📚 A-Level Physics Insert 3 Jan22: Formula Derivations | A-Level物理配套公式表3 Jan22：公式推导
The A-Level Physics Insert 3 (Jan22) provides a comprehensive list of essential formulas for exams. Understanding the derivations behind these equations not only deepens your grasp of physics but also prepares you for questions requiring justification. This article walks through the derivations of key formulas from mechanics, waves, electricity, and nuclear physics, as featured in the insert.
A-Level物理公式表3（2022年1月版）为考试提供了全面的核心公式。理解这些公式的推导不仅能加深对物理学的掌握，也有助于应对需要解释推导过程的题目。本文逐一推导Insert 3中的主要公式，涵盖力学、波、电学和核物理。
1. Deriving the SUVAT Equations | 运动学公式推导
The SUVAT equations link displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t) for constant acceleration. They are derived from the definitions of velocity and acceleration.
对于匀加速直线运动，SUVAT方程将位移(s)、初速度(u)、末速度(v)、加速度(a)和时间(t)联系起来。它们源于速度和加速度的定义。
celeration: a = (v - u)/t. Rearranging gives v = u + at. This is the first SUVAT equation.
从加速度定义开始：a = (v - u)/t。整理得 v = u + at。这是第一个SUVAT方程。
elocity × t, so s = ((u + v)/2) × t. Substitute v = u + at into this to get s = ut + ½at².
匀加速过程中的平均速度为(u + v)/2。位移 s = 平均速度 × 时间，因此 s = ((u + v)/2) × t。将 v = u + at 代入，得到 s = ut + ½at²。
ut + ½at², so v² = u² + 2as. Thus, the four SUVAT equations are derived without calculus.
将 v = u + at 平方得 v² = u² + 2a(ut + ½at²)。注意到 s = ut + ½at²，因此 v² = u² + 2as。这样，四个SUVAT方程无需微积分即可导出。
2. Kinetic Energy Derivation: Ek = ½mv² | 动能公式推导
to accelerate a mass from rest to velocity v. Start with work done W = F × s. Using Newton' d law, F = ma, and the third SUVAT equation v² = u² + 2as with u = 0, we find s = v²/(2a).
动能是将质量为m的物体从静止加速到速度v所做的功。从做功公式 W = F × s 开始。利用牛顿第二定律 F = ma，以及SUVAT第三方程 v² = u² + 2as（令u = 0），可得 s = v²/(2a)。
energy, so Ek = ½mv². This derivation appears in the insert to remind students of the work–energy principle.
代入 W = mas = ma × (v²/(2a)) = ½mv²。这个功转化为动能，因此 Ek = ½mv²。此项推导出现在公式表中，提示学生关注功—能原理。
3. Centripetal Force: F = mv²/r = mω²r | 向心力公式推导
/r. This can be derived from the similarity of the velocity triangle and the displacement triangle over a short time Δt.
物体在半径为r的圆周上以恒定速率v运动时，指向圆心的加速度 a = v²/r。这可以从短时间内速度三角形与位移三角形的相似性导出。
on a, and Δs/Δt = v. So a/v = v/r, hence a = v²/r. With angular velocity ω = v/r, a = ω²r.
由几何关系，Δv / v = Δs / r。两边同除以Δt，得 (Δv/Δt) / v = (Δs/Δt) / r。左边为加速度 a，右边 Δs/Δt = v。因此 a/v = v/r，即 a = v²/r。利用角速度 ω = v/r，得 a = ω²r。
mv²/r = mω²r. This formula is essential for circular motion problems and satellite orbits.
乘以质量得到向心力 F = ma = mv²/r = mω²r。该公式对于圆周运动问题和卫星轨道至关重要。
4. Simple Harmonic Motion Displacement and Velocity Equations | 简谐运动位移与速度方程推导
on. From a = dv/dt = -ω²x and using the chain rule dv/dt = v dv/dx, we get v dv/dx = -ω²x.
在简谐运动中，加速度与位移反向且成正比：a = -ω²x（x为位移，ω为角频率）。这是基本定义方程。由 a = dv/dt = -ω²x，并利用链式法则 dv/dt = v dv/dx，得到 v dv/dx = -ω²x。
tant. When x = amplitude A, v = 0, so constant = ½ω²A². Thus v² = ω²(A² - x²), and v = ± ω√A² - x².
两边积分：∫ v dv = -ω² ∫ x dx，得 ½v² = -½ω²x² + 常数。当 x = 振幅 A 时，v = 0，故常数为 ½ω²A²。因此 v² = ω²(A² - x²)，取平方根得 v = ± ω√A² - x²。
ₘₐₓ = ωA occurs at the equilibrium position. These relationships are central to the insert's SHM section.
微分方程的解给出位移 x = A sin(ωt) 或 x = A cos(ωt)。最大速度 vₘₐₓ = ωA 出现在平衡位置。这些关系是公式表中简谐运动部分的核心。
5. Capacitor Time Constant and Charging/Discharging Equations | 电容器时间常数与充放电方程推导
ince current I = -dQ/dt (discharge), we have -R dQ/dt = Q/C. Rearranging: dQ/dt = -Q/(RC).
对于电容C通过电阻R放电，回路电压定律给出 V = IR，且 Q = CV。由于电流 I = -dQ/dt（放电），有 -R dQ/dt = Q/C。整理得 dQ/dt = -Q/(RC)。
dt/(RC). This yields ln Q = -t/(RC) + constant. Setting Q = Q₀ at t = 0 gives Q = Q₀ e⁻ᵗ/ᴿᶜ. The time constant τ = RC appears in the exponential decay.
分离变量并积分：∫ dQ/Q = -∫ dt/(RC)，得到 ln Q = -t/(RC) + 常数。设 t=0 时 Q = Q₀，得 Q = Q₀ e⁻ᵗ/ᴿᶜ。时间常数 τ = RC 出现在指数衰减中。
Similarly, for charging: Q = Q₀ (1 - e⁻ᵗ/ᴿᶜ) and V = V₀ (1 - e⁻ᵗ/ᴿᶜ ows from solving dQ/dt = (V₀ - Q/C)/R. These formulas are vital for analysing RC circuits.
类似地，充电过程：Q = Q₀ (1 - e⁻ᵗ/ᴿᶜ)，V = V₀ (1 - e⁻ᵗ/ᴿᶜ)。推导源于解方程 dQ/dt = (V₀ - Q/C)/R。这些公式是分析RC电路的基础。
6. Radioactive Decay Law: N = N₀ e⁻ᴸᵗ | 放射性衰变律推导
the number of undecayed nuclei N: A = λN, where λ is the decay constant. Thus -dN/dt = λN.
放射性样品的活度是单位时间内衰变次数：A = -dN/dt。实验表明，A与未衰变原子核数N成正比：A = λN（λ为衰变常数）。因此 -dN/dt = λN。
N₀ e⁻λt. This exponential law is fundamental in nuclear physics, directly from the insert.
分离变量积分：∫ dN/N = -λ ∫ dt，得 ln N = -λt + c。t=0时 N = N₀，因此 c = ln N₀。指数化得到 N = N₀ e⁻λt。这个指数衰变律是核物理的基础，直接来自公式表。
ln 2. Hence T½ = ln 2 / λ. This relationship is often used to find λ from half-life data.
半衰期 T½ 满足 N = N₀/2，即 e⁻λ(T½) = 1/2，得 λ T½ = ln 2。因此 T½ = ln 2 / λ。常用此关系式从半衰期数据求衰变常数。
7. Photon Energy and Matter Waves: E = hf, λ = h/p | 光子能量与物质波公式推导
Einstein' equation established that light consists of photons with energy E = hf, where h is Planck' xperiments: eVₛ = hf - Φ. The insert includes both E = hf and the photon momentum p = h/λ.
爱因斯坦光电方程确立了光由光子组成，能量 E = hf（h为普朗克常数，f为频率）。此关系为基本假设，但通过遏止电势实验 eVₛ = hf - Φ 得到支持。公式表中包含 E = hf 和光子动量 p = h/λ。
has a wavelength: λ = h/p, where p = mv is momentum. This is derived by combining Einstein' entum p = E/c = hf/c = h/λ, and extending it to particles with speed v, giving λ = h/(mv).
德布罗意提出物质也有波长：λ = h/p，p = mv 为动量。将爱因斯坦 E = hf、光子动量 p = E/c = hf/c = h/λ 推广至粒子，得 λ = h/(mv)。
For electrons accelerated through a potential V, kinetic energy ½mv² = eV, so p = √2meV. Then λ = h/√2meV, which matches electron diffraction data. Thus the insert's relationship bridges waves and particles.
对电子经过电势差V加速，动能 ½mv² = eV，因此 p = √2meV，于是 λ = h/√2meV，与电子衍射实验相符。公式表中的这一关系沟通了波与粒子。
8. Ideal Gas Pressure Equation: pV = 1/3 Nm⟨c²⟩ | 理想气体压力方程推导
me wall is 2L/vₓ, so the force on the wall from one molecule is (2mvₓ) / (2L/vₓ) = mvₓ²/L.
考虑边长为L的立方体容器，内含N个质量为m的分子。一个分子沿x方向速度分量为vₓ，与器壁弹性碰撞，动量改变量为2mvₓ。同壁两次碰撞时间间隔为2L/vₓ，因此一个分子对壁的作用力为 (2mvₓ) / (2L/vₓ) = mvₓ²/L。
al force F = (m/L) Σ vₓ². Pressure p = F/A = (m/L)(Σ vₓ²) / L² = (m/V) Σ vₓ². By isotropy, ⟨v²⟩ = ⟨vₓ²⟩ + ⟨vᵧ²⟩ + ⟨vz²⟩ and each mean square component is equal, so ⟨vₓ²⟩ = (1/3)⟨c²⟩, where c is speed. Thus Σ vₓ² = N⟨vₓ²⟩ = (N/3)⟨c²⟩.
对所有分子求和，总力 F = (m/L) Σ vₓ²。压强 p = F/A = (m/L)(Σ vₓ²) / L² = (m/V) Σ vₓ²。由于各向同性，⟨v²⟩ = ⟨vₓ²⟩ + ⟨vᵧ²⟩ + ⟨vz²⟩，各均方分量相等，所以 ⟨vₓ²⟩ = (1/3)⟨c²⟩（c为速率）。因此 Σ vₓ² = N⟨vₓ²⟩ = (N/3)⟨c²⟩。
Substituting yields p = (m/V) × (N/3)⟨c²⟩ = (1/3) (Nm/V) ⟨c²⟩. Hence pV = 1/3 Nm⟨c²⟩. The insert often lists this as pV = 1/3 Nm crms², where crms = √⟨c²⟩.
代入得 p = (m/V) × (N/3)⟨c²⟩ = (1/3) (Nm/V) ⟨c²⟩，因此 pV = 1/3 Nm⟨c²⟩。公式表中常写作 pV = 1/3 Nm crms²，其中 crms = √⟨c²⟩。
9. Radius of a Charged Particle in a Magnetic Field: r = mv/(Bq) | 带电粒子在磁场中的回旋半径推导
ipetal force for circular motion: Bqv = mv²/r. Cancel v (for v ≠ 0) to obtain r = mv/(Bq).
电荷q以速度v垂直于匀强磁场B运动，受洛伦兹力 F = Bqv。该力提供圆周运动的向心力：Bqv = mv²/r。消去v（v≠0）得到 r = mv/(Bq)。
path becomes helical, but the radius is still determined by the perpendicular component v⊥: r = mv⊥ ed in mass spectrometry and cyclotrons. The insert gives this formula for quick reference.
若速度分量平行于磁场，路径为螺旋线，但半径仍由垂直分量v⊥决定：r = mv⊥/(Bq)。此推导用于质谱仪和回旋加速器。公式表提供该公式以便快速查阅。
m/(Bq), independent of speed. The frequency f = 1/T = Bq/(2πm) is the cyclotron frequency.
圆周运动周期 T = 2πr/v = 2πm/(Bq)，与速率无关。频率 f = 1/T = Bq/(2πm)，即为回旋频率。
10. Faraday's Law and Lenz's Law: ε = -N dΦ/dt | 法拉第电磁感应定律与楞次定律推导
Faraday' ge of magnetic flux linkage. For a coil with N turns, flux linkage = NΦ. Experimentally, ε ∝ - d(NΦ)/dt. The minus sign reflects Lenz's law: the induced current opposes the change in flux.
法拉第定律指出，线圈中感应电动势与磁通量链变化率成正比。对于N匝线圈，磁链 = NΦ。实验表明 ε ∝ - d(NΦ)/dt。负号体现楞次定律：感应电流阻碍引起感应的磁通变化。
his matches ε = -dΦ/dt because flux Φ = BA = BLx, where x is displacement, so dΦ/dt = BL dx/dt = BLv. The negative sign indicates direction.
对于导体在磁场中垂直运动，可推导动生电动势：一根长为L的棒以速度v垂直于B运动，电子受洛伦兹力 F = Bev，导致电荷分离，建立电场E直到 eE = Bev，故 E = Bv。电动势 ε = EL = BLv。这与 ε = -dΦ/dt 一致，因为 Φ = BA = BLx（x为位移），dΦ/dt = BL dx/dt = BLv。负号指示方向。
These laws are cornerstone formulas in the insert, essential for generators, transformers, and induction problems.
这两条定律是公式表中的基石，对于发电机、变压器和电磁感应问题必不可少。
11. Equating Gravitational and Centripetal Force for Satellite Motion | 卫星运动的引力与向心力平衡推导
²/r. Equating: GMm/r² = mv²/r. Cancel m and multiply by r: GM/r = v², so orbital speed v = √GM/r.
对于绕质量为M的行星做圆周运动的卫星，引力提供向心力。引力 F = GMm/r²，向心力 mv²/r。令两者相等：GMm/r² = mv²/r。消去m，两边乘r，得 v² = GM/r，因此轨道速率 v = √GM/r。
The orbital period T = 2πr/v, so substituting v gives T² = (4π²/GM) r³, which is Kepler' ently required when using data from the insert, which lists both F = GMm/r² and g = GM/r².
轨道周期 T = 2πr/v，代入v得 T² = (4π²/GM) r³，即开普勒第三定律。使用公式表中 F = GMm/r² 和 g = GM/r² 时，常需要进行这些推导。
12. Energy Stored in a Capacitor: E = ½QV = ½CV² = ½Q²/C | 电容器储存的能量公式推导
Q: W = ∫₀ᵠ (q/C) dq = ½ Q²/C = ½ QV = ½ CV². This energy is stored in the electric field.
电容器充电时，克服逐渐增大的电势差移动电荷做功。因为 V = Q/C，添加微小电荷 dq 所做的功 dW = V dq = (q/C) dq。从0到Q积分：W = ∫₀ᵠ (q/C) dq = ½ Q²/C = ½ QV = ½ CV²。此能量储存在电场中。
This derivation explains why half the energy supplied by a battery is dissipated in a simple RC circuit: the battery delivers energy QV, but only ½QV is stored; the rest is lost as heat in the resistor, independent of R. The insert includes these energy formulas, often prompting consideration of efficiency.
这项推导解释了为何简单RC电路中电池供给的能量一半被耗散：电池输出能量 QV，但仅 ½QV 得以储存；其余以热的形式耗散在电阻上，且与R无关。公式表中包含这些能量公式，常引发对效率的思考。
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