PH03-INS Experimental Investigation: Mastering the Resistivity of a Wire | PH03-INS实验探究：掌握导线电阻率的测定

📚 PH03-INS Experimental Investigation: Mastering the Resistivity of a Wire | PH03-INS实验探究：掌握导线电阻率的测定

In the 30 May 2023 International A-Level Physics (PH03) examination, candidates were tasked with an experimental investigation centred around determining the resistivity of a metal wire. This article unpacks every critical dimension of that core practical – from precise measurement of length and diameter to the V-I graphical analysis, uncertainty compounding, and subtle sources of error that examiners love to test. By working through the step-by-step logic, you will acquire the kind of robust practical insight that turns a standard practical write‑up into a high‑mark response.

在2023年5月30日的国际A-Level物理(PH03)考试中，考生需要完成一项以测定金属导线电阻率为核心的实验探究。本文逐一拆解该核心实验的每一个关键维度——从长度与直径的精密测量、V-I图像分析，到不确定度的合成，以及考官喜欢考查的细微误差来源。通过逐步推演实验逻辑，你将获得那种能把普通的实验报告转化为高分答案的扎实实验洞察力。

1. Experiment Overview | 实验概述

Resistivity (ρ) is an intrinsic material property given by ρ = RA / l, where R is the resistance of a wire of length l and cross‑sectional area A. In the PH03 investigation, the nominal goal is to determine ρ for a nichrome or constantan wire. This demands a high‑precision approach to measuring R (via a V-I graph to minimise random errors), l (using a metre rule) and A (computed from multiple diameter readings taken with a micrometer screw gauge). The experiment is a classic amalgam of electrical measurement, metrology and uncertainty analysis – exactly the skill set examined in Unit 3.

电阻率(ρ)是一个由公式 ρ = RA / l 给出的材料本征属性，其中R是长度为l、横截面积为A的导线的电阻。在PH03探究中，名义上的目标是测定一根镍铬合金或康铜丝的ρ值。这要求用高精度的方法测量R（借助V-I图像以减小随机误差）、l（使用米尺）以及A（根据用千分尺多次测量的直径计算得出）。该实验是电学测量、计量学与不确定度分析的经典融合——正是Unit 3所考查的那一套核心技能。

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2. Key Apparatus and Their Precision | 关键仪器及其精度

The quality of your measured data starts with knowing the resolution and appropriate usage of every instrument. A typical PH03 apparatus list includes: a 1.5 m nichrome wire (26 swg or similar) mounted on a wooden metre rule with large crocodile clips to define the test length; a micrometer screw gauge (resolution 0.01 mm); a digital multimeter or separate analogue ammeter (0–1 A, resolution 0.01 A) and voltmeter (0–5 V, resolution 0.1 V); a low‑voltage DC power supply or battery pack; a rheostat (0–10 Ω, 2 A); connecting leads and a switch. A wooden board and G‑clamps can reduce wire movement.

测量数据的质量起始于了解每件仪器的分辨率并恰当使用。一份典型的PH03器材清单包括：一根约1.5 m的镍铬丝(26 swg或类似规格)，用大型鳄鱼夹固定在木制米尺上以确定测试长度；一把千分尺(分辨率0.01 mm)；一只数字万用表或分立的模拟电流表(0–1 A，分辨率0.01 A)与电压表(0–5 V，分辨率0.1 V)；低压直流电源或电池组；一个滑动变阻器(0–10 Ω，2 A)；连接导线和开关。一块木板和G形夹可以减少导线的移动。

Instrument / 仪器	Range / 量程	Resolution / 分辨率	Typical uncertainty / 典型不确定度
Micrometer screw gauge	0–25 mm	0.01 mm	±0.01 mm
Metre rule	0–1.000 m	1 mm	±1 mm (with parallax error)
Analogue ammeter	0–1 A	0.01 A	±0.005 A
Analogue voltmeter	0–5 V	0.1 V	±0.05 V

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3. Detailed Experimental Procedure | 详细实验步骤

Step 1: Securely clamp the nichrome wire on top of a metre rule. Use crocodile clips to define a test length l of about 1.000 m. Measure the exact length l three times with the metre rule, ensuring your eye is perpendicular to the scale to avoid parallax. Record the values and calculate the mean.

步骤1：将镍铬丝牢靠地夹在米尺上方。用鳄鱼夹界定约1.000 m的测试长度l。用米尺精确测量长度l共三次，确保视线与刻度垂直以避免视差。记录数值并计算平均值。

Step 2: Using the micrometer screw gauge, measure the diameter d of the wire at six different positions along its length, rotating the wire slightly between readings. Check for zero error before starting and record it. Subtract any zero error from your readings. Calculate the mean diameter d_avg.

步骤2：使用千分尺，在导线长度方向上选取六个不同位置测量直径d，并在读数之间轻微转动导线。开始前检查零点误差并记录。从你的读数中减去零点误差。计算平均直径 d_avg。

Step 3: Set up the circuit: connect the test wire, ammeter, voltmeter, rheostat, switch and power supply in series‑parallel configuration. The voltmeter must be connected directly across the test length of wire. Initially set the rheostat to its maximum resistance to protect the wire from excessive current.

步骤3：连接电路：将待测导线、电流表、电压表、滑动变阻器、开关和电源按串并联方式连接。电压表必须直接并联在导线的测试长度两端。最初将滑动变阻器调至最大电阻值，以保护导线免受过电流冲击。

Step 4: Close the switch briefly and adjust the rheostat to vary the current. Record pairs of voltmeter (V) and ammeter (I) readings for at least six different currents, ensuring the current does not exceed 0.5 A to minimise heating. Open the switch between readings to allow the wire to cool.

步骤4：短暂闭合开关并调节变阻器以改变电流。记录至少六组不同电流下的电压表(V)与电流表(I)读数，确保电流不超过0.5 A以减小发热效应。在两次读数之间断开开关让导线冷却。

Step 5: Repeat the entire V-I data set once more to give two independent sets. This allows you to judge the random scatter and calculate a mean value for the resistance from the combined graph.

步骤5：将整套V-I数据重复测量一次，获得两套独立数据。这让你能够评估随机离散度，并从合并图像中计算出平均电阻值。

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4. Recording Data – Diameter and Length | 记录数据——直径与长度

Sample diameter readings (after zero correction): 0.45, 0.44, 0.46, 0.45, 0.44, 0.45 mm. The mean d_avg = (0.45 + 0.44 + 0.46 + 0.45 + 0.44 + 0.45) / 6 = 0.4483 ≈ 0.45 mm. The range is 0.02 mm, so the absolute uncertainty in d can be taken as half the range = 0.01 mm. However, since the micrometer resolution is also 0.01 mm, the uncertainty is at least ±0.01 mm. We quote d = 0.45 ± 0.01 mm.

直径读数示例（零点修正后）：0.45、0.44、0.46、0.45、0.44、0.45 mm。平均值 d_avg = (0.45 + 0.44 + 0.46 + 0.45 + 0.44 + 0.45) / 6 = 0.4483 ≈ 0.45 mm。极差为0.02 mm，因此d的绝对不确定度可取半极差 = 0.01 mm。但是，由于千分尺的分辨率也是0.01 mm，不确定度至少为 ±0.01 mm。我们给出 d = 0.45 ± 0.01 mm。

The length l is measured three times: 1.000, 1.001, 0.999 m. Mean l = 1.000 m. The metre rule has a resolution of 1 mm, and the dominant uncertainty often includes the difficulty of accurately positioning crocodile clips, so we take Δl = ±2 mm. Thus l = 1.000 ± 0.002 m.

长度l测量三次：1.000、1.001、0.999 m。平均值 l = 1.000 m。米尺的分辨率为1 mm，主要不确定度经常包括鳄鱼夹精确定位的困难，因此我们取 Δl = ±2 mm。故 l = 1.000 ± 0.002 m。

The cross‑sectional area A = π d² / 4. Using d = 0.45 × 10⁻³ m, A = π (0.45×10⁻³)² / 4 ≈ 1.59 × 10⁻⁷ m². The fractional uncertainty in A is twice that of d, so ΔA/A = 2 × (0.01/0.45) ≈ 0.044 or 4.4%.

横截面积 A = π d² / 4。代入 d = 0.45 × 10⁻³ m，得 A = π (0.45×10⁻³)² / 4 ≈ 1.59 × 10⁻⁷ m²。A的相对不确定度是d的两倍，因此 ΔA/A = 2 × (0.01/0.45) ≈ 0.044，即4.4%。

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5. Electrical Measurements and the V-I Graph | 电学测量与V-I图

A sample set of readings is given below. The current is deliberately kept low to avoid significant heating; the wire’s temperature rise stays below 5 °C, preserving near‑constant resistivity.

下面给出了一组示例读数。电流被有意控制在低水平以避免显著发热；导线的温升保持在5 °C以下，使电阻率近乎恒定。

V / V	I / A	V / V (repeat)	I / A (repeat)
0.52	0.12	0.51	0.12
0.98	0.23	1.00	0.23
1.55	0.36	1.53	0.36
2.08	0.48	2.10	0.49
2.54	0.59	2.53	0.59
3.05	0.71	3.06	0.71

When plotted on a graph of V against I, the points should lie close to a straight line through the origin. The resistance R is the gradient. From the above combined data, a best‑fit line gives R ≈ 4.30 Ω. To estimate the uncertainty ΔR, draw the steepest and shallowest acceptable lines; if the gradients are 4.40 Ω and 4.20 Ω, then ΔR = (4.40 – 4.20)/2 = 0.10 Ω. Thus R = 4.30 ± 0.10 Ω.

当绘制V对I的图像时，数据点应聚集在一条通过原点的直线附近。电阻R是图像的斜率。根据上述合并数据，最佳拟合线给出的 R ≈ 4.30 Ω。为了估算不确定度 ΔR，画出可接受的最陡和最浅直线；若两条直线的斜率分别为4.40 Ω和4.20 Ω，则 ΔR = (4.40 – 4.20)/2 = 0.10 Ω。因此 R = 4.30 ± 0.10 Ω。

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6. Calculating Resistivity and Compounding Uncertainties | 计算电阻率与合成不确定度

Resistivity formula: ρ = R A / l. Substituting A = π d²/4 gives:

电阻率公式：ρ = R A / l。代入 A = π d²/4 得到：

ρ = π d² R / (4 l)

Using our values (d = 4.5×10⁻⁴ m, R = 4.30 Ω, l = 1.000 m):

代入我们的数值 (d = 4.5×10⁻⁴ m, R = 4.30 Ω, l = 1.000 m)：

ρ = π × (4.5×10⁻⁴)² × 4.30 / (4 × 1.000) ≈ 1.37 × 10⁻⁶ Ω m

The fractional uncertainty in ρ is obtained by adding the fractional uncertainties in each factor, remembering that d appears squared: Δρ/ρ = 2(Δd/d) + ΔR/R + Δl/l. Insert the numbers: 2×(0.01/0.45) = 0.0444; ΔR/R = 0.10/4.30 = 0.0233; Δl/l = 0.002/1.000 = 0.002. Sum = 0.0697 ≈ 7.0%. Hence Δρ = 0.070 × 1.37×10⁻⁶ ≈ 1.0×10⁻⁷ Ω m. The final quoted result is ρ = (1.37 ± 0.10) × 10⁻⁶ Ω m.

ρ的相对不确定度通过各影响量的相对不确定度相加得到，注意d以平方形式出现：Δρ/ρ = 2(Δd/d) + ΔR/R + Δl/l。代入数字：2×(0.01/0.45) = 0.0444；ΔR/R = 0.10/4.30 = 0.0233；Δl/l = 0.002/1.000 = 0.002。总和 = 0.0697 ≈ 7.0%。因此 Δρ = 0.070 × 1.37×10⁻⁶ ≈ 1.0×10⁻⁷ Ω m。最终报告的结果为 ρ = (1.37 ± 0.10) × 10⁻⁶ Ω m。

The dominant uncertainty source is clearly the diameter measurement – a small absolute error in d gets magnified by the squared factor and the tiny cross‑sectional area. This is always a key discussion point in PH03 answers.

显然，最大的不确定度来源是直径测量——d的微小绝对误差因平方效应和极小的横截面积而被放大。这一直都是PH03答案中的一个关键讨论点。

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7. Identifying and Minimising Systematic Errors | 识别与减小系统误差

Systematic errors shift all readings in one direction. For the resistivity experiment, the most insidious one is the zero error of the micrometer. If the micrometer reads +0.02 mm when fully closed, all diameter measurements will be too large, leading to an overestimated A and an underestimated ρ. Always check zero error and subtract it. Another systematic issue is the contact resistance at the crocodile clips. By measuring the voltage directly across the wire with a voltmeter (separate from the current‑carrying clips), the voltmeter’s high resistance ensures the measured V truly reflects the potential difference across the wire, not the contacts.

系统误差使所有读数朝同一个方向偏移。对电阻率实验而言，最隐蔽的系统误差是千分尺的零点误差。若千分尺在完全闭合时读数为 +0.02 mm，所有直径测量值都会偏大，导致A被高估、ρ被低估。务必检查零点误差并将其扣除。另一个系统性问题在于鳄鱼夹处的接触电阻。用电压表直接测量导线两端电压（与传导电流的夹子分开），电压表的高阻抗保证了测得的V真实反映导线两端的电势差，而非接触电阻上的压降。

Heating of the wire by the current raises its temperature and hence its resistivity. Keeping I ≤ 0.5 A and opening the circuit between readings limits this temperature rise. A quick “open‑read‑close” routine can reduce systematic temperature drift. Additionally, using a larger‑diameter wire lowers the resistance and reduces Ohmic heating per unit volume, but then the smaller resistance demands more sensitive meters – a trade‑off that exam questions delight in exploring.

电流对导线的加热使其温度升高，从而电阻率增大。将电流保持在 ≤ 0.5 A 且在读数间隙断开电路可抑制温升。一个快速的“断开‑读数‑闭合”操作流程可以减少系统性的温度漂移。此外，使用更大直径的导线虽能降低电阻并减小单位体积的欧姆发热，但此时更小的电阻又要求更高灵敏度的仪表——这种权衡取舍正是考卷乐于探索的命题点。

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8. Random Errors and How to Refine the Procedure | 随机误差与细化步骤

Random uncertainties originate from fluctuating contact resistance, vibration, and the observer’s ability to read analogue scales consistently. Repeating the entire V-I sweep twice, as done here, allows you to spot outliers and estimate the precision of your mean resistance. Additionally, taking the diameter at six distributed positions with rotation accounts for variation in circularity and wire thickness. If the wire is heavily kinked, discard that section or increase the number of readings.

随机不确定度来源于波动的接触电阻、振动以及观察者一致读取模拟刻度盘的能力。如我们这般把整套V-I扫描重复一遍，能让你发现离群值并估计平均电阻的精密程度。此外，在转动导线的同时于六个分布位置测量直径，可以顾及圆度和线径的不均匀性。如果导线严重弯折，应弃用该段或增加读取次数。

One under‑used refinement is to plot a graph of R against the reciprocal of cross‑sectional area for different wire samples. While this goes beyond the typical single‑wire PH03 task, the concept – minimising the percentage uncertainty in R by using a long length of wire – is standard: a longer wire gives a larger resistance, so the gradient of the V-I graph becomes steeper and easier to measure accurately. In the context of the 2023 paper, if candidates were asked to suggest an improvement, “increase the length l to 2 m while keeping current low” would be a robust answer.

一种常被忽视的细化方法是，对不同导线样品绘制R与横截面积倒数的关系图像。尽管这超越了典型单导线PH03任务的范围，但其核心概念——通过使用长导线来减小R的百分不确定度——是标准做法：更长的导线产生更大的电阻，使得V-I图像的斜率更陡、更易于精确测量。在2023年试卷的语境下，如果考生被要求提出一项改进，回答“将长度l增大到2 m同时保持低电流”将是一个有力的答案。

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9. Analysing the V-I Graph – Shape and Anomalies | 分析V-I图像——形状与异常

A straight V-I line passing through the origin confirms Ohm’s law and implies constant resistance (and constant temperature). If the line begins to curve at higher currents, the wire’s temperature has risen enough to increase its resistivity. The 2023 PH03 paper possibly included a graph with slight curvature at high V, asking students to identify that the

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