📚 Formula Derivations from the OxfordAQA PH03 Jan 2023 Mark Scheme | 来自OxfordAQA PH03 2023年1月评分方案的公式推导
The OxfordAQA AS Physics Unit 3 (PH03) paper tests investigative and practical skills, demanding not only experimental technique but also rigorous data analysis. The January 2023 mark scheme provides a fascinating window into the precise derivations expected of candidates. By dissecting the reasoning behind each allocated mark, students can move beyond mechanical plug‑and‑chug to a genuine understanding of how physical laws are expressed through graphed data and associated uncertainties. This article extracts the core formula derivations embedded in that mark scheme and explains each step, pairing a clear derivation in English with an equivalent explanation in Chinese.
OxfordAQA AS物理第三单元（PH03）考查探究与实验技能，不仅要求过硬的动手能力，还需要严谨的数据分析。2023年1月的评分方案为考生必须掌握的公式推导提供了清晰的窗口。通过剖析每个得分点背后的逻辑，你可以从机械的“代入公式”跃升至真正理解物理规律如何通过图像与不确定度表达出来。本文提炼了该评分方案中核心的公式推导过程，每一步推导都配以中英双语解读。
1. Percentage and Compound Uncertainty | 百分比与组合不确定度
All measured quantities carry an absolute uncertainty ΔX. The mark scheme consistently expects the percentage uncertainty to be written as (ΔX / X) × 100%. When two measurements are added or subtracted, the absolute uncertainties add in quadrature: ΔZ = √(ΔA² + ΔB²). For multiplication or division, the percentage uncertainties add. The rationale is that fractional changes propagate directly.
任何测量量都带有绝对不确定度ΔX。评分方案一贯要求将百分比不确定度写为(ΔX / X) × 100%。当两个量相加或相减时，绝对不确定度按平方和根方式合成：ΔZ = √(ΔA² + ΔB²)。对于乘除运算，百分比不确定度直接相加，这是因为相对变化会线性传递。
If Z = A × B or Z = A / B, then %UZ = %UA + %UB
如果Z = A × B 或 Z = A / B，那么 Z 的百分比不确定度是 A 和 B 百分比不确定度之和。
In the PH03 scheme, a typical question might ask for the percentage uncertainty in the volume of a wire V = (πd²L)/4. The derived percentage uncertainty is 2(%Ud) + %UL, because d is squared and L appears to the first power. The mark scheme awards the mark only when the factor of 2 is correctly included alongside the expression.
在PH03方案中，典型题目可能要求计算导线体积 V = (πd²L)/4 的百分比不确定度。推导结果为 2(%Ud) + %UL，因为d以平方出现，而L是一次方。只有当表达式中正确包含因数2时，评分方案才给予得分。
2. Linearising Equations Using Logarithms | 用对数线性化方程
When a relationship is suspected to be of the form y = kxn, the mark scheme frequently requires taking natural logs to obtain ln y = ln k + n ln x. Plotting ln y against ln x yields a straight line of gradient n and vertical intercept ln k. The uncertainty in the gradient can then be directly linked to the uncertainty in the exponent n.
当怀疑关系为 y = kxn 时，评分方案通常要求取自然对数，得到 ln y = ln k + n ln x。以ln y为纵轴、ln x为横轴作图，可得一条斜率为n、截距为ln k的直线。斜率的误差则直接与指数n的不确定度相关。
log y = log k + n log x → straight line with gradient = n
The January 2023 PH03 mark scheme rewarded explicit substitution of logarithmic values into the straight‑line equation and then using the gradient and intercept to solve for k and n. A common derived expression is n = (ln y₂ − ln y₁) / (ln x₂ − ln x₁), taken directly from two well‑separated points on the best‑fit line.
2023年1月PH03评分方案对将对数值直接代入直线方程、并利用斜率和截距求出k和n的步骤给予得分。常用的推导表达式为 n = (ln y₂ − ln y₁) / (ln x₂ − ln x₁)，直接从最佳拟合线上两个相距较远的点获取。
3. Deriving g from a Simple Pendulum Graph | 从单摆图像推导重力加速度g
The period T of a simple pendulum is given by T = 2π√(L/g). The mark scheme expects candidates to square both sides, giving T² = (4π²/g)L. Plotting T² on the y‑axis against L on the x‑axis yields a straight line through the origin with gradient = 4π²/g. Therefore, g = 4π² / gradient.
单摆周期公式为 T = 2π√(L/g)。评分方案要求考生将两边平方，得到 T² = (4π²/g)L。以T²为纵轴、L为横轴作图，可得到一条过原点的直线，斜率等于4π²/g。由此，g = 4π² / 斜率。
g = 4π² / (ΔT²/ΔL)
The mark scheme often checks whether the candidate identifies the uncertainty in g by combining the percentage uncertainty in the gradient with the uncertainty in any systematic corrections. The absolute uncertainty in g is then Δg = g × (%Ugradient / 100). This derived formula appears repeatedly in examiners’ reports.
评分方案通常会检查考生是否通过组合斜率百分比不确定度与任何系统修正的不确定度来求得g的不确定度。g的绝对不确定度为 Δg = g × (%U斜率 / 100)。这个推导公式在考官报告中反复出现。
4. Resistivity and the Gradient of V–I Graph | 电阻率与V–I图像斜率
For a uniform wire, resistance R = ρL/A, where A = πd²/4. The PH03 mark scheme derives an expression for ρ by combining Ohm’s law with the wire’s dimensions. Measuring V and I for a fixed length gives R = V/I, but more powerfully, multiple lengths allow a plot. The scheme shows that if V is plotted against I for different lengths, the gradient gives R for each length. Then plotting R against L yields a straight line of gradient ρ/A.
对于均匀导线，电阻 R = ρL/A，其中A = πd²/4。PH03评分方案通过组合欧姆定律与导线尺寸推导出ρ的表达式。测量固定长度下的V和I可得R = V/I，但更有效的方法是利用不同长度作图。方案表明，若对不同长度绘制V–I图，斜率给出各长度对应的R。然后以R为纵轴、L为横轴作图，得到斜率为ρ/A的直线。
ρ = gradient × A = gradient × (πd²/4)
Thus, the derived uncertainty in ρ requires the percentage uncertainty in the gradient added to twice the percentage uncertainty in the diameter and the percentage uncertainty in the length measurement. The mark scheme meticulously weights the contribution of d because of the squared term.
因此，ρ的推导不确定度需要将斜率的百分比不确定度加上直径百分比不确定度的两倍以及长度测量百分比不确定度。由于存在平方项，评分方案极其细致地加上了d的贡献。
5. Young Modulus and Stress–Strain Gradient | 杨氏模量与应力–应变斜率
The definition of Young modulus E is tensile stress / tensile strain. In a typical wire extension experiment, stress = F/A and strain = e/L. The mark scheme rearranges this to F = (EA/L) × e, so that a graph of F against e (force vs. extension) has gradient = EA/L. Hence E = (gradient × L) / A.
杨氏模量 E 的定义为拉伸应力除以拉伸应变。在典型的金属丝伸长实验中，应力 = F/A，应变 = e/L。评分方案将公式变形为 F = (EA/L) × e，这样F对e（力–伸长量）图的斜率即为EA/L。因此，E = (斜率 × L) / A。
E = (ΔF/Δe) × (L / (πd²/4))
The mark scheme may also require the candidate to propagate the uncertainty from the diameter, original length, and gradient. The percentage uncertainty in E becomes %Ugradient + %UL + 2(%Ud), once again highlighting the dominance of the diameter term.
评分方案还可能要求考生传递直径、原始长度和斜率的不确定度。E的百分比不确定度变为 %U斜率 + %UL + 2(%Ud)，再一次凸显直径项的支配性影响。
6. Uncertainty in Intercept from a Graph | 图像截距的不确定度
When the dependent variable is expected to be zero for zero input, the mark scheme often asks for the y‑intercept to be compared with the expected value (usually zero). The absolute uncertainty in the intercept can be estimated from the scatter of points about the best‑fit line. One accepted method, referenced in the scheme, is to draw worst‑acceptable lines and find the difference in intercepts.
当因变量在输入为零时预期为零时，评分方案常要求将y轴截距与预期值（通常为零）进行比较。截距的绝对不确定度可以根据数据点围绕最佳拟合线的离散程度来估计。方案中引用的一种可接受方法是绘制最差可接受斜线并求出截距之差。
Δintercept = (interceptmax − interceptmin) / 2
The January 2023 mark scheme awarded marks for explicitly stating that a non‑zero intercept within this uncertainty range still supports the proportional relationship. The derivation of the intercept uncertainty is based purely on the spread of plotted data and the extremes of plausible straight lines.
2023年1月的评分方案对明确说明在该不确定度范围内的非零截距仍然支持正比关系这一做法给予了加分。截距不确定度的推导完全基于所绘数据的离散程度和合理直线的极限范围。
7. Repeated Measurements and Standard Error | 重复测量与标准误差
The PH03 practical paper values precision, and the mark scheme consistently expects the use of repeat readings to reduce random error. For a set of N values, the mean X̄ is the best estimate. The absolute uncertainty in the mean is often taken as the standard error, which is sample standard deviation s divided by √N. The scheme awards marks for recognising that increasing N reduces this uncertainty.
PH03实验卷高度重视精密度，评分方案一直期望通过重复读数减小随机误差。对于N个数值，平均值X̄是最佳估计值。平均值的绝对不确定度通常取为标准误差，即样本标准差s除以√N。方案对认识到增大N可减小该不确定度给予得分。
ΔX̄ = s / √N, where s = √( Σ(xᵢ − X̄)² / (N−1) )
When the derived quantity depends on this mean, such as the period of a pendulum, the fractional uncertainty in the period is then ΔT/T, which feeds into the final propagated uncertainty. The mark scheme shows that a single reading uncertainty is never as good as the standard error from repeats.
当推导量依赖于该平均值时，比如单摆周期，周期的相对不确定度为ΔT/T，该值会进入最终传递的不确定度。评分方案表明，单次读数的不确定度远不如来自重复测量序列的标准误差可靠。
8. Combining Uncertainties for Power Functions | 幂函数不确定度的合成
For a relationship Q = k am bn, the general rule for percentage uncertainty is %UQ = |m| × %Ua + |n| × %Ub. This appears explicitly in the PH03 mark scheme when candidates analyse data that follows a power law, such as the period of an oscillating mass–spring system where T ∝ m½.
对于关系式 Q = k am bn，百分比不确定度的一般规则为 %UQ = |m| × %Ua + |n| × %Ub。当考生分析符合幂律的数据，例如弹簧振子周期 T ∝ m½ 时，该规则明确出现在PH03评分方案中。
%UT = ½ × %Um + %Uk (if k is also measured)
The mark scheme then tests whether students can invert the relationship to solve for an unknown exponent n by using logs as shown in Section 2. The combined use of the power‑rule and log‑linearisation is a hallmark of high‑level analysis in PH03.
随后评分方案会考查学生能否通过取对数反解出未知指数n，如第2节所示。幂规则与对数线性化的组合运用是PH03高水平数据分析的标志。
9. Using Log–Log Plots to Determine Exponent | 利用双对数图确定指数
The mark scheme often includes a task where students are given raw data and must decide whether a power law or exponential behaviour fits. For an exponential decay y = A e⁻λᵗ, the natural log gives ln y = ln A − λt. A plot of ln y vs t yields a straight line of gradient −λ. The absolute uncertainty in λ is derived from the gradient uncertainty.
评分方案常包含一项任务，让学生根据原始数据判断是幂律还是指数行为更为合适。对于指数衰减 y = A e⁻λᵗ，取自然对数得 ln y = ln A − λt。ln y对t作图可得一条斜率为−λ的直线。λ的绝对不确定度由斜率不确定度导出。
λ = −gradient; Δλ = worst‑accept‑gradient range / 2
A specific example from the January 2023 scheme concerned the discharge of a capacitor, where voltage V = V₀ e⁻t/RC. The derived time constant RC is calculated from the reciprocal of the gradient, and its uncertainty is found by combining relative uncertainties. The mark scheme insists on checking the consistency of the relationship through the intercept.
2023年1月方案中的一个具体实例涉及电容放电，电压 V = V₀ e⁻t/RC。推导的时间常数 RC 由斜率倒数计算得出，其不确定度通过组合相对不确定度求得。评分方案坚持通过截距检验关系的一致性。
10. Extracting Values from Exponential Decay Graphs | 从指数衰减图像中提取数值
The final common derivation in the PH03 mark scheme is to determine the half‑life from an exponential graph without using natural logs explicitly. By reading the time taken for the dependent variable to halve from any starting value, the effective decay constant can be obtained via λ = ln 2 / T½. The mark scheme expects the candidate to show that the half‑life is constant, confirming exponential behaviour.
PH03评分方案中最后一种常见推导是通过不显式使用自然对数而从指数图像中求半衰期。从任意起始值读取因变量减半所需的时间，即可通过 λ = ln 2 / T½ 得到有效衰变常数。评分方案期望考生证明半衰期是常数，从而确认指数行为。
T½ = ln 2 / λ, verified by equal half‑lives
Uncertainty in T½ is usually taken as the spread of several half‑life readings from the graph. The derived λ uncertainty then follows from the percentage uncertainty in T½. This concise logic, when clearly laid out, consistently earns full marks according to the PH03 mark scheme.
T½的不确定度通常取从图像读取的多个半衰期值的范围。推导的λ不确定度则源自T½的百分比不确定度。这一简洁的逻辑若表述清楚，按照PH03评分方案总能获得满分。
Published by TutorHao | Physics Revision Series | aleveler.com
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