📚 A-Level Physics: Formula Derivations from the Practical Handbook | A-Level物理：实验手册公式推导
The A-Level Physics practical handbook contains a series of core experiments, each requiring a clear understanding of the relevant formulas. Rather than simply memorising these equations, students are expected to derive them from fundamental principles and to appreciate how experimental data lead to the calculated quantities. This article walks you through the derivations of the key formulas encountered in the most common A-Level physics practicals, helping you master both the theory and the practical skills.
A-Level 物理实验手册涵盖一系列核心实验，每个实验都需要清晰理解相关公式。与其单纯记忆这些方程，不如从基本原理出发推导它们，并理解实验数据如何得出最终结果。本文带你逐一推导最常见A-Level物理实验中涉及的公式，帮助你同时掌握理论与实验技能。
1. Simple Pendulum: Determining g | 单摆实验：测定重力加速度g
provided by the component of weight tangential to the arc, giving an angular frequency ω = √g/L.
在小角度摆动下，单摆作简谐运动。回复力由重力沿弧切线方向的分量提供，得到角频率 ω = √g/L。
Since period T = 2π/ω, we directly obtain T = 2π √L/g /g) L. Therefore, plotting T² against L produces a straight line through the origin with gradient 4π²/g.
由于周期 T = 2π/ω，直接得到 T = 2π √L/g。两边平方得 T² = (4π²/g) L。因此绘制 T²-L 图时，可得一条过原点的直线，斜率为 4π²/g。
g = 4π² / gradient
In the experiment, you measure the time for multiple oscillations to reduce timing uncertainty, and vary the length L systematically. The derived value of g can be compared with 9.81 m s⁻².
实验中，通过测量多个周期的时间以减少计时误差，并系统地改变摆长 L。求得的 g 值可与标准值 9.81 m s⁻² 比较。
2. Free Fall using a Trapdoor or Light Gates | 自由落体实验：使用落板或光电门
n g. The kinematic equation s = ut + ½at² reduces to s = ½gt² when initial velocity u = 0.
从静止释放的物体做匀加速直线运动，加速度为 g。运动学方程 s = ut + ½at² 在初速度 u = 0 时简化为 s = ½gt²。
r each trial. More reliably, a graph of s against t² has gradient ½g, so g = 2 × gradient.
测量下落距离 s 与相应的下落时间 t，即可根据 g = 2s/t² 计算每一次的重力加速度。更可靠的方法是绘制 s–t² 图，其斜率为 ½g，因此 g = 2 × 斜率。
g = 2s / t² or g = 2 × (gradient of s vs t²)
e instantaneous speed at two positions, the equation v² = u² + 2as can be applied. With v² – u² = 2gs, plotting v² against s yields gradient 2g.
若使用光电门测量两个位置的瞬时速度，可用 v² = u² + 2as。由 v² – u² = 2gs，绘制 v²-s 图得到斜率 2g。
3. Young Modulus of a Wire | 金属丝的杨氏模量
stretching force, A the cross-sectional area, e the extension, and L the original length.
杨氏模量 E 定义为弹性限度内拉伸应力与拉伸应变的比值。应力 = F/A，应变 = e/L，其中 F 为拉力，A 为横截面积，e 为伸长量，L 为初始长度。
E = (F/A) / (e/L) = FL / (eA)
so a graph of extension e against load F has slope L/(AE), from which E can be extracted.
典型实验中，在金属丝上施加砝码，用游标尺测量伸长量。用千分尺测量直径 d，计算 A = πd²/4。重排公式得 e = (L/(AE)) F，因此绘制伸长量 e–负荷 F 图，斜率为 L/(AE)，从而可求出 E。
E = L / (A × slope of e vs F)
Common error sources include measuring the initial length accurately and ensuring the wire remains within its elastic limit. The formula derivation directly follows from the definition of stress and strain.
常见误差来源包括准确测量初始长度以及确保金属丝处于弹性限度内。公式推导直接源于应力和应变的定义。
4. Resistivity of a Metal Wire | 金属丝的电阻率
R = ρL/A. By measuring the resistance of a wire of known dimensions, ρ can be determined.
电阻率 ρ 是材料的固有属性，通过关系式 R = ρL/A 将电阻 R、长度 L 和横截面积 A 联系起来。测量已知尺寸导线的电阻，即可确定 ρ。
R = ρL / A ⇒ ρ = RA / L
In the experiment, the wire' L should be a straight line through the origin with gradient ρ/A, hence ρ = gradient × A.
实验中，用千分尺测出导线直径 d，得 A = πd²/4。电阻 R 通过伏特计和安培计利用 V/I 求得。若系统地改变 L，绘制 R-L 图应为一条过原点的直线，斜率为 ρ/A，因此 ρ = 斜率 × A。
ρ = (gradient of R vs L) × (πd²/4)
This expression shows that resistivity can be calculated without measuring R directly for each length if the gradient is used, reducing random errors.
此表达式表明，如果利用斜率，可不必单独测量每一长度下的 R 来计算电阻率，从而减少随机误差。
5. Spring Constant from Hooke's Law | 通过胡克定律测定弹簧劲度系数
Hooke' rce F, provided the elastic limit is not exceeded: F = kx, where k is the spring constant.
胡克定律指出，在弹性限度内，弹簧的伸长量 x 与所施加的力 F 成正比：F = kx，其中 k 为劲度系数。
a graph of extension x against mass m is a straight line with slope g/k, so k = g / slope.
当在弹簧上悬挂重物时，力 F = mg。因此 mg = kx，重排得 x = (g/k) m。这表明伸长量 x 与质量 m 的关系图为一条直线，斜率为 g/k，从而 k = g / 斜率。
k = g / (gradient of x vs m)
Alternatively, if force F is plotted directly against extension x, the gradient is simply k. The derivation follows directly from the proportionality statement, and the experiment reinforces understanding of linear relationships and systematic measurement.
另外，若直接绘制力 F 与伸长量 x 的关系图，斜率即为 k。推导直接来自比例关系，实验加深对线性关系和系统测量的理解。
6. Refractive Index using Snell's Law | 利用斯涅尔定律测量折射率
When light passes from one medium to another, Snell' ass n = sin i / sin r, where i is the angle of incidence and r is the angle of refraction.
当光从一种介质进入另一种时，斯涅尔定律描述为：n₁ sin θ₁ = n₂ sin θ₂。对于从空气射入玻璃块的光线，n₁ ≈ 1，因此玻璃的折射率 n = sin i / sin r，其中 i 为入射角，r 为折射角。
n = sin i / sin r
In the practical, a ray box and a semi-circular or rectangular block are used. By measuring a series of angles i and r, a graph of sin i against sin r is plotted. The gradient gives the refractive index n directly, and the linearity confirms Snell's law.
实验中，使用光源盒和半圆形或矩形玻璃块。通过测量一系列 i 和 r，绘制 sin i–sin r 图，斜率直接给出折射率 n，线性关系验证了斯涅尔定律。
rom glass to air, n = 1 / sin C, where C is the critical angle. This is derived from Snell's law by setting r = 90° and n₂ = 1.
对于利用全反射的临界角方法，光从玻璃射向空气时，n = 1 / sin C，其中 C 为临界角。此式由斯涅尔定律设定 r = 90°、n₂ = 1 推导而来。
n = 1 / sin C
7. Thin Lens Equation for a Convex Lens | 凸透镜的薄透镜方程
ere u is the object distance, v is the image distance, and f is the focal length. This formula is fundamental in optical experiments, though its derivation from ray diagrams and similar triangles is assumed knowledge.
凸透镜成像遵循透镜公式 1/f = 1/u + 1/v，其中 u 为物距，v 为像距，f 为焦距。该公式是光学实验的基础，虽然其几何推导通常属于基础理论。
ot a graph of 1/v against 1/u: both the x- and y-intercepts equal 1/f, so f = 1/intercept.
实验中，测量不同物距下的 u 和 v。对于每一对数据，焦距可由 f = uv/(u+v) 计算。为减小误差，通常绘制 1/v–1/u 图：两个截距均为 1/f，因此 f = 1/截距。
f = uv / (u+v) and f = 1 / (intercept of 1/v vs 1/u graph)
The rearrangement demonstrates algebraic manipulation skills expected in A-Level physics. The graph approach also verifies that the lens formula is valid for a range of object distances, providing evidence for the thin lens model.
这一重排体现了A-Level物理所要求的代数处理能力。图示法也验证了透镜公式在一系列物距下的有效性，为薄透镜模型提供证据。
8. Specific Heat Capacity of a Solid | 固体的比热容
pplied by an immersion heater: Q = VIt, where V is voltage, I current, and t heating time.
材料的比热容 c 由方程 Q = mcΔθ 定义，其中 Q 为传递的热量，m 为质量，Δθ 为温度变化。在电加热实验中，热量由浸入式电热器提供：Q = VIt，V 为电压，I 为电流，t 为加热时间。
c = Q / (m Δθ) = VIt / (m Δθ)
By measuring the temperature rise Δθ over a timed interval and recording the electrical power, c can be calculated directly. To improve accuracy, the experiment is often performed with a graph of temperature θ against time t, and the initial rate of temperature rise is used to minimise heat loss corrections.
通过测量定时区间内的温升 Δθ 并记录电功率，可直接计算 c。为提高准确性，通常绘制温度 θ–时间 t 图，利用初始温升速率以最小化热损失修正。
linearised form is derived from the definition by assuming minimal heat loss at the start.
若功率 P = VI 恒定，则 Δθ = (P/(mc)) t，因此初始 θ-t 图斜率为 P/(mc)，得 c = P/(m × 斜率)。这种线性化形式由定义推导而来，假设初始阶段热损失极小。
9. Boyle's Law: Pressure-Volume Relationship for a Gas | 玻意耳定律：气体的压强–体积关系
Boyle' = k(1/V), indicating that a graph of p against 1/V is a straight line through the origin.
玻意耳定律指出，对于一定质量、温度恒定的气体，压强与体积的乘积为常数：pV = k。可改写为 p = k(1/V)，表明 p–1/V 图为一条过原点的直线。
pV = constant ⇒ p = constant / V
In the practical, a gas syringe or a trapped air column with a pressure sensor is used. By recording p and V and plotting either p against 1/V or calculating pV for each trial, the law can be verified. The constant k is proportional to nRT from the ideal gas equation, linking macroscopic measurements to the mole concept.
实验中，使用气体注射器或带压强传感器的密封空气柱。记录 p 和 V，绘制 p–1/V 图或计算各次的 pV 值，即可验证该定律。常数 k 与理想气体状态方程中的 nRT 成正比，将宏观测量与摩尔概念联系起来。
The derivation does not require calculus at this stage; it is an empirical law. However, the mathematical rearrangement highlights the inverse proportionality, a key skill in data analysis.
现阶段推导不需要微积分；这是一个经验定律。然而，数学重排突显了反比例关系，这是数据分析中的关键技能。
10. Capacitor Discharge and Time Constant | 电容放电与时间常数
rges through a resistor, the voltage V across the capacitor decays exponentially: V = V₀ e–t/RC, where V₀ is the initial voltage, R the resistance, C the capacitance, and t the time. The product RC is called the time constant τ.
当电容通过电阻放电时，电容两端的电压 V 呈指数衰减：V = V₀ e–t/RC，其中 V₀ 为初始电压，R 为电阻，C 为电容，t 为时间。乘积 RC 称为时间常数 τ。
V = V₀ e–t / RC and τ = RC
Taking the natural logarithm of both sides gives ln V = ln V₀ – (t / RC). Hence a graph of ln V against t is a straight line with gradient –1/RC, from which τ can be found.
两边取自然对数得 ln V = ln V₀ – (t / RC)。因此 ln V–t 图为一条直线，斜率为 –1/RC，由此可求得 τ。
gradient = –1 / (RC) ⇒ RC = –1 / gradient
This logarithmic linearisation is derived directly from the exponential law. In the practical, a data logger or a voltmeter and stopwatch are used to record V at regular intervals. The experiment reinforces understanding of exponential processes and the use of natural logarithms in physics.
这种对数线性化由指数定律直接推导而来。实验中，使用数据记录器或伏特计和秒表定期记录 V。该实验强化了对指数过程的理解及自然对数在物理中的应用。
lve, can be shown to be T½ = RC ln 2, another useful derived result from setting V = V₀/2.
此外，放电半衰期 T½（电压减半所需时间）可推导为 T½ = RC ln 2，这是令 V = V₀/2 得到的另一个实用推导结果。
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图