📚 AS Physics Unit 4 Insert June 2019 Formula Derivations | AS物理单元4 2019年6月公式推导
This article derives the key formulas found in the June 2019 insert for Unit 4 of the AS Physics specification, covering momentum, circular motion, electric fields, capacitance, magnetic fields and electromagnetic induction. Each derivation strengthens conceptual understanding and prepares you for challenging exam questions.
本文推导2019年6月AS物理单元4公式表中的关键公式，涉及动量、圆周运动、电场、电容、磁场和电磁感应。每条推导都能加深你的概念理解，为应对高难度考题做好准备。
1. Momentum and Impulse | 动量与冲量
Momentum p is defined as the product of an object’s mass m and its velocity v, giving p = m v. This vector quantity points in the same direction as velocity.
动量 p 定义为物体质量 m 和速度 v 的乘积，即 p = m v。这个矢量方向与速度一致。
Newton’s second law states that the resultant force F is equal to the rate of change of momentum: F = Δp / Δt. Multiplying both sides by Δt yields the impulse-momentum theorem: Δp = F Δt, where F Δt is the impulse.
牛顿第二定律指出合力 F 等于动量的变化率：F = Δp / Δt。两边同乘 Δt 得到冲量 – 动量定理：Δp = F Δt，其中 F Δt 是冲量。
p = m v
F = Δp / Δt → Δp = F Δt
When the force is constant, impulse equals the area under a force–time graph. This derivation is fundamental to collision and explosion problems.
当力恒定时，冲量等于力–时间图下的面积。该推导是处理碰撞和爆炸问题的基础。
2. Centripetal Acceleration | 向心加速度
An object moving in a circle of radius r with constant speed v changes direction continuously. In a small time Δt, it sweeps out an angle Δθ = v Δt / r. The change in velocity Δv is perpendicular to v and has magnitude v Δθ, producing an acceleration towards the centre.
物体以恒定速率 v 沿半径 r 做圆周运动时方向不断改变。在很短时间 Δt 内，扫过的角度 Δθ = v Δt / r。速度变化量 Δv 垂直于 v，大小为 v Δθ，指向圆心产生加速度。
Centripetal acceleration a is given by a = Δv / Δt = v Δθ / Δt = v ω, where ω = Δθ / Δt is the angular speed. Substituting ω = v / r yields two equivalent forms.
向心加速度 a 为 a = Δv / Δt = v Δθ / Δt = v ω，其中 ω = Δθ / Δt 是角速度。代入 ω = v / r 得到两个等价表达式。
a = v² / r = ω² r
These formulas are valid only for uniform circular motion; the acceleration vector always points towards the centre.
这两个公式仅适用于匀速圆周运动；加速度矢量始终指向圆心。
3. Centripetal Force | 向心力
According to Newton’s second law, the net force causing centripetal acceleration is the centripetal force F = m a. Replacing a with v² / r or ω² r gives the two standard expressions.
根据牛顿第二定律，产生向心加速度的合力就是向心力 F = m a。用 v² / r 或 ω² r 代换 a 得到两个标准表达式。
F = m v² / r = m ω² r
This force is not a new type of force—it is provided by tension, gravity, friction or the normal contact force. For example, a car rounding a bend relies on lateral friction to supply m v² / r.
该力并非新型力——它可以由张力、重力、摩擦力或支持力充当。例如，汽车转弯时依赖侧向摩擦力提供 m v² / r。
4. Electric Field Strength | 电场强度
Electric field strength E is defined as the force per unit positive charge: E = F / q. Its SI unit is N C⁻¹. Substituting Coulomb’s law F = k Q q / r² for a point charge Q gives the field due to that point charge.
电场强度 E 定义为单位正电荷受到的力：E = F / q，SI 单位是 N C⁻¹。将点电荷 Q 的库仑定律 F = k Q q / r² 代入，得到点电荷激发的电场。
E = k Q / r²
Here k = 1 / (4 π ε₀) and the field is radial: it points away from a positive Q and towards a negative Q.
式中 k = 1 / (4 π ε₀)，此电场为径向场：从正电荷 Q 发出，指向负电荷 Q。
5. Uniform Electric Field | 匀强电场
Between two parallel plates separated by distance d with potential difference V, the work done on a charge q moving from one plate to the other is W = q V. In terms of the electric force F = q E acting over distance d, the work is also W = F d = q E d.
在两块相距 d 的平行板之间加上电势差 V，电荷 q 从一板移至另一板做功 W = q V。就电场力 F = q E 在距离 d 上做功而言，W = F d = q E d。
Equating the two expressions for work gives q V = q E d, so E = V / d. This shows the field is uniform and independent of position between the plates.
令两个做功表达式相等得 q V = q E d，因此 E = V / d。这表明板间电场均匀，与位置无关。
E = V / d
6. Capacitance and Energy Stored | 电容与储能
Capacitance C is defined as the charge stored per unit potential difference: C = Q / V. Its unit is the farad (F). For a parallel-plate capacitor, the capacitance depends on plate area A and separation d through the relation C = ε₀ A / d.
电容 C 定义为单位电势差下的储电量：C = Q / V，单位是法拉 (F)。对于平行板电容器，电容取决于板面积 A 和间距 d，关系式为 C = ε₀ A / d。
To store charge, work must be done to move electrons against the growing potential difference. The incremental work dW = V dq, and since V = q / C, we integrate from q = 0 to Q:
储存电荷时，需要克服逐渐升高的电势差做功。微小功 dW = V dq，由于 V = q / C，从 q = 0 积分到 Q：
W = ∫₀ᴼ (q / C) dq = ½ Q² / C
Using C = Q / V, this can be expressed in three equivalent forms:
利用 C = Q / V，可表示为三种等价形式：
E = ½ Q V = ½ C V² = ½ Q² / C
This stored energy resides in the electric field between the plates.
这部分储存的能量存在于两板之间的电场中。
7. Time Constant for RC Circuit | RC 电路时间常数
For a discharging capacitor through a fixed resistor R, the charge decays exponentially: q = Q₀ e^(–t / RC). The product RC appears in the exponent and has dimensions of time. It is called the time constant τ.
电容器通过固定电阻 R 放电时，电荷按指数衰减：q = Q₀ e^(–t / RC)。乘积 RC 出现在指数中，具有时间量纲，称为时间常数 τ。
Differentiating the charge equation gives the current: i = dq / dt = – (Q₀ / RC) e^(–t / RC). At t = 0, the initial current is I₀ = – V₀ / R, confirming that RC governs the decay rate.
对电荷方程求导得到电流：i = dq / dt = – (Q₀ / RC) e^(–t / RC)。在 t = 0 时，初始电流为 I₀ = – V₀ / R，证明 RC 决定了衰减速率的快慢。
τ = R C
After one time constant, the charge falls to about 37% of its initial value. The same constant governs charging behaviour.
经过一个时间常数后，电荷降至初始值的约 37%。该常数同样适用于充电过程。
8. Force on a Moving Charge in Magnetic Field | 磁场中运动电荷的力
A charge q moving with velocity v in a magnetic field B experiences a magnetic force. The direction is perpendicular to both v and B, given by Fleming’s left-hand rule (or the right-hand rule for positive charges). The magnitude depends on the angle θ between v and B.
电荷 q 以速度 v 在磁感应强度 B 中运动时会受到磁力，方向垂直于 v 和 B，遵循左手定则（正电荷用右手）。力的大小取决于 v 和 B 之间的夹角 θ。
F = B q v sin θ
When the velocity is perpendicular to the field (θ = 90°), sin θ = 1 and F = B q v. This relationship is derived from the Lorentz force law and is applied in mass spectrometers and particle detectors.
当速度与磁场垂直 (θ = 90°) 时，sin θ = 1，F = B q v。该关系源自洛伦兹力定律，用于质谱仪和粒子探测器。
9. Radius of Circular Path in Magnetic Field | 磁场中圆周运动半径
If a charged particle moves perpendicular to a uniform magnetic field, the magnetic force provides the centripetal force required for circular motion. Setting B q v equal to m v² / r allows us to solve for the radius r.
若带电粒子垂直于匀强磁场运动，磁力充当向心力。令 B q v = m v² / r，可解出轨道半径 r。
B q v = m v² / r → r = m v / (B q)
The radius is proportional to momentum m v and inversely proportional to the magnetic field strength and charge. This formula is central to analysing cyclotrons and the deflection of charged particles.
半径与动量 m v 成正比，与磁场强度和电荷量成反比。该公式是分析回旋加速器和带电粒子偏转的核心。
10. Faraday’s Law of Induction | 法拉第电磁感应定律
When the magnetic flux Φ through a coil changes, an electromotive force (emf) is induced. The magnitude of the induced emf equals the rate of change of flux linkage N Φ. Lenz’s law gives the negative sign, indicating opposition to the change.
当穿过线圈的磁通量 Φ 变化时，会感应出电动势。感应电动势的大小等于磁链 N Φ 的变化率。楞次定律给出负号，表明感应方向反抗变化。
ε = – N (ΔΦ / Δt)
For a rectangular coil rotating at constant angular speed ω in a uniform magnetic field, the flux linkage varies sinusoidally: N Φ = N B A cos(ω t). Differentiating gives ε = N B A ω sin(ω t), which is the standard result for an AC generator.
对于在匀强磁场中以恒定角速度 ω 转动的矩形线圈，磁链随时间正弦变化：N Φ = N B A cos(ω t)。求导得到 ε = N B A ω sin(ω t)，这是交流发电机的标准表达。
This equation underpins all electromagnetic induction calculations, including transformers and back emf in motors.
该方程是所有电磁感应计算的基础，包括变压器和电动机反电动势。
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