📚 Pre-U WJEC Physics: Case Study Practical Exercises | Pre-U WJEC 物理：案例分析实战演练
In Pre-U WJEC Physics, the case study component requires you to analyse a real-world physical situation, apply appropriate models, process experimental data, and evaluate the reliability of conclusions. This article provides a structured approach to tackling these exercises, illustrated through an investigation into the resistivity of a metal wire and its temperature dependence.
在 Pre-U WJEC 物理中，案例分析部分要求你分析一个真实世界的物理情境，应用适当的模型，处理实验数据，并评估结论的可靠性。本文提供一个结构化的方法来应对这些练习，并通过一个关于金属导线电阻率及其温度依赖性的探究来加以说明。
1. Deconstructing the Problem Statement | 解构问题陈述
Read the case study prompt carefully, identifying the key physical quantity to be investigated, the variables that may be changed, and any constraints provided. For example, you may be asked: ‘Investigate how the resistance of a metallic conductor depends on its length and temperature.’
仔细阅读案例研究的提示，识别需要研究的关键物理量、可能改变的变量以及给出的任何限制条件。例如，你可能会被问到：“研究金属导体的电阻如何取决于其长度和温度。”
Extract from the prompt a list of independent and dependent variables, and note whether the investigation is qualitative or quantitative. Often, you will be required to design an experiment or analyse given data.
从提示中提取出自变量和因变量的清单，并注意该研究是定性还是定量的。通常，你将被要求设计一个实验或分析给定的数据。
2. Identifying Relevant Physics Principles | 识别相关物理原理
Link the problem to core equations. For a wire, the resistance R is given by R = ρL / A, where ρ is resistivity, L is length, and A is cross-sectional area. Temperature dependence introduces R = R₀[1 + α(T − T₀)], where α is the temperature coefficient of resistance.
将问题与核心方程联系起来。对于导线，电阻 R 由 R = ρL / A 给出，其中 ρ 是电阻率，L 是长度，A 是横截面积。温度依赖性则引入 R = R₀[1 + α(T − T₀)]，其中 α 是电阻温度系数。
You must recognise which principles are relevant: Ohm’s law, resistivity definition, thermal expansion (minor effect), and energy dissipation. Write down the applicable equations and identify any assumptions (e.g., uniform wire, constant current).
你必须识别哪些原理相关：欧姆定律、电阻率定义、热膨胀（次要影响）和能量耗散。写下可用的方程，并识别任何假设（例如，均匀导线，恒定电流）。
3. Formalising Models and Assumptions | 建立模型与假设
State clear assumptions: the wire is cylindrical with constant cross‑section, the current is small enough to avoid significant self‑heating (or account for it), and the temperature distribution is uniform. These assumptions define the validity limits of your model.
明确陈述假设：导线是圆柱形且横截面恒定，电流足够小以避免显著的自发热（或考虑其影响），并且温度分布均匀。这些假设定义了模型的有效范围。
For temperature dependence, you may assume that α is constant over a narrow temperature range. If the range is wide, a polynomial model might be more appropriate, but the linear approximation often suffices in pre‑U contexts.
对于温度依赖性，你可以假设 α 在较窄的温度范围内是常数。如果温度范围较宽，多项式模型可能更合适，但在 Pre‑U 情境下，线性近似通常就足够了。
4. Designing the Experiment and Controlling Variables | 设计实验与控制变量
For the length dependence, you would use a metre‑bridge or four‑point probe to measure R for different lengths L of wire while keeping temperature constant. The diameter must be measured with a micrometer to calculate A.
对于长度依赖性，你可以使用滑线电桥或四端法来测量不同长度 L 导线的电阻 R，同时保持温度恒定。必须用千分尺测量直径以计算横截面积 A。
For temperature dependence, a coil of wire is placed in a water bath whose temperature is varied from about 20 °C to 80 °C. A digital thermometer and an ohmmeter record T and R simultaneously. Stirring ensures uniform temperature.
对于温度依赖性，将导线线圈放入水浴中，温度从约 20 °C 变化到 80 °C。使用数字温度计和欧姆表同时记录 T 和 R。搅拌确保温度均匀。
Control variables: use the same wire sample, avoid mechanical strain, and keep current small to minimise Joule heating. Identify these controls explicitly in your plan.
控制变量：使用同一样品导线，避免机械应变，并保持小电流以最小化焦耳热。在你的方案中明确列出这些控制措施。
5. Data Collection and Processing | 数据收集与处理
Present raw data in a table with correct headings and units. For length investigation: columns for L/m, V/V, I/A, R/Ω (calculated as V/I). For temperature: T/°C, R/Ω. Calculate derived quantities such as R and resistivity ρ = RA/L, estimating A from diameter.
在表格中呈现原始数据，使用正确的标题和单位。对于长度研究：列有 L/m，V/V，I/A，R/Ω（由 V/I 计算）。对于温度：T/°C，R/Ω。计算导出量，如 R 和电阻率 ρ = RA/L，根据直径估算 A。
Use appropriate significant figures, reflecting the precision of instruments. For example, if a micrometer can read to 0.01 mm, then diameter might be 0.25 × 10⁻³ m, giving A to two significant figures.
使用适当有效数字，反映仪器的精度。例如，如果千分尺可读至 0.01 mm，那么直径可能为 0.25 × 10⁻³ m，横截面积 A 取两位有效数字。
6. Graphical Methods and Linearisation | 图形方法与线性化
Plot a graph of R (y‑axis) against L (x‑axis) to test the linear relationship R = (ρ/A)L. The gradient is ρ/A. Since A is known, ρ can be determined. For temperature, plot R against T to check the linear model R = R₀(1 + α(T − T₀)) = R₀αT + R₀(1 − αT₀).
绘制 R（y 轴）对 L（x 轴）的图，以检验线性关系 R = (ρ/A)L。斜率为 ρ/A。由于 A 已知，可求得 ρ。对于温度，绘制 R 对 T 的图，以检验线性模型 R = R₀(1 + α(T − T₀)) = R₀αT + R₀(1 − αT₀)。
Use a line of best fit; do not simply join the dots. Calculate gradient and intercept with appropriate units. For the temperature graph, the ratio gradient / intercept yields α/(1 − αT₀) ≈ α for small α.
使用最佳拟合线，不要简单地连接点。计算斜率和截距并附上适当单位。对于温度图，斜率与截距的比值给出 α/(1 − αT₀) ≈ α（因为 α 很小）。
slope = ΔR / ΔL
α = slope / (R₀ − slope × T₀) (or simplified)
7. Uncertainty Analysis | 不确定度分析
Quantify uncertainties in measurements: length (ruler ±1 mm), diameter (micrometer ±0.01 mm), voltage and current (digital multimeter ±0.5% + 1 digit). Combine these to find uncertainty in R and ρ using standard propagation formulas.
量化测量中的不确定度：长度（刻度尺 ±1 mm），直径（千分尺 ±0.01 mm），电压和电流（数字万用表 ±0.5% + 1 位）。使用标准传播公式合成这些不确定度，求得 R 和 ρ 的不确定度。
For a product/quotient, percentage uncertainty adds. For ρ = RA/L = (V/I) × (πd²/4) / L, you can compute %U(ρ) = %U(V) + %U(I) + 2×%U(d) + %U(L). Clearly show your working.
对于乘除关系，百分比不确定度相加。对于 ρ = RA/L = (V/I) × (πd²/4) / L，你可以计算 %U(ρ) = %U(V) + %U(I) + 2×%U(d) + %U(L)。清晰地展示你的计算过程。
| Quantity | Value | Absolute Uncertainty | % Uncertainty |
|---|---|---|---|
| L | 1.000 m | ±0.001 m | 0.1% |
| d | 0.50 × 10⁻³ m | ±0.01 × 10⁻³ m | 2% |
| V | 1.50 V | ±0.01 V | 0.67% |
| I | 0.200 A | ±0.001 A | 0.5% |
For the temperature experiment, the uncertainty in α derived from graph gradients can be estimated using the spread of data points about the best‑fit line. Draw worst‑fit lines to find Δslope.
对于温度实验，α 的不确定度可由数据点围绕最佳拟合线的离散程度来估计。绘制最差拟合线以求得 Δ斜率。
8. Interpreting Results and Drawing Conclusions | 解释结果与得出结论
Compare your experimental ρ value with the accepted reference value for the material (e.g., copper ρ = 1.72 × 10⁻⁸ Ω·m). State whether the difference is within experimental uncertainty. If not, discuss possible systematic errors.
将实验 ρ 值与材料的公认参考值（例如，铜 ρ = 1.72 × 10⁻⁸ Ω·m）进行比较。说明差异是否在实验不确定度范围内。如果不在，讨论可能的系统误差。
For α, check if the value makes physical sense (positive for metals, typically 10⁻³ to 10⁻² K⁻¹). Conclude by summarising the relationship verified and the reliability of the findings.
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