problems.
不含时间的形式，常用于能量相关问题。
2. Newton’s Laws and Impulse | 牛顿定律与冲量
Newton’s laws govern the relationship between force and motion. The second law is central to dynamics; the concept of impulse links force to change in momentum.
牛顿定律描述了力与运动的关系，第二定律是动力学的核心；冲量概念则将力与动量变化联系起来。
F = dp/dt (net force = rate of change of momentum)
For constant mass, this simplifies to F = ma. In Advanced Higher, you must be able to handle situations where mass changes, e.g. rocket motion.
当质量恒定时简化为 F = ma。在高级物理中需处理质量变化的情形，如火箭运动。
Impulse = ∫ F dt = Δp = mΔv
Impulse equals area under a force–time graph and is a vector quantity.
冲量等于力－时间图下的面积，是矢量。
3. Circular Motion & Gravitation | 圆周运动与引力
An object moving in a circle at constant speed still has an acceleration directed towards the centre. Gravitational force provides the centripetal force for satellites and planetary orbits.
匀速圆周运动的物体仍具有指向圆心的加速度。引力为人造卫星和行星轨道提供向心力。
a = v²/r = ω²r
a: centripetal acceleration, v: linear speed, ω: angular speed (rad s⁻¹), r: radius.
a：向心加速度，v：线速率，ω：角速率 (rad s⁻¹)，r：半径。
F = m v²/r = mω²r
Centripetal force is the resultant force towards the centre, not a separate ‘new’ force.
向心力是指向圆心的合力，并非一种新的独立力。
F = G M m / r²
Newton’s law of universal gravitation. G = 6.674 × 10⁻¹¹ N m² kg⁻².
牛顿万有引力定律，G = 6.674 × 10⁻¹¹ N m² kg⁻²。
T² = (4π²/GM) r³
Kepler’s third law derived from equating gravitational and centripetal force.
由引力与向心力相等导出的开普勒第三定律。
4. Simple Harmonic Motion (SHM) | 简谐运动
SHM occurs when the restoring force is proportional to the negative of displacement. The motion is sinusoidal and is characterised by amplitude A, angular frequency ω, and period T.
当回复力与位移负值成正比时发生简谐运动，其运动规律呈正弦形式，由振幅 A、角频率 ω 和周期 T 描述。
F = -k x
k: force constant (e.g. spring constant). This leads to a = – (k/m) x.
k：力常数（如弹簧劲度系数），从而有 a = – (k/m) x。
d²x/dt² = – ω² x
The defining differential equation of SHM. ω = 2πf = 2π/T.
SHM 的定义微分方程，ω = 2πf = 2π/T。
x = A sin(ωt + φ) or x = A cos(ωt + φ)
Solutions to the SHM equation; φ is the phase constant.
SHM 方程的解，φ 为初相位。
v = ± ω √(A² – x²), v_max = ωA
Velocity at a given displacement; the maximum speed occurs at the equilibrium position.
给定位移下的速度；最大速率发生在平衡位置。
a = – ω² x, a_max = ω²A
Acceleration is proportional to displacement and opposite in direction.
加速度与位移成正比，方向相反。
T = 2π √(m/k) for mass-spring system
m: mass, k: spring constant.
m：质量，k：弹簧劲度系数。
T = 2π √(L/g) for simple pendulum (small angles)
L: length of pendulum, g: acceleration due to gravity.
L：摆长，g：重力加速度。
5. Damped and Forced Oscillations, Resonance | 阻尼振动、受迫振动与共振
Damping removes energy from an oscillating system. When a periodic driving force is applied, resonance occurs if the driving frequency matches the natural frequency of the system.
阻尼使振动系统能量耗散。当施加周期性驱动力且其频率等于系统固有频率时，发生共振。
For light damping, the amplitude decays exponentially. The logarithmic decrement is used to quantify damping: δ = ln(xₙ/xₙ₊₁).
弱阻尼时振幅呈指数衰减，对数减缩 δ = ln(xₙ/xₙ₊₁) 用于量化阻尼。
At resonance, amplitude is maximum. Sharpness of resonance is related to the quality factor Q = ω₀/Δω, where Δω is the bandwidth at half-power points.
共振时振幅最大，共振的尖锐程度与品质因子 Q = ω₀/Δω 有关，其中 Δω 是半功率点处的带宽。
6. Waves and Interference | 波与干涉
Waves transfer energy without net particle displacement. They are described by the wave equation and exhibit superposition leading to interference and standing waves.
波传递能量而无介质的整体移动，由波动方程描述，叠加原理导致干涉和驻波。
v = f λ
v: wave speed, f: frequency, λ: wavelength.
v：波速，f：频率，λ：波长。
y = A sin(ωt – kx)
Equation of a travelling wave. k = 2π/λ is the wave number.
行波方程，k = 2π/λ 为波数。
Path difference = mλ ⇒ constructive interference (maxima)
Path difference = (m + ½)λ ⇒ destructive interference (minima)
m: integer (0, 1, 2…). Used in Young’s double-slit and grating experiments.
m 为整数 (0, 1, 2…)，用于杨氏双缝和光栅实验。
d sinθ = mλ
Grating equation for normal incidence, where d is the slit spacing and θ is the angle to the m-th order maximum.
正入射时的光栅方程，d 为缝间距，θ 为第 m 级亮纹角位置。
Δx = λL / d (Young’s double-slit fringe separation for small angles)
Δx: fringe spacing, L: distance to screen, d: slit separation.
Δx：条纹间距，L：到屏幕的距离，d：双缝间距。
7. Standing Waves and Harmonics | 驻波与谐波
Standing waves form when two identical travelling waves moving in opposite directions superpose. They are characterised by nodes (zero amplitude) and antinodes (maximum amplitude).
两列相同但传播方向相反的波叠加形成驻波，其特征是波节（振幅为零）和波腹（振幅最大）。
For a string fixed at both ends: L = n λ/2, where n = 1, 2, 3…. Frequencies are f_n = n v/(2L).
对于两端固定的弦：L = n λ/2，n = 1, 2, 3…，频率 f_n = n v/(2L)。
For a pipe open at both ends: L = n λ/2 (same harmonic pattern). For a pipe closed at one end: L = n λ/4 with odd n only (1, 3, 5…).
两端开口管：L = n λ/2（同弦）；一端闭管：L = n λ/4，n 只能取奇数 (1, 3, 5…)。
8. Electric Fields and Forces | 电场与力
Electric fields describe the force experienced by a unit positive charge. Key formulas relate field, potential, and force for both point charges and uniform fields.
电场描述单位正电荷所受的力，核心公式涉及点电荷和匀强电场中场、势与力的关系。
F = QE
Force on a charge Q in an electric field E.
电荷 Q 在电场 E 中所受的力。
E = V / d (uniform field between parallel plates)
V: potential difference, d: plate separation.
V：电势差，d：板间距。
E = k Q / r² (radial field due to point charge)
where k = 1/(4πε₀) ≈ 8.99 × 10⁹ N m² C⁻². ε₀: permittivity of free space.
其中 k = 1/(4πε₀) ≈ 8.99 × 10⁹ N m² C⁻²，ε₀ 为真空介电常数。
V = k Q / r (electric potential at distance r from point charge)
Potential is a scalar; work done in moving a charge q between two points = qΔV.
电势是标量；两点间移动电荷 q 所需功 = qΔV。
C = Q / V and C = ε₀ A / d (parallel-plate capacitor)
C: capacitance. Energy stored in a capacitor: U = ½ QV = ½ CV² = ½ Q²/C.
C：电容。电容器储能：U = ½ QV = ½ CV² = ½ Q²/C。
9. Magnetic Fields and Motion of Charged Particles | 磁场与带电粒子运动
A magnetic field exerts a force on moving charges and current-carrying conductors. The direction is given by the right-hand rule for the cross product.
磁场对运动电荷和载流导体施加力，方向由叉积的右手定则确定。
F = B I L sinθ (force on a straight current-carrying wire)
θ: angle between wire and magnetic field B.
θ：导线与磁场 B 的夹角。
F = q v B sinθ (Lorentz force on a moving charge)
When motion is perpendicular to uniform B, the charge follows a circular path: r = mv/(qB).
当运动方向垂直于匀强磁场时，电荷做圆周运动：r = mv/(qB)。
10. Electromagnetic Induction | 电磁感应
Faraday’s and Lenz’s laws describe how a changing magnetic flux induces an emf. This is the principle behind generators, transformers, and many sensors.
法拉第定律和楞次定律描述了变化的磁通量如何感应出电动势，这是发电机、变压器及许多传感器的基础原理。
ε = – dΦ/dt
Induced emf equals the negative rate of change of magnetic flux Φ = BA cosθ. The negative sign (Lenz’s law) indicates the induced current opposes the change in flux.
感应电动势等于磁通量 Φ = BA cosθ 变化率的负值。负号（楞次定律）表示感应电流反抗磁通量的变化。
For a conductor of length L moving perpendicular to B with speed v: ε = B L v.
对于长度为 L 的导体垂直于磁场以速度 v 运动：ε = B L v。
V_s / V_p = N_s / N_p (ideal transformer)
Turns ratio equation. For an ideal transformer, power input = power output, so I_p V_p = I_s V_s.
理想变压器匝数比方程，且输入功率等于输出功率：I_p V_p = I_s V_s。
11. Photons, Photoelectric Effect, and Wave-Particle Duality | 光子、光电效应与波粒二象性
The photoelectric effect provides evidence for the particle nature of light. Einstein’s equation relates photon energy to work function and maximum kinetic energy. The de Broglie hypothesis extends wave–particle duality to matter.
光电效应为光的粒子性提供了证据，爱因斯坦方程联系光子能量、功函数与最大动能。德布罗意假说将波粒二象性推广到实物粒子。
E = h f = h c / λ
h: Planck constant (6.63 × 10⁻³⁴ J s), f: frequency, λ: wavelength.
h：普朗克常数 (6.63 × 10⁻³⁴ J s)，f：频率，λ：波长。
h f = φ + ½ m v²_max (Einstein’s photoelectric equation)
φ: work function of the metal. The stopping potential V_s is given by e V_s = ½ m v²_max.
φ：金属的功函数，截止电压满足 e V_s = ½ m v²_max。
λ = h / p (de Broglie wavelength)
p: momentum = mv. Electrons can exhibit diffraction, confirming their wave nature.
p：动量 = mv；电子可发生衍射，证实其波动性。
12. Special Relativity – Time Dilation and Length Contraction | 狭义相对论 – 时间膨胀与长度收缩
At speeds approaching the speed of light c, measurement of time and length differs between inertial frames. These effects are described by the Lorentz factor γ.
当速度接近光速 c 时，不同惯性系中时间和长度的测量不同，由洛伦兹因子 γ 描述。
γ = 1 / √(1 – v²/c²)
v: relative speed between two inertial frames, c = 3.00 × 10⁸ m s⁻¹.
v：两惯性系相对速度，c = 3.00 × 10⁸ m s⁻¹。
t’ = γ t₀ (time dilation)
The proper time t₀ is the time interval measured in the frame where the events occur at the same location. The dilated time t’ is always longer than the proper time.
原时 t₀ 是在事件发生地静止的参考系中测得的时间间隔，膨胀后的时间 t’ 总是大于原时。
L’ = L₀ / γ (length contraction)
The proper length L₀ is the length measured in the rest frame of the object. The contracted length L’ is measured by an observer moving relative to the object.
原长 L₀ 是在物体静止的参考系中测得的长度，收缩后的长度 L’ 由相对物体运动的观察者测得。
The relativistic momentum is p = γ m₀ v and total energy E = γ m₀ c² with rest energy E₀ = m₀ c². The kinetic energy is KE = (γ – 1) m₀ c².
相对论动量为 p = γ m₀ v，总能量 E = γ m₀ c²，静能 E₀ = m₀ c²，动能 KE = (γ – 1) m₀ c²。
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