📚 Deriving Key Formulas from A-Level Physics Unit 5 (Jan 2019 Insert) | A-Level 物理 Unit 5 2019年1月公式推导
The Edexcel IAL Physics Unit 5 examination (January 2019) centres on topics from further mechanics, gravitational and electric fields, capacitors, magnetic fields, thermodynamics, and nuclear physics. The insert booklet supplies a condensed reference of key equations, but genuine mastery of the subject demands an understanding of where these formulas come from. This article presents step‑by‑step derivations of the most important relationships found in that insert, moving from first principles to the final expressions you will use in the exam. Working through each derivation will strengthen your physical intuition and equip you to answer even the most challenging synoptic questions.
Edexcel IAL 物理 Unit 5 考试（2019 年 1 月）围绕进阶力学、引力场与电场、电容器、磁场、热力学以及核物理展开。试题册中的公式表提供了一份精简的核心方程参考，但真正掌握这门学科需要理解这些公式从何而来。本文从基本原理出发，逐步推导该公式表中最重要的关系式，直至考试中直接使用的最终形式。逐一推导每一个公式不仅能增强你的物理直觉，也能让你游刃有余地应对最具挑战的综合型考题。
1. Centripetal Acceleration | 向心加速度
Consider an object moving with constant speed v along a circular path of radius r. In a small time interval Δt, it sweeps out an angle Δθ = vΔt / r. The velocity vector rotates through the same angle while its magnitude remains constant. The vector change in velocity, Δv, is almost perpendicular to the instantaneous velocity and has magnitude vΔθ for small Δθ. Therefore, the instantaneous acceleration towards the centre is a = vΔθ / Δt = v × (v / r) = v² / r.
考虑一个沿半径为 r 的圆形路径以恒定速率 v 运动的物体。在极短的时间 Δt 内，物体扫过的圆心角 Δθ = vΔt / r。速度矢量在同一时间内转过相同的角度，而大小保持不变。速度的变化量 Δv 近似垂直于瞬时速度，当 Δθ 很小时其大小为 vΔθ。因此，指向圆心的瞬时加速度 a = vΔθ / Δt = v × (v / r) = v² / r。
Substituting the angular velocity ω = v / r gives the equivalent forms a = ω²r and a = ωv. This relationship is purely geometrical and does not involve any force; it is the essential starting point for all uniform circular motion analysis.
代入角速度 ω = v / r，可得到等价形式 a = ω²r 以及 a = ωv。这一关系纯粹来自几何，与力无关；它是分析一切匀速圆周运动的基本出发点。
2. Centripetal Force | 向心力
Newton’s second law, F = ma, is applied directly to the instantaneous centripetal acceleration derived above. For an object of mass m moving in a circle, there must be a net force directed towards the centre of magnitude F = m × (v² / r). Using ω, the same force can be written as F = mω²r.
将牛顿第二定律 F = ma 直接应用于前面推导出的瞬时向心加速度。对于质量为 m 作圆周运动的物体，必须存在一个指向圆心的合力，大小为 F = m × (v² / r)。利用角速度 ω，该力也可写作 F = mω²r。
This centripetal force is not a new type of force; it is provided by tension, friction, gravity, or electromagnetic interactions depending on the scenario. The formula enables calculation of the orbit speed of satellites, the tension in a string whirl, or the banking angle of a vehicle.
向心力并非一种新的力，而是根据具体情景由张力、摩擦力、引力或电磁力提供。该公式可用于计算卫星的轨道速度、绳摆中的张力或交通工具的倾斜角。
3. Gravitational Field Strength at a Point | 空间点的引力场强
Newton’s law of universal gravitation states that two point masses M and m separated by distance r attract each other with a force F = GMm / r², where G is the gravitational constant. Gravitational field strength g at the location of m is defined as the force per unit mass: g = F / m. Therefore, g = GM / r².
牛顿万有引力定律指出，两个相距 r 的点质量 M 和 m 以力 F = GMm / r² 互相吸引，其中 G 为万有引力常数。在 m 所在位置的引力场强 g 定义为单位质量所受的力：g = F / m。因此，g = GM / r²。
For a spherically symmetric body such as a planet, the result holds for points outside the body if r is taken as the distance from the centre. This formula explains why g decreases with altitude and allows the calculation of planetary mass from surface g and radius.
对于球对称天体（如行星），只要把 r 取为距中心的距离，该结果对球外各点成立。该公式解释了 g 随高度增加而减小，并使得由地面 g 和半径计算行星质量成为可能。
4. Electric Field Due to a Point Charge | 点电荷的电场
Coulomb’s law gives the force between two point charges Q and q in a vacuum: F = kQq / r², where k = 1 / (4πε₀). The electric field strength E at the position of q is the force per unit positive charge, E = F / q, so E = kQ / r². In a uniform electric field between parallel plates, the field is constant and given by E = V / d, where V is the potential difference and d is the plate separation.
库仑定律给出真空中两点电荷 Q 与 q 之间的作用力：F = kQq / r²，其中 k = 1 / (4πε₀)。q 所在处的电场强度 E 定义为单位正电荷所受的力，即 E = F / q，因此 E = kQ / r²。在平行板间的匀强电场中，场强恒定，并由 E = V / d 给出，其中 V 为电势差，d 为板间距。
Equating the two expressions for the force on a charge in a uniform field, E = V / d can be derived by considering the work done moving a charge from one plate to the other: Fd = qV, and since F = qE, we obtain qEd = qV, hence E = V / d.
令电荷在匀强电场中受力的两种表达式相等，可以导出 E = V / d：移动电荷从一板至另一板做功 Fd = qV，又因 F = qE，有 qEd = qV，因此 E = V / d。
5. Energy Stored in a Capacitor | 电容器储存的能量
As a capacitor of capacitance C is charged, the potential difference v across it is proportional to the stored charge q: v = q / C. Moving an additional small charge dq from one plate to the other requires work dW = v dq = (q / C) dq. The total work done (and energy stored) from zero charge to final charge Q is the integral W = ∫₀ᵯ (q / C) dq = (1/2) Q² / C.
当电容为 C 的电容器充电时，其两端电势差 v 与已储存电荷 q 成正比：v = q / C。将额外微小电荷 dq 从一板移动至另一板需做功 dW = v dq = (q / C) dq。从零电荷充至最终电荷 Q 所作总功（即储存的能量）为积分 W = ∫₀ᵯ (q / C) dq = (1/2) Q² / C。
Using the definition C = Q / V, the energy can be expressed in the three familiar forms: W = ½ QV = ½ CV² = ½ Q² / C. Which version is most convenient depends on the known quantities in a problem.
利用定义 C = Q / V，该能量可表示为三种常见形式：W = ½ QV = ½ CV² = ½ Q² / C。具体采用哪种形式取决于题目中已知量。
6. Time Constant for an RC Circuit | RC 电路的时间常数
For a discharging capacitor, Kirchhoff’s loop rule gives V_C = V_R, i.e., q / C = −R dq/dt (since current I = −dq/dt). Rearranging: dq / q = −(1 / RC) dt. Integrating between initial charge Q₀ and charge q at time t yields ln(q / Q₀) = −t / RC, or q = Q₀ e^(−t / RC). The product τ = RC appears naturally as the time constant, the time for the charge to fall to about 37% of its original value.
对于放电的电容器，基尔霍夫回路定则给出 V_C = V_R，即 q / C = −R dq/dt（因为电流 I = −dq/dt）。重排得：dq / q = −(1 / RC) dt。对时间从 0 到 t 积分，初始电荷为 Q₀，得到 ln(q / Q₀) = −t / RC，即 q = Q₀ e^(−t / RC)。乘积 τ = RC 自然地作为时间常数出现，是电荷降至约初始值 37% 所需的时间。
Similarly, for charging, q = Q₀(1 − e^(−t / RC)). The time constant governs how quickly a capacitor charges or discharges and is independent of the applied voltage, depending only on the resistance and capacitance values.
同样地，对于充电过程，q = Q₀(1 − e^(−t / RC))。时间常数决定了电容器充放电的快慢，与外加电压无关，仅取决于电阻与电容值。
7. Magnetic Force on a Moving Charge | 运动电荷所受磁场力
Experiment shows that a charge q moving with velocity v through a magnetic field B experiences a force perpendicular to both v and B. The magnitude is given by F = Bqv sin θ, where θ is the angle between v and B. This is the Lorentz force for a charged particle in a magnetic field alone. The direction is determined by Fleming’s left‑hand rule, reversing the current direction for a negative charge.
实验表明，以速度 v 在磁场 B 中运动的电荷 q 会受到同时垂直于 v 和 B 的力。力的大小为 F = Bqv sin θ，其中 θ 为 v 与 B 之间的夹角。这就是带电粒子仅受磁场作用时的洛伦兹力。方向由弗莱明左手定则判断，负电荷需反转电流方向。
When a charged particle moves perpendicular to a uniform magnetic field, the force provides the centripetal force: Bqv = mv² / r, leading to the radius of the circular path r = mv / (Bq). This principle is used in mass spectrometers and particle accelerators.
当带电粒子垂直进入匀强磁场时，该力充当向心力：Bqv = mv² / r，从而得圆形路径半径 r = mv / (Bq)。这一原理应用于质谱仪和粒子加速器。
8. Faraday’s Law of Electromagnetic Induction | 法拉第电磁感应定律
Faraday’s law states that the magnitude of the induced electromotive force (e.m.f.) in a circuit is equal to the rate of change of magnetic flux linkage: ε = −N dΦ / dt, where Φ = BA cos θ is the magnetic flux through one turn and N is the number of turns. The negative sign indicates Lenz’s law: the induced e.m.f. opposes the change in flux.
法拉第定律指出，电路中感应电动势的大小等于磁通匝链数的变化率：ε = −N dΦ / dt，其中 Φ = BA cos θ 是穿过一匝线圈的磁通量，N 是线圈匝数。负号体现了楞次定律：感应电动势总是阻碍磁通量的变化。
The flux can change because of variation in B (transformer effect), A (motional effect), or the angle θ (rotation in a generator). For a rod of length l moving at speed v perpendicular to a uniform field B, the e.m.f. simplifies to ε = Blv, derived by considering the area swept out per unit time.
磁通量可以因 B 变化（变压器效应）、A 变化（动生效应）或角度 θ 变化（发电机中转动）而改变。对于一根长度为 l 的导体棒以速度 v 垂直于匀强磁场 B 运动，感应电动势简化为 ε = Blv，此式可通过考虑单位时间扫过的面积来导出。
9. Kinetic Theory and the Ideal Gas Equation | 分子动理论与理想气体方程
Kinetic theory models a gas as a large number of identical molecules in random motion. Considering a cube of side L containing N molecules each of mass m, collisions with a wall produce a pressure p. The momentum transfer per collision is 2m v_x, and the number of collisions per unit time on one wall is (N / L³) × (L² v_x) × ½ (averaging direction). Summing over all molecules and using the average squared speed
分子动理论将气体视为大量无规运动的相同分子。考虑边长为 L 的立方体，含有 N 个质量均为 m 的分子，分子与器壁的碰撞产生压强 p。每次碰撞的动量传递为 2m v_x，单位时间内在某一器壁上的碰撞次数为 (N / L³) × (L² v_x) × ½（方向平均）。对所有分子求和，并利用均方速率
Combining this with the empirical ideal gas equation pV = NkT (where k is the Boltzmann constant) gives ½ m
将该式与经验中的理想气体状态方程 pV = NkT（k 为玻尔兹曼常量）结合，可得 ½ m
10. First Law of Thermodynamics | 热力学第一定律
The first law is a statement of energy conservation for a system: ΔU = Q − W, where ΔU is the increase in internal energy, Q is the heat supplied to the system, and W is the work done by the system on its surroundings. This sign convention is standard in physics; the insert may sometimes write the law as ΔU = Q + W with a note on signs.
热力学第一定律是对系统能量守恒的表述：ΔU = Q − W，其中 ΔU 为内能的增量，Q 为向系统提供的热量，W 为系统对外界所作的功。这一正负号约定是物理学中的标准；公式表中有时也写作 ΔU = Q + W 并附正负号说明。
For an ideal gas, the internal energy depends only on temperature. In an isothermal expansion, ΔU = 0, so Q = W. In an adiabatic process, Q = 0, so ΔU = −W. Understanding these special cases is crucial for analysing heat engines and thermodynamic cycles such as the Carnot cycle.
对于理想气体，内能仅依赖于温度。在等温膨胀中，ΔU = 0，因此 Q = W。在绝热过程中，Q = 0，因此 ΔU = −W。理解这些特例对分析热机和卡诺循环等热力学循环至关重要。
11. Radioactive Decay Law | 放射性衰变定律
Radioactive decay is a random process; for a large number of unstable nuclei, the activity A (decays per second) is proportional to the number N of undecayed nuclei present: A = λN, where λ is the decay constant. Since activity is the rate of decrease of N, we write dN/dt = −λN. Separating variables and integrating gives N = N₀ e^(−λt), where N₀ is the initial number of nuclei.
放射性衰变是一种随机过程；对于大量不稳定的原子核，活度 A（每秒衰变数）与现存未衰变的核数目 N 成正比：A = λN，其中 λ 为衰变常量。因活度即 N 的减少率，可写出 dN/dt = −λN。分离变量并积分得到 N = N₀ e^(−λt)，其中 N₀ 为初始核数目。
The half‑life T½ is the time taken for half the original nuclei to decay. Setting N = N₀ / 2 at t = T½ yields (1/2) = e^(−λ T½), so ln(1/2) = −λ T½, giving T½ = ln 2 / λ. This relationship links the decay constant with a measurable macroscopic quantity.
半衰期 T½ 是原有核衰变一半所需的时间。令 t = T½ 时 N = N₀ / 2，得 (1/2) = e^(−λ T½)，故 ln(1/2) = −λ T½，于是 T½ = ln 2 / λ。此关系式将衰变常量与可测量的宏观量联系了起来。
12. Mass–Energy Equivalence | 质能方程
Einstein’s famous equation E = mc² states that mass m and energy E are equivalent, with c being the speed of light in vacuum. In nuclear reactions, the mass of the products is slightly different from the mass of the reactants, and the energy released (or absorbed) is Q = Δm c², where Δm is the mass defect. This formula appears in the Unit 5 insert for calculations of binding energy and reaction energies.
爱因斯坦著名的方程 E = mc² 表明质量 m 与能量 E 是等价的，c 为真空中的光速。在核反应中，产物的总质量与反应物的总质量略有差异，释放（或吸收）的能量为 Q = Δm c²，其中 Δm 为质量亏损。该公式出现在 Unit 5 公式表中，用于结合能及反应能的计算。
For example, the binding energy of a nucleus can be found by comparing the nucleus’ actual mass with the sum of the masses of its constituent nucleons. The difference, expressed as energy, accounts for the stability of the nucleus and explains features of the atomic mass unit (u) energy equivalent, 1 u ≈ 931.5 MeV.
例如，原子核的结合能可通过比较核的实际质量与组成它的全部核子质量之和来求得。该差值以能量形式表达，解释了原子核的稳定性以及原子质量单位 u 的能量当量 1 u ≈ 931.5 MeV。
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