Formula Derivations for International A-Level Physics Unit 4 (PH04) | 国际A-Level物理PH04单元公式推导

📚 Formula Derivations for International A-Level Physics Unit 4 (PH04) | 国际A-Level物理PH04单元公式推导

The International A-Level Physics Unit 4 (PH04) specimen paper covers advanced mechanics, fields, and particles. Understanding the derivations of key formulas is essential for solving complex problems and achieving top grades. This article presents step-by-step derivations for important equations, including circular motion, simple harmonic motion, gravitational and electric fields, capacitance, and electromagnetic induction.

国际A-Level物理单元4(PH04)样卷涵盖进阶力学、场和粒子物理。掌握核心公式的推导过程对于解决复杂问题、取得高分至关重要。本文逐步推导重要方程，包括圆周运动、简谐运动、引力场和电场、电容以及电磁感应。

1. Centripetal Acceleration and Force | 向心加速度与向心力

Consider an object moving in a circle of radius r with constant speed v. In a short time Δt, the object moves from point P to Q, covering an angle Δθ. The velocity vectors at P and Q have equal magnitude but different directions. The change in velocity Δv is directed towards the centre O.

考虑一个物体以恒定速率 v 在半径为 r 的圆周上运动。在短时间 Δt 内，物体从点 P 移动到 Q，转过角度 Δθ。P 和 Q 处的速度矢量大小相等但方向不同。速度变化量 Δv 指向圆心 O。

From geometry, the magnitude of Δv is approximately vΔθ. For small angles this becomes exact. The angular displacement Δθ = vΔt / r, so Δv = v²Δt / r. Acceleration a = Δv / Δt = v² / r, always directed radially inward.

根据几何关系，Δv 的大小近似为 vΔθ，对微小角度精确成立。角位移 Δθ = vΔt / r，因此 Δv = v²Δt / r。加速度 a = Δv / Δt = v² / r，方向始终沿径向向内。

Using Newton’s second law, the centripetal force is F = ma = mv² / r. Since v = ωr, we obtain alternative forms: a = ω²r and F = mω²r. These equations are fundamental for any object in uniform circular motion.

根据牛顿第二定律，向心力 F = ma = mv² / r。因为 v = ωr，可得其他形式：a = ω²r，F = mω²r。这些方程对任何做匀速圆周运动的物体都成立。

a = v² / r = ω²r F = mv² / r = mω²r

2. Simple Harmonic Motion Equations | 简谐运动方程

Simple harmonic motion (SHM) can be defined as the projection of uniform circular motion onto a diameter. An object rotating with angular velocity ω on a circle of radius A has horizontal coordinate x = A cos θ. Setting θ = ωt gives the displacement x = A cos(ωt), where A is the amplitude.

简谐运动可定义为匀速圆周运动在直径上的投影。以角速度 ω 在半径为 A 的圆周上旋转的物体的水平坐标为 x = A cos θ。令 θ = ωt，得到位移 x = A cos(ωt)，其中 A 为振幅。

Differentiating with respect to time yields velocity: v = dx/dt = −Aω sin(ωt). The maximum speed is Aω. Differentiating again gives acceleration: a = dv/dt = −Aω² cos(ωt) = −ω²x. This shows that acceleration is proportional to the negative of displacement.

对时间求导得速度：v = dx/dt = −Aω sin(ωt)，最大速率为 Aω。再次求导得加速度：a = dv/dt = −Aω² cos(ωt) = −ω²x。这表明加速度与位移成正比且方向相反。

The defining equation of SHM is thus a = −ω²x. The time period T is related to angular frequency by T = 2π/ω. The frequency f = 1/T = ω/(2π). For a mass-spring system, ω = √(k/m), giving T = 2π√(m/k). For a simple pendulum, T = 2π√(L/g) for small angles.

简谐运动的定义方程为 a = −ω²x。周期 T 与角频率的关系为 T = 2π/ω。频率 f = 1/T = ω/(2π)。对于弹簧-质量系统，ω = √(k/m)，得 T = 2π√(m/k)。对于单摆在小角度下，T = 2π√(L/g)。

x = A cos(ωt) v = −Aω sin(ωt) a = −ω²x

3. Gravitational Field of a Point Mass | 质点引力场

Newton’s law of gravitation states that the force between two point masses M and m separated by distance r is F = GMm / r². The gravitational field strength g at a point is the force per unit mass on a small test mass placed there: g = F/m.

牛顿引力定律表明，两个点质量 M 和 m 相距 r 时的引力为 F = GMm / r²。引力场强度 g 定义为在该点放置的检验质量所受的力与其质量之比：g = F/m。

Substituting the force expression gives g = GM / r². The direction of g is towards the mass M. The field is radial and follows an inverse square law. The gravitational potential V at a point is the work done per unit mass to bring a test mass from infinity to that point. Since F = GMm / r², work done W = ∫(infinity to r) −(GMm / r²) dr = −GMm / r. Per unit mass, V = −GM / r.

代入力的表达式得 g = GM / r²，方向指向质量 M。该场为径向场，遵循平方反比定律。引力势 V 定义为将单位质量从无穷远处移到该点所做的功。因为 F = GMm / r²，做功 W = ∫(∞→r) −(GMm / r²) dr = −GMm / r。因此单位质量的势 V = −GM / r。

The escape speed from a planet of mass M and radius R is found by equating kinetic energy to the magnitude of gravitational potential energy: ½mv² = GMm / R, giving vₑ = √(2GM / R).

从质量为 M 半径为 R 的行星逃逸所需的速率，由动能等于引力势能的大小得出：½mv² = GMm / R，解得 vₑ = √(2GM / R)。

g = GM / r² V = −GM / r vₑ = √(2GM / R)

4. Electric Field of a Point Charge | 点电荷电场

Coulomb’s law states that the force between two point charges Q and q separated by distance r is F = kQq / r², where k = 1/(4πε₀) in a vacuum. The electric field strength E is the force per unit positive charge: E = F/q.

库仑定律指出，两个点电荷 Q 和 q 相距 r 时的力为 F = kQq / r²，真空中 k = 1/(4πε₀)。电场强度 E 定义为单位正电荷所受的力：E = F/q。

Substituting gives E = kQ / r² = Q / (4πε₀ r²). The direction is radially outward from a positive charge. The electric potential V at a point is the work done per unit charge to bring a test charge from infinity to that point. Integration along the radial path yields V = kQ / r = Q / (4πε₀ r). Unlike gravitational potential, electric potential can be positive or negative.

代入可得 E = kQ / r² = Q / (4πε₀ r²)，方向由正电荷沿径向向外。电势 V 定义为将单位正电荷从无穷远处移到该点所做的功。沿径向积分得 V = kQ / r = Q / (4πε₀ r)。与引力势不同，电势可正可负。

For a uniform electric field between parallel plates separated by distance d with potential difference V, the field strength is E = V / d. The work done moving a charge q through a potential difference V is W = qV.

对于平行板之间的匀强电场，板间距 d，电势差 V，则场强 E = V / d。移动电荷 q 经过电势差 V 所做的功为 W = qV。

E = Q / (4πε₀ r²) Vₑ = Q / (4πε₀ r) E = V / d

5. Capacitor Discharge Equation | 电容放电方程

When a capacitor of capacitance C discharges through a resistor R, the current I and charge Q decrease with time. By definition, I = dQ/dt, but the current is the rate of decrease of charge, so I = −dQ/dt. The potential difference across the resistor is IR = Q/C (from V = Q/C).

当电容为 C 的电容器通过电阻 R 放电时，电流 I 和电荷 Q 随时间减少。由定义，I = dQ/dt，但这里电流是电荷的减少率，因此 I = −dQ/dt。电阻上的电势差 IR = Q/C（来自 V = Q/C）。

Combining these: −R dQ/dt = Q/C, which rearranges to dQ/dt = −Q / (RC). This first-order differential equation has the solution Q = Q₀ exp(−t/(RC)), where Q₀ is the initial charge. The product RC is called the time constant τ.

联立得 −R dQ/dt = Q/C，整理为 dQ/dt = −Q / (RC)。这个一阶微分方程的解为 Q = Q₀ exp(−t/(RC))，其中 Q₀ 为初始电荷。乘积 RC 称为时间常数 τ。

The voltage across the capacitor decays similarly: V = V₀ exp(−t/τ). The current also decays as I = I₀ exp(−t/τ), where I₀ = V₀/R. The half-life t½, the time for the charge or voltage to halve, satisfies ½ = exp(−t½/τ), giving t½ = τ ln 2 ≈ 0.693 τ.

电容两端电压作类似衰减：V = V₀ exp(−t/τ)。电流也以 I = I₀ exp(−t/τ) 衰减，其中 I₀ = V₀/R。半衰期 t½，即电荷或电压减半所需的时间，满足 ½ = exp(−t½/τ)，得 t½ = τ ln 2 ≈ 0.693 τ。

Q = Q₀ exp(−t / RC) τ = RC t½ = RC ln 2

6. Energy Stored in a Capacitor | 电容器储存的能量

To charge a capacitor, work must be done to move charge against the growing potential difference. When a small charge dq is added, the p.d. is v = q/C, so the work done dW = v dq = (q/C) dq. Integrating from 0 to Q gives total stored energy W = ∫₀Q (q/C) dq = ½ Q²/C.

对电容器充电需要克服逐渐增大的电势差做功。当加入微小电荷 dq 时，电势差为 v = q/C，所做功 dW = v dq = (q/C) dq。从 0 到 Q 积分得到总储能 W = ∫₀Q (q/C) dq = ½ Q²/C。

Using the relation Q = CV, the energy can also be expressed as W = ½ CV² or W = ½ QV. This energy is stored in the electric field between the plates. For a parallel plate capacitor of area A and separation d, C = ε₀ A/d, the energy density (energy per unit volume) is ½ ε₀ E², where E = V/d.

利用 Q = CV，能量也可表示为 W = ½ CV² 或 W = ½ QV。该能量储存在两极板间的电场中。对于面积为 A、间距为 d 的平行板电容器，C = ε₀ A/d，能量密度（单位体积能量）为 ½ ε₀ E²，其中 E = V/d。

W = ½ Q² / C = ½ CV² = ½ QV

7. Magnetic Force on a Moving Charge | 运动电荷在磁场中的力

A charge q moving with velocity v in a magnetic field B experiences a magnetic force F = Bqv sin θ, where θ is the angle between v and B. The direction is given by Fleming’s left-hand rule (for positive charge) or the right-hand slap rule. In vector form, F = q (v × B).

以速度 v 在磁场 B 中运动的电荷 q 受到磁力 F = Bqv sin θ，其中 θ 是 v 与 B 之间的夹角。力的方向由左手定则（正电荷）或右手螺旋定则给出。矢量形式为 F = q (v × B)。

When the charge moves perpendicular to a uniform magnetic field (θ = 90°), F = Bqv. This force provides the centripetal force for circular motion: Bqv = mv² / r. Solving for the radius gives r = mv / (Bq). The time period of the circular motion is T = 2πr / v = 2πm / (Bq), independent of speed.

当电荷垂直于匀强磁场运动时（θ = 90°），F = Bqv。这个力提供圆周运动所需的向心力：Bqv = mv² / r。解出半径得 r = mv / (Bq)。圆周运动的周期 T = 2πr / v = 2πm / (Bq)，与速率无关。

For a current-carrying conductor of length L carrying current I perpendicular to a magnetic field, the force on all moving charges results in F = BIL. If the conductor makes an angle θ with the field, F = BIL sin θ.

对于长度为 L、载流 I 且垂直于磁场的导体，所有运动电荷所受的总力为 F = BIL。若导体与磁场夹角为 θ，则 F = BIL sin θ。

F = Bqv sin θ r = mv / (Bq) T = 2πm / (Bq)

8. Faraday’s Law of Electromagnetic Induction | 法拉第电磁感应定律

Faraday’s law states that the induced emf in a circuit is equal to the rate of change of magnetic flux linkage. Flux Φ through an area A in a uniform magnetic field B is Φ = BA cos θ, where θ is the angle between the field and the normal to the area. For a coil of N turns, flux linkage = NΦ.

法拉第电磁感应定律指出，电路中产生的感应电动势等于磁通链的变化率。通过面积 A 的匀强磁场 B 中的磁通量 Φ = BA cos θ，其中 θ 为磁场与面积法线间的夹角。对于 N 匝线圈，磁通链 = NΦ。

The induced emf ε is given by ε = −N ΔΦ / Δt, and instantaneously ε = −N dΦ/dt. The minus sign reflects Lenz’s law: the induced current flows so as to oppose the change that produced it.

感应电动势 ε 由 ε = −N ΔΦ / Δt 给出，瞬时值 ε = −N dΦ/dt。负号体现楞次定律：感应电流的方向总是阻碍引起它的变化。

A common example is a conductor of length L moving perpendicularly through a field B with speed v. The flux cut per unit time is BLv, so ε = BLv. If the motion is at angle θ to the field, ε = BLv sin θ. This can be derived from the motional emf concept using F = qvB.

常见例子是长度为 L 的导体以速度 v 垂直于磁场 B 运动。单位时间内切割的磁通量为 BLv，因此 ε = BLv。若运动方向与磁场夹角为 θ，则 ε = BLv sin θ。这也可由动生电动势的概念通过 F = qvB 推导得出。

ε = −N ΔΦ / Δt Φ = BA cos θ ε = BLv (perpendicular)

9. Time Constant and Half-life in Capacitor Discharge | 电容放电的时间常数与半衰期

The time constant τ = RC is a measure of how quickly a capacitor discharges. After a time t = τ, the charge falls to Q₀ exp(−1) ≈ 0.368 Q₀. After 3τ, the charge drops to about 5% of the initial value, and the capacitor is often considered discharged.

时间常数 τ = RC 是衡量电容器放电快慢的指标。经过时间 t = τ，电荷降至 Q₀ exp(−1) ≈ 0.368 Q₀。经过 3τ，电荷降至初始值的约 5%，通常认为电容器已放电完毕。

Another useful measure is the half-life t½, the time for the charge to decrease to half its initial value. Setting Q = Q₀/2 in the decay equation gives ½ = exp(−t½/τ). Taking natural logarithms yields t½ = τ ln 2. This relation is independent of Q₀ and is characteristic of exponential decay processes.

另一个有用指标是半衰期 t½，即电荷减少到初始值一半所需的时间。在衰减方程中令 Q = Q₀/2，得 ½ = exp(−t½/τ)。取自然对数得 t½ = τ ln 2。该关系与 Q₀ 无关，是指数衰减过程的特征。

Log-linear graphs of ln Q against t give a straight line with slope −1/τ, allowing experimental determination of τ. The intercept is ln Q₀. This technique is widely used in analyzing capacitor discharge experiments.

作 ln Q 对 t 的半对数图可得一条斜率为 −1/τ 的直线，从而可由实验测定 τ。截距为 ln Q₀。这种方法广泛用于分析电容放电实验。

τ = RC t½ = τ ln 2 Q = Q₀ exp(−t/τ)

10. Gravitational Potential and Orbital Mechanics | 引力势与轨道力学

The gravitational potential

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Formula Derivations for International A-Level Physics Unit 4 (PH04) | 国际A-Level物理PH04单元公式推导