A-Level Physics: Practical and Analytical Skills – Formula Derivation | A-Level物理实验与分析技能：公式推导

📚 A-Level Physics: Practical and Analytical Skills – Formula Derivation | A-Level物理实验与分析技能：公式推导

Practical investigations in A-Level Physics demand more than just collecting data; they require a deep understanding of how to manipulate theoretical equations to extract meaningful quantities from graphs. This article explores essential formula derivations, linearisation techniques, and uncertainty analysis that appear throughout the Oxford AQA International A-Level specification.

A-Level 物理的实验探究不止于收集数据，更要求深刻理解如何通过理论公式的变换，从图像中提取有意义的物理量。本文围绕牛津 AQA 国际 A-Level 考纲，解析核心的公式推导、线性化方法以及不确定度分析。

1. Why Linearise? | 为何要线性化？

Most physical relationships are not directly proportional, yet plotting a straight line remains the most reliable way to determine constants and validate models. By rearranging an equation into the form y = mx + c, we transform curvature into linearity, making it straightforward to use the gradient and intercept.

大多数物理关系并非直接成正比，但绘制直线仍然是确定常数和验证模型最可靠的方式。通过将方程整理为y = mx + c的形式，我们将非线性扭转为线性，从而能够直接使用斜率和截距。

For example, the period T of a simple pendulum depends on the square root of length L. Squaring both sides gives T² proportional to L, a linear form. Similarly, exponential decay can be linearised by taking natural logarithms. This skill is central to practical papers and the analysis of experimental errors.

例如，单摆的周期T与摆长L的平方根成正比。两边平方后得到T² 与 L 成正比，即线性关系。同样，指数衰减可以通过取自然对数来线性化。这一技能是实验卷和误差分析的核心。

2. Simple Pendulum – Deriving the Linear Equation | 单摆 – 直线方程的推导

Start with the well-known formula for the period of a simple pendulum at small amplitudes:

从小振幅单摆的周期公式出发：

T = 2π √(L / g)

Square both sides to remove the square root:

两边平方消去根号：

T² = 4π² L / g

This can be written in the straight-line form y = mx (with zero intercept) if we let y = T² and x = L. The gradient m is then:

若令 y = T²，x = L，上式即可写为 y = mx 的形式（截距为零），斜率 m 为：

m = 4π² / g

Thus, by measuring T for various values of L and plotting T² against L, we obtain a straight line through the origin. The value of g is simply g = 4π² / m.

因此，测量不同摆长 L 下的周期 T，并以 T² 对 L 作图，即可得到一条过原点的直线。g 可直接由 g = 4π² / m 求得。

3. Graphical Determination of g | 利用图像测定重力加速度

In practice, data points will show some scatter. Draw a best-fit straight line that passes through the origin. Calculate the gradient m from a large triangle on the graph. Then:

实际数据点会有些分散。画出过原点的最佳拟合直线，并在图上取一个大的三角形计算斜率 m。然后：

g = 4π² / m

If the line does not quite pass through the origin, a non-zero intercept may indicate a systematic error, such as an incorrect zero for length measurement. The consistency of g with the accepted value of 9.81 m s⁻² tests the quality of the experiment.

若直线不完全过原点，非零截距可能说明存在系统误差，比如长度测量的零位不准。将求出的 g 与公认值 9.81 m s⁻² 对比，可检验实验质量。

Using error bars on T² and worst-acceptable lines (steepest and shallowest reasonable fits) provides the uncertainty in gradient, Δm. Then the absolute uncertainty in g is Δg = (4π² / m²) Δm, or more simply Δg/g = Δm/m.

在 T² 上添加误差棒，并绘制最陡和最浅的合理拟合线，可得到斜率的不确定度 Δm。g 的绝对不确定度则为 Δg = (4π² / m²) Δm，或更简单地 Δg/g = Δm/m。

4. Combining Uncertainties in Derived g | 合成重力加速度的不确定度

If the raw measurements of L and T have associated uncertainties ΔL and ΔT, we can estimate the fractional uncertainty in g directly from the formula g = 4π²L / T². Since g depends on L and T⁻², the fractional uncertainty is:

若直接测量量L和T分别带有不确定度ΔL和ΔT，我们可以从公式 g = 4π²L / T² 直接估算 g 的相对不确定度。因 g 依赖于 L 和 T⁻²，相对不确定度为：

Δg/g = √[(ΔL/L)² + (2 ΔT/T)²]

This equation comes from the standard propagation of uncertainties: for a product, we add relative uncertainties in quadrature, and for a power n, the relative uncertainty is multiplied by |n|. The factor 2 arises from the square in T².

该式来自标准的不确定度传递公式：对于乘积，相对不确定度以平方和根号形式合成；对于 n 次幂，相对不确定度乘以 |n|。因子 2 即源于 T²。

If ΔL = ±0.001 m and ΔT = ±0.01 s, both must be converted to relative forms and then combined.

若 ΔL=±0.001 m，ΔT=±0.01 s，两者需转换为相对形式后合成。
This analytical approach can be compared with the graphical uncertainty found from worst-fit lines, and the larger value is usually quoted.

这种分析方式可与由最劣拟合线得到的图像不确定度作比较，通常取两者中较大者作为最终不确定度。

5. Example 2: Resistivity of a Wire | 实例2: 导线电阻率

The resistance of a metallic wire at constant temperature is given by:

恒温下金属导线的电阻由下式给出：

R = ρ L / A

where ρ is resistivity, L is length, and A is cross-sectional area (A = πr² = πd²/4). Insert the area expression:

其中 ρ 为电阻率，L 为长度，A 为横截面积 (A = πr² = πd²/4)。代入面积表达式：

R = 4ρL / (π d²)

For a wire of constant diameter, we can treat the coefficient as constant. Plotting R (on y-axis) against L (on x-axis) yields a straight line with gradient m = 4ρ/(π d²). Hence resistivity is:

对于直径不变的导线，系数可视为常数。以R (y轴) 对 L (x轴) 作图，得到斜率为 m = 4ρ/(π d²) 的直线。故电阻率为：

ρ = m π d² / 4

6. Obtaining Resistivity and Its Uncertainty | 电阻率的求取及其不确定度

Measure the diameter d of the wire at several points using a micrometer, then calculate the mean and its uncertainty Δd (from the spread or the instrument limit, whichever larger). Plot R vs L, draw the best-fit line, and find its gradient m ± Δm.

用千分尺在导线多处测量直径 d，计算平均值及不确定度 Δd（取自多次测量分散性或仪器允差中较大者）。作 R-L 图，画出最佳拟合线，求出斜率 m ± Δm。

The fractional uncertainty in ρ can be expressed as:

ρ 的相对不确定度可表示为：

Δρ/ρ = √[(Δm/m)² + (2 Δd/d)²]

The factor 2 with Δd/d appears because d appears squared in the formula. This combined uncertainty gives a reliability range for the resistivity value, to be compared with standard tables (e.g., ρ of copper ~ 1.7 × 10⁻⁸ Ω m).

式中 Δd/d 前有因子 2，是因为公式中出现 d²。合成的不确定度给出了电阻率值的可靠性区间，可与标准数据表（如铜的 ρ ~ 1.7×10⁻⁸ Ω m）比较。

7. Capacitor Discharge – An Exponential Model | 电容器放电 – 指数模型

When a capacitor discharges through a fixed resistor, the voltage across it follows an exponential decay:

当电容器通过固定电阻放电时，其两端电压遵循指数衰减规律：

V = V₀ e^{-t / RC}

where V₀ is the initial voltage at t = 0, R is resistance, C is capacitance, and RC is the time constant. Taking the natural logarithm of both sides linearises the equation:

其中 V₀ 为 t=0 时的初始电压，R 为电阻，C 为电容，RC 为时间常数。两边取自然对数即可线性化：

ln V = ln V₀ − (1 / RC) t

This is of the form y = c + m x, with y = ln V, x = t, gradient m = −1/RC, and intercept c = ln V₀.

此式即为 y = c + m x 形式，其中 y = ln V，x = t，斜率 m = −1/RC，截距 c = ln V₀。

8. Graphical Analysis for RC Circuits | RC 电路的图像分析

Record voltage values V at various times during discharge. Compute the natural logarithms and plot ln V against t. The points should lie on a straight line of negative slope. Determine the gradient m and hence the time constant:

记录放电过程中不同时刻的电压值 V，计算自然对数并绘制 ln V 对 t 图。数据点应落在负斜率的直线上。求出斜率 m，由此得到时间常数：

RC = −1 / m

If R is known, the capacitance can be calculated as C = −1/(m R). The intercept ln V₀ should match the natural log of the initial recorded voltage, serving as a check against systematic errors.

若 R 已知，电容可计算为 C = −1/(m R)。截距 ln V₀ 应与初始记录电压的自然对数相符，可作为系统误差的检验。

Uncertainty analysis here often uses the spread of the gradient from alternative fit lines. The fractional uncertainty in C then follows ΔC/C = Δm/m (if R has negligible uncertainty), again allowing comparison with component tolerance.

此处的不确定度分析常用不同拟合线斜率的范围。C 的相对不确定度则为 ΔC/C = Δm/m（若 R 的不确定度可忽略），也可与元件容差比较。

9. Power Laws and Logarithmic Plots | 幂律关系与对数坐标

Another important linearisation applies when a variable y depends on a power of x: y = k xⁿ. Taking logarithms (base 10 or natural) gives:

另一种重要的线性化适用于 y 与 x 的幂次关系：y = k xⁿ。取对数（以10为底或自然对数都可）得到：

log y = log k + n log x

A plot of log y against log x produces a straight line whose gradient is the exponent n and whose intercept is log k. This technique is useful, for instance, to verify whether the period of a pendulum truly depends on √L, or to determine the exponent in an unknown relationship.

以 log y 对 log x 作图可得一条直线，其斜率即为指数 n，截距为 log k。此方法可用于验证单摆周期是否确实与 √L 有关，或确定未知关系中的指数。

In exam practicals, candidates may be asked to test a relationship by converting data to logarithms, plotting them, and using the gradient to find unknown constants. The ability to switch between raw form, linearised form, and logarithmic form is a key analytical skill.

在考试实验中，考生可能被要求将数据转换为对数形式，作图后利用斜率求未知常数。能够在原始形式、线性化形式和对数形式之间灵活切换，是一项关键的分析技能。

10. Conclusion and Exam Tips | 结论与应考提示

Mastering the art of formula derivation and linearisation transforms a set of raw numbers into physically meaningful constants. Always identify what should be plotted on each axis to achieve a straight line, clearly state the relationship between gradient/intercept and the desired quantities, and present uncertainties with appropriate significant figures.

掌握公式推导和线性化的技巧，能将原始数据转化为有物理意义的常数。务必明确每个坐标轴应作什么变量以得到直线，清晰地阐述斜率/截距与所求物理量的关系，并以适当的有效数字呈现不确定度。

Common pitfalls include forgetting to square the period, omitting the factor 2 when propagating uncertainty for a squared term, and failing to use enough significant figures in the logarithmic calculations. Practice with past-paper data sets builds confidence in handling these derivations under timed conditions.

常见错误包括忘记周期平方、在平方项的不确定度合成中遗漏因子2，以及对数运算中有效数字位数不足。通过历年真题数据进行练习，能够提升在限时条件下处理这类推导的信心。

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A-Level Physics: Practical and Analytical Skills – Formula Derivation | A-Level物理实验与分析技能：公式推导