A-Level Physics Jun 18 Insert 2: Resistivity of a Metal Wire | A-Level 物理实验探究：金属丝电阻率的测定（2018年6月插入材料2）

📚 A-Level Physics Jun 18 Insert 2: Resistivity of a Metal Wire | A-Level 物理实验探究：金属丝电阻率的测定（2018年6月插入材料2）

In this experimental investigation, typically featured as Insert 2 in the June 2018 A-Level Physics practical paper, students aim to determine the resistivity of a nichrome or constantan wire by studying how its resistance varies with length. This task integrates circuit analysis, measurement techniques, graphical analysis and uncertainty evaluation, which are core skills for A-Level physics.

本实验探究常见于2018年6月A-Level物理实验卷的插入材料2，要求学生通过研究导线电阻随长度的变化，测定镍铬合金或康铜丝的电阻率。该任务综合了电路分析、测量技术、图像分析以及不确定度评估，这些都是A-Level物理的核心技能。

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1. Introduction and Theory | 引言与理论依据

The resistance R of a wire of length L and uniform cross-sectional area A is given by R = ρL/A, where ρ is the resistivity. If the diameter d is known, the area is A = πd²/4. Therefore a graph of R against L should be a straight line through the origin with gradient k = ρ/A, allowing ρ to be determined from ρ = kA.

均匀横截面积 A、长度为 L 的导线电阻 R 由 R = ρL/A 给出，其中 ρ 为电阻率。若已知直径 d，则面积 A = πd²/4。因此 R 对 L 的图像应是一条过原点的直线，斜率 k = ρ/A，由此可求得 ρ = kA。

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2. Apparatus and Setup | 实验仪器与装置

The apparatus includes a 1-metre length of thin constantan wire fixed to a metre rule with tape, a power supply, a rheostat, an ammeter, a voltmeter, a micrometer screw gauge, connecting leads, and crocodile clips. One crocodile clip is fixed at the zero end of the wire (with good contact), and the other can be moved to make contact at different lengths.

仪器包括一根固定在米尺上的1米长康铜细丝、电源、变阻器、电流表、电压表、螺旋测微器、连接导线和鳄鱼夹。一个鳄鱼夹固定在导线的零刻度端（良好接触），另一个鳄鱼夹可移动以在不同长度处接通电路。

• Micrometer screw gauge (0.01 mm resolution) – to measure the wire diameter.
• Spiral micrometer – to measure the wire diameter.
• Digital multimeters used as ammeter and voltmeter – reduce zero error.
• 螺旋测微器（分度值0.01 mm）– 测量导线直径。
• 数字万用表用作电流表和电压表 – 减少零位误差。

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3. Methodology | 实验方法

The wire diameter was measured at five different points along the wire using the micrometer, and the mean diameter was recorded. The circuit was connected with the fixed crocodile clip at 0 cm and the sliding clip set at 10.0 cm. The rheostat was adjusted to keep the current below 0.5 A to avoid heating. The current I and the potential difference V were recorded, and the resistance was calculated as R = V/I. This was repeated for lengths up to 100.0 cm in 10.0 cm increments.

用螺旋测微器在导线五个不同位置测量直径，记录平均值。电路连接时固定鳄鱼夹于0 cm处，滑动夹置于10.0 cm处。调节变阻器使电流低于0.5 A以避免发热。记录电流 I 和电位差 V，电阻由 R = V/I 计算。以10.0 cm为增量重复测量，直至长度达100.0 cm。

For each length, the voltage and current were recorded twice with reversed polarity to cancel any thermal emf, and the average V and I were used.

每个长度均通过反向连接测量两次电压和电流，以消除热电动势，并取平均值用于计算。

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4. Raw Data | 原始数据

Length L / cm	V₁ / V	I₁ / A	V₂ (reversed) / V	I₂ / A	Mean V / V	Mean I / A
10.0	0.21	0.48	0.21	0.47	0.210	0.475
20.0	0.42	0.47	0.43	0.48	0.425	0.475
30.0	0.63	0.47	0.64	0.48	0.635	0.475
40.0	0.84	0.48	0.85	0.48	0.845	0.480
50.0	1.03	0.47	1.04	0.47	1.035	0.470
60.0	1.24	0.48	1.25	0.48	1.245	0.480
70.0	1.45	0.48	1.46	0.49	1.455	0.485
80.0	1.66	0.49	1.67	0.49	1.665	0.490
90.0	1.86	0.48	1.87	0.49	1.865	0.485
100.0	2.07	0.48	2.08	0.48	2.075	0.480

Diameter readings (micrometer, 0.01 mm): 0.27 mm, 0.26 mm, 0.27 mm, 0.28 mm, 0.27 mm → Mean d = 0.270 mm = 2.70 × 10⁻⁴ m.

直径读数（螺旋测微器，分度0.01 mm）：0.27、0.26、0.27、0.28、0.27 mm → 平均值 d = 0.270 mm = 2.70 × 10⁻⁴ m。

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5. Data Analysis – Resistance Calculation | 数据分析 – 电阻计算

Resistance R for each length is found by R = V / I using the mean values. The consistency of current being approximately 0.48 A shows good control of temperature. Sample: for L = 50.0 cm, R = 1.035 V / 0.470 A ≈ 2.20 Ω.

每个长度下的电阻 R 由平均电压与平均电流之比 V/I 算出。电流基本稳定在0.48 A附近，说明温度控制良好。示例：L = 50.0 cm时，R = 1.035 V / 0.470 A ≈ 2.20 Ω。

All calculated R values are tabulated:

所有计算的 R 值列表如下：

L / cm	10.0	20.0	30.0	40.0	50.0	60.0	70.0	80.0	90.0	100.0
R / Ω	0.442	0.895	1.337	1.760	2.202	2.594	3.000	3.398	3.845	4.323

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6. Graphical Analysis | 图像分析

A graph of R (Ω) on the y-axis against L (m) on the x-axis was plotted. Error bars for R were estimated as ±2% from meter readings; L uncertainty was ±0.002 m. The best-fit straight line shows a clear linear relationship, with a small positive intercept of about 0.02 Ω, suggesting contact resistance.

绘制以 R (Ω) 为纵轴、L (m) 为横轴的图像。R 的不确定度由仪表读数估计为 ±2%；L 的不确定度为 ±0.002 m。最佳拟合直线显示清晰的线性关系，存在约 0.02 Ω 的正截距，表明存在接触电阻。

The gradient k was calculated using a large triangle: (4.32 – 0.44) Ω / (1.00 – 0.10) m = 3.88 / 0.90 = 4.31 Ω m⁻¹.

使用大三角形计算斜率：k = (4.32 – 0.44) Ω / (1.00 – 0.10) m = 3.88 / 0.90 = 4.31 Ω m⁻¹。

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7. Determining Resistivity from Gradient | 由斜率求电阻率

Cross-sectional area: A = πd²/4 = π × (2.70 × 10⁻⁴ m)² / 4 = 5.73 × 10⁻⁸ m². Therefore ρ = k × A = 4.31 Ω m⁻¹ × 5.73 × 10⁻⁸ m² ≈ 2.47 × 10⁻⁷ Ω m.

横截面积：A = πd²/4 = π × (2.70 × 10⁻⁴ m)² / 4 = 5.73 × 10⁻⁸ m²。因此 ρ = k × A = 4.31 Ω m⁻¹ × 5.73 × 10⁻⁸ m² ≈ 2.47 × 10⁻⁷ Ω m。

This value is comparable to the accepted resistivity of nichrome (≈ 1.10 × 10⁻⁶ Ω m) or possibly constantan (≈ 4.9 × 10⁻⁷ Ω m). The measured value indicates the wire might be constantan with some deviation due to temperature or non-uniformity.

该值与镍铬合金（约 1.10 × 10⁻⁶ Ω m）或康铜（约 4.9 × 10⁻⁷ Ω m）的公认电阻率相当。测量值表明该导线可能是康铜，但存在因温度或不均匀性引起的偏差。

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8. Uncertainty Analysis | 不确定度分析

The percentage uncertainty in diameter is (0.01 mm / 0.27 mm) × 100% ≈ 3.7%. Since area depends on d², the uncertainty in A is 2 × 3.7% = 7.4%. The gradient uncertainty from max–min slopes was approximately 4.5%. Combining these gives a total ρ uncertainty ≈ √(7.4² + 4.5²) % ≈ 8.7%. Hence ρ = (2.47 ± 0.21) × 10⁻⁷ Ω m.

直径的百分不确定度 = (0.01 mm / 0.27 mm) × 100% ≈ 3.7%。由于面积与 d² 有关，面积的不确定度为 2 × 3.7% = 7.4%。由最大 – 最小斜率法得到斜率的不确定度约为 4.5%。合并得到 ρ 的总不确定度 ≈ √(7.4² + 4.5²) % ≈ 8.7%。因此 ρ = (2.47 ± 0.21) × 10⁻⁷ Ω m。

The largest contribution to uncertainty comes from the diameter measurement, highlighting the importance of a precise micrometer and multiple readings.

不确定度最大的贡献来自直径测量，凸显了精密螺旋测微器和多次读数的重要性。

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9. Sources of Error | 误差来源

• Zero error in the metre rule – the crocodile clip might not be exactly at the zero mark. / 米尺的零位误差 – 鳄鱼夹可能未精确对准零刻度。
• Heating effect even with low current, causing resistance to rise slightly at longer times. / 即使电流较低，发热效应仍会导致电阻随时间略有上升。
• Non-uniform wire thickness along its length; the diameter can vary, affecting local R. / 导线沿长度方向粗细不均，局部电阻值受影响。
• Contact resistance at crocodile clips adding a constant offset to R; this is why the intercept is not zero. / 鳄鱼夹接触电阻给 R 增加一恒定偏置，导致截距不为零。
• Voltmeter and ammeter calibration or internal resistance effects. / 电压表和电流表的校准或内阻影响。

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10. Improvements and Evaluation | 改进与评估

To eliminate the contact-resistance intercept, a four-point probe measurement could be used, where separate current and voltage contacts are attached. To reduce heating, the current could be kept smaller (e.g., 0.2 A), and waiting time between readings could be minimized. A digital calliper could supplement the micrometer to detect diameter variation more conveniently along the wire. Taking more data points at smaller length increments (every 5 cm) would improve the gradient accuracy.

为消除接触电阻带来的截距，可采用四探针测量法，使电流和电压触点分离。为减少发热影响，可进一步减小电流（如 0.2 A），并缩短两次读数之间的等候时间。除螺旋测微器外，可配合数显卡尺更便捷地检测导线沿长度的直径变化。以更小的长度间隔（如每 5 cm）采集更多数据点，可提高斜率的准确性。

Repeating the experiment with a wire of known composition, such as pure constantan, and comparing with standard values would validate the method.

使用已知成分的导线（如纯康铜）重复实验，并与标准值比较，可验证本方法的有效性。

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11. Conclusion | 结论

The resistivity of the metal wire was successfully determined as (2.47 ± 0.21) × 10⁻⁷ Ω m, consistent with constantan wire. The experiment effectively demonstrates the linear relationship between resistance and length and reinforces core practical skills, including the use of micrometres, circuit setup, graphical analysis, and uncertainty propagation. The presence of a small intercept highlights systematic errors, offering a rich discussion point on experimental refinement.

成功测定该金属丝的电阻率为 (2.47 ± 0.21) × 10⁻⁷ Ω m，与康铜丝相符。该实验有效展示了电阻与长度的线性关系，并强化了核心实验技能，包括螺旋测微器的使用、电路搭建、图像分析以及不确定度的传递。微小截距的存在凸显了系统误差，为实验改进提供了丰富的讨论点。

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