A-Level Physics Unit 3 Formula Derivations (Jan 2019) | A-Level物理单元3公式推导（2019年1月评分方案）

📚 A-Level Physics Unit 3 Formula Derivations (Jan 2019) | A-Level物理单元3公式推导（2019年1月评分方案）

In A-Level Physics examinations, particularly the Unit 3 practical paper, candidates are often required not only to perform calculations but also to derive key formulas from first principles or from graphical analysis. The January 2019 mark scheme provides clear guidance on how such derivations should be structured and the awarding of marks. This article revisits the essential derivations commonly tested, referencing the style of the Jan 2019 Unit 3 paper, and explains each step in a way that aligns with examiner expectations.

在A-Level物理考试中，特别是单元3实验卷，考生不仅需要进行计算，还常需从基本原理或图形分析出发推导关键公式。2019年1月的评分方案明确说明了这类推导应如何构建以及得分方式。本文重温了常考的核心推导，参照2019年1月单元3试卷的风格，并以符合考官预期的方式逐步解释每一步。

1. Understanding the Mark Scheme and Formula Derivation | 理解评分方案与公式推导

The mark scheme for A-Level Physics Unit 3 allocates specific marks for showing the derivation process, not just the final formula. Examiners look for logical steps, correct use of symbols, and clear algebraic manipulation. A formula that is simply stated without derivation typically gains no credit in these questions.

A-Level物理单元3的评分方案为展示推导过程分配了专门分数，而不仅仅是最终公式。考官看重逻辑步骤、符号的正确使用和清晰的代数操作。在这些问题中，直接写出公式而不展示推导通常不会得分。

In the Jan 2019 paper, candidates were asked to derive relationships like the gradient of a straight-line graph in terms of physical constants. Each step—such as rearranging an equation, taking logarithms, or substituting expressions—had to be explicitly shown.

在2019年1月的试卷中，考生被要求推导如直线图斜率用物理常量表示的关系。每一步——例如重新排列方程、取对数或代入表达式——都必须明确展示。

2. Deriving the Resistivity Formula from Basic Definitions | 从基本定义推导电阻率公式

Resistivity ρ is defined by the equation R = ρL/A, where R is resistance, L is length, and A is cross-sectional area. This formula can be rearranged to ρ = RA/L. In an experiment, you may measure R, L, and diameter d to find A = πd²/4. The derivation of ρ in terms of measured quantities is straightforward but essential: ρ = R × (πd²/4) / L = (πRd²)/(4L).

电阻率ρ由方程R = ρL/A定义，其中R是电阻，L是长度，A是横截面积。该公式可重新排列为ρ = RA/L。实验中可能测量R、L和直径d以求出A = πd²/4。用测量量表示的ρ推导直接但至关重要：ρ = R × (πd²/4) / L = (πRd²)/(4L)。

Often, the mark scheme expects candidates to derive the expression for the gradient of a graph. For example, if you plot R against 1/A (or against L/A), you can obtain ρ from the slope. Starting from R = (ρL)/A, a plot of R (y-axis) against L/A (x-axis) yields a straight line through the origin with gradient = ρ.

评分方案常期待考生推导出图形梯度的表达式。例如，若绘制R与1/A图（或L/A图），可从斜率求得ρ。由R = (ρL)/A出发，若以R为y轴，L/A为x轴，则得到一条过原点的直线，其梯度 = ρ。

R = ρ (L/A)

When the area is expressed in terms of diameter, A = πd²/4, the equation becomes R = (4ρL)/(πd²). A plot of R vs L will have a gradient of (4ρ)/(πd²), provided d is constant. This derivation is a typical mark scheme requirement.

当面积用直径表示A = πd²/4时，方程变为R = (4ρL)/(πd²)。若绘制R对L的图，其斜率将是(4ρ)/(πd²)，前提是d恒定。这一推导是评分方案中的典型要求。

3. Propagation of Uncertainty for ρ: Step-by-Step Derivation | 电阻率不确定度传播的逐步推导

To find the percentage uncertainty in ρ, we combine the uncertainties of R, d, and L. Since ρ = (πR d²)/(4L), and π and 4 are exact constants, we apply the rule: for a product/quotient, percentage uncertainty is the sum of absolute percentage uncertainties, with powers multiplied. Thus, %U(ρ) = %U(R) + 2×%U(d) + %U(L).

要求出ρ的百分不确定度，需合并R、d和L的不确定度。由于ρ = (πR d²)/(4L)，且π和4为精确常数，应用规则：对于乘除运算，百分不确定度为各绝对百分不确定度之和，其中指数需乘以相应倍数。因此，%U(ρ) = %U(R) + 2×

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