📚 Experimental Investigation of Resistivity of a Metal Wire | 金属丝电阻率的实验探究
Resistivity is a core concept in electricity that describes how strongly a material opposes the flow of electric current. In this classic A-level physics investigation, students determine the resistivity of a metal wire by measuring its resistance at different lengths. This article walks through the experimental procedure, data handling, and uncertainty analysis, equipping learners with the skills needed to tackle practical exam questions confidently.
电阻率是电学中描述材料对电流阻碍作用的核心概念。在这个经典的A-level物理实验中，学生通过测量不同长度下金属丝的电阻来确定其电阻率。本文详细讲解实验步骤、数据处理与不确定度分析，帮助学习者自信应对实验类考题。
1. Theoretical Background of Resistivity | 电阻率的理论背景
Resistivity (ρ) is an intrinsic material property that remains constant for a given substance under fixed temperature. The resistance R of a wire is related to its length L and cross-sectional area A through the formula:
R = ρ L / A
电阻率（ρ）是材料的内禀属性，在温度一定时保持不变。导线电阻 R 与长度 L 和横截面积 A 的关系为：
R = ρ L / A
Rearranging this equation gives ρ = R A / L. By keeping the material and diameter constant and varying the length, one can determine ρ from the gradient of an R versus L graph multiplied by the cross-sectional area. For a wire of circular cross-section, A = π d² / 4, where d is the diameter measured with a micrometer.
变换公式得 ρ = R A / L。在材料和直径不变的前提下，通过改变长度并绘制 R-L 图，其斜率乘以横截面积即可求得 ρ。对于圆形截面导线，A = π d² / 4，其中 d 使用千分尺测量。
2. Aims of the Experimental Investigation | 实验探究的目的
The primary aim is to determine the resistivity of a nichrome wire by measuring resistance for at least six different lengths. Additionally, the investigation aims to evaluate measurement uncertainties and suggest improvements, thereby strengthening students’ understanding of experimental physics and data validity.
主要目的是通过测量至少六组不同长度下镍铬合金丝的电阻，确定其电阻率。同时，评估测量不确定度并提出改进建议，从而深化对实验物理和数据有效性的理解。
3. Apparatus and Setup | 实验仪器与装置
The following equipment is typically required:
- Nichrome wire (approx. 1 m, fixed on a metre rule)
- Power supply (d.c., low voltage)
- Ammeter (0-1 A) and voltmeter (0-5 V)
- Micrometer screw gauge (0-10 mm, resolution 0.01 mm)
- Metre rule with crocodile clips and connecting leads
- Switch (to minimise heating)
通常需要以下仪器：
- 镍铬合金丝（约1米，固定在米尺上）
- 直流低压电源
- 电流表（0-1 A）和电压表（0-5 V）
- 千分尺（0-10 mm，精度0.01 mm）
- 米尺、鳄鱼夹和连接导线
- 开关（以减少发热）
The wire is taped along a metre rule, and a crocodile clip acts as a sliding contact to vary the effective length L. The ammeter is connected in series and the voltmeter in parallel across the test length.
导线被固定在米尺上，一个鳄鱼夹作为滑动接头改变有效长度 L。电流表串联，电压表并联在测试长度两端。
4. Measuring the Diameter of the Wire | 测量导线直径
Use a micrometer screw gauge to measure the diameter at several points along the wire. Record six readings, check for zero error, and calculate the mean diameter d. The cross-sectional area is then A = π d² / 4. This step is critical because a small error in d leads to a doubled fractional error in A (since A ∝ d²), heavily influencing the final resistivity value.
使用千分尺在导线不同位置多次测量直径。记录六个读数，检查零误差，计算平均直径 d。横截面积 A = π d² / 4。这一步至关重要，因为 d 的微小误差会导致 A 的双倍相对误差（A ∝ d²），严重影响最终电阻率值。
5. Circuit and Resistance Measurement | 电路与电阻测量
Set up the circuit with the ammeter in series and the voltmeter across the test length. Start with the maximum length (e.g., 1.00 m). Close the switch only long enough to take readings, then open it to avoid heating the wire. Record the voltage V and the current I, and compute R = V / I. Repeat for decreasing lengths, e.g., 0.80 m, 0.60 m, 0.40 m, 0.20 m, and 0.10 m.
连接电路，电流表串联，电压表并联在待测长度两端。从最大长度（如1.00 m）开始。闭合开关仅需短暂的读数时间，随即断开以避免导线发热。记录电压 V 和电流 I，计算 R = V / I。依次减小长度重复测量，如0.80 m、0.60 m、0.40 m、0.20 m和0.10 m。
6. Data Collection Table | 数据记录表
Organise the measurements in a table. A typical layout is shown below. Ensure all raw readings and calculated values are recorded with correct units and consistent significant figures.
将测量数据整理成表格。典型的表格布局如下。确保所有原始读数和计算值均带正确单位，有效数字保持一致。
| L / m | V / V | I / A | R = V/I / Ω |
|---|---|---|---|
| 1.000 | 2.40 | 0.48 | 5.00 |
| 0.800 | 2.05 | 0.51 | 4.02 |
| 0.600 | 1.67 | 0.56 | 2.98 |
| 0.400 | 1.22 | 0.61 | 2.00 |
| 0.200 | 0.67 | 0.67 | 1.00 |
The resistance values decrease roughly in proportion to length, which is expected from the resistivity equation. Anomalous points, if any, should be repeated.
电阻值大致与长度成正比，符合电阻率公式的预期。若有异常点，应重复测量。
7. Graphical Analysis | 图像分析
Plot a graph of resistance R (y-axis) against length L (x-axis) on graph paper or using software. The plot should yield a straight line passing through the origin. According to R = (ρ / A) L, the gradient m = ρ / A. Draw a line of best fit, and also plot maximum and minimum gradient lines to aid uncertainty evaluation.
在坐标纸或软件上绘制电阻 R（y轴）对长度 L（x轴）的图像。图像应为一条通过原点的直线。根据 R = (ρ / A) L，斜率 m = ρ / A。画出最佳拟合线，并绘制最大梯度和最小梯度线以帮助评估不确定度。
8. Calculating Resistivity and Its Uncertainty | 计算电阻率及其不确定度
Determine the gradient m from the best-fit line. The resistivity is then ρ = m × A. For example, if the mean diameter d = 0.38 mm = 3.8 × 10⁻⁴ m, then A = π (3.8 × 10⁻⁴)² / 4 = 1.13 × 10⁻⁷ m². If the gradient m = 1.25 Ω m⁻¹, ρ = 1.25 × 1.13 × 10⁻⁷ = 1.41 × 10⁻⁷ Ω m. Compare this with the accepted value for nichrome (approx. 1.10 × 10⁻⁶ Ω m) – be prepared to explain discrepancies.
从最佳拟合线求出斜率 m，电阻率 ρ = m × A。例如，若平均直径 d = 0.38 mm = 3.8 × 10⁻⁴ m，则 A = π (3.8 × 10⁻⁴)² / 4 = 1.13 × 10⁻⁷ m²。若斜率 m = 1.25 Ω m⁻¹，ρ = 1.25 × 1.13 × 10⁻⁷ = 1.41 × 10⁻⁷ Ω m。将该值与镍铬合金的公认值（约1.10 × 10⁻⁶ Ω m）比较，需能解释偏差。
For uncertainties, calculate Δm from half the difference of maximum and minimum gradients. The fractional uncertainty in ρ combines those in m and A: Δρ/ρ = Δm/m + 2 Δd/d (since A ∝ d²). Record the final result as ρ ± Δρ.
对于不确定度，用最大与最小斜率差值的一半计算 Δm。ρ 的相对不确定度由 m 和 A 的不确定度合成：Δρ/ρ = Δm/m + 2 Δd/d（因 A ∝ d²）。最终结果记为 ρ ± Δρ。
9. Principal Sources of Error | 主要误差来源
- Heating effect: Current flowing through the wire raises its temperature, increasing resistance. Keeping the switch closed briefly minimises this.
- Diameter inconsistencies: Variation in wire thickness along its length causes scatter in data.
- Zero error in micrometer: A systematic error that shifts all diameter readings.
- Contact resistance: Crocodile clips may introduce extra resistance at connections.
- Parallax errors in reading analogue meters.
- 热效应：电流通过导线使其升温，电阻增大。短暂闭合开关可减少此影响。
- 直径不均匀：导线沿途厚度变化导致数据分散。
- 千分尺零误差：系统误差，使所有直径读数偏移。
- 接触电阻：鳄鱼夹在连接处可能引入额外电阻。
- 读取模拟表时的视差误差。
10. Improvements and Further Refinements | 改进与优化
Use a four-point probe (Kelvin connection) to eliminate the effect of contact resistance. Perform the experiment in a temperature-controlled environment or let the wire cool between readings. Take diameter measurements at more positions and use a digital micrometer for higher precision. Additionally, use a data logger to record V and I simultaneously, reducing human reaction errors.
采用四端法（开尔文连接）消除接触电阻影响；在恒温环境下实验或让导线在两次读数间充分冷却；增加直径测量点并使用数字千分尺提高精度；使用数据采集器同步记录 V 和 I，减少人为反应误差。
11. Conclusion and Link to Examination Success | 结论与考试成功之道
This investigation develops essential practical skills: using a micrometer, setting up a potential divider circuit, exploiting graphical methods, and quantifying uncertainty. When writing up such an experiment in an exam, always relate your results to the expected equation, evaluate the reliability of your data, and suggest realistic improvements. Mastery of these elements will set your practical write-up apart.
本实验培养了关键动手能力：千分尺的使用、分压电路搭建、图像分析及不确定度量化。在考试中撰写此类实验报告时，务必将结果与理论公式关联，评估数据可靠性，并提出切实的改进。掌握这些要素能让你的实验作答脱颖而出。
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