📚 Formula Derivation in the A-Level Physics Unit 3 Mark Scheme (Jan22) | A-Level物理第三单元2022年1月评分方案中的公式推导
This article unpacks a typical formula derivation question from the A-Level Physics Unit 3 (Practical Skills) mark scheme for the January 2022 exam series. By focusing on an experiment to determine the resistivity of a metal wire, we explore how candidates are expected to manipulate equations, linearize data and propagate uncertainties. The step-by-step breakdown mirrors the logic required in the official mark scheme, helping you master the derivation techniques that can secure high marks.
本文解析了2022年1月A-Level物理第三单元（实验技能）评分方案中的一道典型公式推导题。我们以测量金属丝电阻率的实验为例，探讨考生如何变换方程、将数据线性化以及进行不确定度传递。分步解析完全对应官方评分方案的逻辑，帮助你掌握这些能拿下高分的推导技巧。
1. Role of Formula Derivation in Unit 3 | 第三单元公式推导的作用
Unit 3 of the International A-Level Physics course assesses practical skills, including data analysis and error evaluation. A common task in the January 2022 paper required learners to derive a physical quantity from an experimental graph, applying linear relationships and algebraic manipulation. The mark scheme rewards logical reasoning, correct algebraic steps, and clear uncertainty treatment.
国际A-Level物理第三单元考查实验技能，包含数据分析与误差评估。2022年1月试卷中一道常见题型要求学生从实验图像中推导一个物理量，涉及线性关系与代数变换。评分方案对逻辑推理、正确代数步骤以及明确的的不确定度处理均有给分。
2. The Experimental Context: Resistivity of a Wire | 实验背景：金属丝电阻率
The examination paper described an investigation where the resistance R of a constantan wire was measured for different lengths L, with the cross-sectional area A kept constant. Candidates needed to determine the resistivity ρ of the alloy using the fundamental relation R = ρL / A. The mark scheme required students to derive an expression for ρ from the gradient of a suitable straight-line graph.
试卷描述了一个实验：测量不同长度L的一段康铜丝的电阻R，横截面积A保持不变。考生需要利用基本关系式 R = ρL / A 测定该合金的电阻率ρ。评分方案要求学生从一条合适直线图像的斜率推导出ρ的表达式。
3. Core Equation and Its Rearrangement | 核心方程及其变形
The starting point is the resistivity equation:
R = ρL / A
To obtain a straight-line plot, this must be rearranged into the form y = mx + c. If R is plotted on the y-axis and L on the x-axis, the equation becomes:
R = (ρ / A) L + 0
The gradient m is therefore equal to ρ / A, and the intercept is zero. The derivation task then focuses on extracting ρ from the gradient.
出发点是电阻率方程：
R = ρL / A
为了得到直线图，需将方程化为 y = mx + c 的形式。若将R画在y轴、L画在x轴，方程变为：
R = (ρ / A) L + 0
因此斜率m等于ρ / A，截距为零。推导任务的核心就是从斜率求出ρ。
4. Deriving Resistivity from the Graph Gradient | 从图像斜率推导电阻率
Let the experimentally determined gradient be mexp. From the linear relationship:
mexp = ΔR / ΔL = ρ / A
Multiply both sides by A to isolate ρ:
ρ = mexp × A
The cross-sectional area A must be calculated from the wire’s diameter d, measured with a micrometer. Using A = πd²/4, the final resistivity expression becomes:
ρ = mexp × (πd² / 4)
This logical, step-by-step isolation of the target quantity is exactly what the Jan22 mark scheme credits under formula derivation.
设实验测得的斜率为 mexp。由线性关系：
mexp = ΔR / ΔL = ρ / A
两边同乘以A即可分离出ρ：
ρ = mexp × A
横截面积A须通过螺旋测微器测得的直径d计算，A = πd²/4。最终电阻率表达式为：
ρ = mexp × (πd² / 4)
这种逐步分离目标量的逻辑推导，正是Jan22评分方案在公式推导题中给予分值的地方。
5. Tackling Logarithmic Derivations (Alternative Graph) | 处理对数推导（另一种图像方法）
Some Jan22 candidates encountered a variation where they had to verify that R ∝ L. By plotting a log-log graph: log R = log L + log(ρ / A). The gradient of this graph was expected to be 1, confirming direct proportionality. Deriving ρ then involved using the intercept log(ρ / A) = c, giving ρ = A × 10c. This exercise tested candidates’ ability to manipulate logarithmic expressions, a skill highlighted in the mark scheme.
一些Jan22的考生会遇到变体：需要验证R ∝ L。通过绘制双对数图：log R = log L + log(ρ / A)。该图像的梯度应为1，以确认正比关系。而后从截距c = log(ρ / A)推导ρ，即ρ = A × 10c。这一练习考查考生处理对数表达式的能力，也是评分方案中强调的要点。
6. Uncertainty Propagation in Derivation | 推导中的不确定度传递
A critical component of the mark scheme is the derivation of the absolute uncertainty in ρ. Given that ρ = (π/4) mexp d², and the measurements of mexp and d have uncertainties Δm and Δd, the fractional uncertainty in ρ is:
Δρ / ρ = √[ (Δm / mexp)² + (2 × Δd / d)² ]
The factor 2 appears because d is squared. The absolute uncertainty is then Δρ = (Δρ / ρ) × ρ. Candidates must show the combination of percentage uncertainties in quadrature to be awarded the full marks.
评分方案中的一个关键部分是推导ρ的绝对不确定度。由于ρ = (π/4) mexp d²，且测量值 mexp 和 d 的不确定度分别为Δm和Δd，ρ的相对不确定度为：
Δρ / ρ = √[ (Δm / mexp)² + (2 × Δd / d)² ]
平方项引入因子2是因为d被平方。绝对不确定度则为 Δρ = (Δρ / ρ) × ρ。考生必须展示百分不确定度的平方和根组合，才能获得满分。
7. Worked Example: Numerical Derivation | 实例演算：数值推导
Suppose a student obtains the following data from the R vs L graph:
| Quantity | Value | Absolute Uncertainty |
| Gradient mexp | 3.50 Ω m⁻¹ | ± 0.12 Ω m⁻¹ |
| Diameter d | 0.274 × 10⁻³ m | ± 0.004 × 10⁻³ m |
The resistivity ρ is calculated as:
ρ = 3.50 × (π × (0.274 × 10⁻³)² / 4) ≈ 2.06 × 10⁻⁷ Ω m
For uncertainty: Δm/m = 0.12/3.50 ≈ 0.0343 (3.43%), Δd/d = 0.004/0.274 ≈ 0.0146 (1.46%). Then Δρ/ρ = √(0.0343² + (2×0.0146)²) = √(0.001176 + 0.000853) ≈ 0.0450. Hence Δρ = 2.06 × 10⁻⁷ × 0.0450 ≈ 0.09 × 10⁻⁷ Ω m. Final result: ρ = (2.06 ± 0.09) × 10⁻⁷ Ω m.
假设学生从R-L图得到如下数据：
| 物理量 | 数值 | 绝对不确定度 |
| 斜率 mexp | 3.50 Ω m⁻¹ | ± 0.12 Ω m⁻¹ |
| 直径 d | 0.274 × 10⁻³ m | ± 0.004 × 10⁻³ m |
电阻率ρ计算如下：
ρ = 3.50 × (π × (0.274 × 10⁻³)² / 4) ≈ 2.06 × 10⁻⁷ Ω m
不确定度：Δm/m = 0.12/3.50 ≈ 0.0343 (3.43%)，Δd/d = 0.004/0.274 ≈ 0.0146 (1.46%)。则Δρ/ρ = √(0.0343² + (2×0.0146)²) ≈ 0.0450。因此Δρ = 2.06 × 10⁻⁷ × 0.0450 ≈ 0.09 × 10⁻⁷ Ω m。最终结果：ρ = (2.06 ± 0.09) × 10⁻⁷ Ω m。
8. Mark Scheme Commentary on Derivation Steps | 评分方案对推导步骤的评注
The Jan22 mark scheme explicitly allocated marks for: (i) correctly identifying the gradient expression, (ii) rearranging the equation to make ρ the subject, (iii) substituting area in terms of diameter, and (iv) computing the combined uncertainty. Each algebraic manipulation needed to be clearly shown; omitting intermediate steps led to a loss of ‘method’ marks even if the final formula was correct.
Jan22评分方案明确将分数分配给：(i) 正确写出斜率表达式，(ii) 移项使ρ成为公式主体，(iii) 将面积用直径表示并代入，(iv) 计算组合不确定度。每一个代数变换都需要清晰展示；省略中间步骤即使最终公式正确也会导致“方法”分数丢失。
9. Common Mistakes in Deriving Formulas | 公式推导中的常见错误
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Misidentifying the gradient: Students often write m = ρL / A instead of ρ / A. Remember, the factor multiplying the x‑axis variable must be the gradient.
斜率识别错误：学生常写成 m = ρL / A 而非 ρ / A。务必记住，乘以x轴变量的因子才是斜率。
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Forgetting to square the diameter in A = πd²/4, which leads to a dimensional error.
忘记在 A = πd²/4 中将直径平方，导致量纲错误。
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Linear uncertainty addition vs quadrature: Using simple addition Δρ/ρ = Δm/m + 2Δd/d instead of the root-sum-square method penalised heavily in the mark scheme.
线性不确定度相加与方和根法混淆：使用简单相加 Δρ/ρ = Δm/m + 2Δd/d 而非平方和根法，在评分方案中会被严重扣分。
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Poor presentation of the final answer: The mark scheme expects the derived formula to be stated clearly, often with the numerical value quoted to an appropriate number of significant figures.
最终答案表述不清：评分方案要求清晰陈述推导出的公式，且数值通常需要引用到合适的有效数字位数。
10. Derivation in Other Jan22 Practical Contexts | Jan22其他实验情境中的推导
While the resistivity derivation is emblematic, the Jan22 Unit 3 paper also featured derivations for the acceleration of free fall using a trap-door switch and for the Young modulus from a stress-strain graph. The underlying pattern remains: start with a theoretical model, identify linearised variables, obtain the gradient/slope expression, and reverse-engineer the target quantity. The mark scheme consistently rewards this systematic approach.
电阻率推导虽具代表性，Jan22第三单元试卷还出现了利用门控开关测定自由落体加速度，以及从应力-应变图推导杨氏模量的题目。其基本模式始终是：从理论模型出发，确定线性化变量，得到斜率表达式，并逆向求出目标物理量。评分方案对此类系统性方法始终给予分数奖励。
11. Checklist for Securing Full Marks on Derivation Questions | 完整获得推导题满分的检查清单
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Write down the fundamental relationship clearly.
清晰写出基本关系式。
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Express the dependent variable in terms of the independent variable, matching y = mx + c.
将因变量表示为自变量的函数，与 y = mx + c 匹配。
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State gradient m and intercept c explicitly; label which experimental quantities they represent.
明确写出斜率m和截距c，并标明它们代表哪些实验量。
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Algebraically isolate the required physical quantity.
通过代数运算分离出所需物理量。
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If uncertainties are required, write the formula for fractional or absolute uncertainty propagation using quadrature for independent measurements.
若要求不确定度，写出相对或绝对不确定度的传递公式，对独立测量量使用平方和根法。
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Substitute numerical values only after the final formula is derived, and quote the result with proper units and significant figures.
仅在推导出最终公式后才代入数值，结果给出正确单位和有效数字。
12. Summary: Master the Mark Scheme Logic | 总结：掌握评分方案的逻辑
Formula derivation in the Jan22 Unit 3 mark scheme is not about memorising a single result but about demonstrating a clear, logical pathway from raw data to a processed physical constant. Practice with graphs, algebraic rearrangement and uncertainty propagation will equip you to tackle any similar question confidently. Use the mark scheme as a revision tool to internalise the structure and precision required.
Jan22第三单元评分方案中的公式推导，不在于记住某一个结果，而在于展现一条从原始数据到处理后物理常数的清晰逻辑路径。通过图像、代数变换和不确定度传递的练习，你将能够自信应对任何类似问题。将评分方案作为复习工具，内化所需的结构与严谨性。
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