Resistivity of a wire

What you are trying to do

Determine the resistivity of the metal of a wire from its resistance, length and cross-sectional area.

Apparatus

• A metre of bare resistance wire (nichrome or constantan) taped along a metre rule
• Micrometer screw gauge for the diameter
• Low-voltage supply, ammeter and voltmeter (or a calibrated ohmmeter)
• Crocodile clip to make the moving contact

Variables

• Independent: the length of wire in the circuit
• Dependent: its resistance
• Control: the wire's diameter and temperature: same wire, low current, readings taken briefly

Method

1. Measure the diameter at several places along the wire and in two perpendicular directions at each place; take the mean, and check the micrometer's zero error first.

1. Clip onto the wire at a series of measured lengths, and record V and I (or R directly) at each. Keep the current small and switch off between readings so the wire does not warm up: resistivity rises with temperature.

Analysis

1. R = ρL/A, so a graph of R against L is a straight line through the origin with gradient ρ/A.
2. Compute the area from the mean diameter, A = πd²/4, then ρ = gradient × A.
3. A small intercept is contact resistance at the clip; the gradient shrugs it off, and that immunity is what makes the graph beat a single reading.
4. How good an answer this is. The comparison worth making is with the book value for the metal you think you have. Nichrome sits near 1.1 × 10⁻⁶ Ω m and constantan near 4.9 × 10⁻⁷ Ω m, more than a factor of two apart, so even a ten per cent measurement settles which of them is taped to your rule. That is a conclusion your data support. Landing within one per cent of a book value is not, unless your own uncertainty is smaller than that, and a result quoted more precisely than the measurement allows invites the obvious question about how you knew.
5. What the meters cannot separate. A voltmeter and an ammeter across a clipped length read everything in the loop: the wire, the clip, the leads and the meters themselves. Taking the gradient discards whatever stays constant, which is why the intercept is harmless, but nothing in this procedure separates the wire's resistance from the contact's at a single length. The four-terminal arrangement does, with one pair of contacts carrying the current and a separate pair reading the pd, so the clip's resistance never appears in the measurement at all.

A worked set of readings

Nichrome wire of mean diameter 0.36 mm (A = 1.02 × 10⁻⁷ m²):

Idealised illustrative data, chosen so the working is easy to follow. Real readings scatter about the line rather than sitting on it, and your own graph will have points either side of the best fit.

The gradient of R against L is 10.8 Ω m⁻¹, so ρ = gradient × A = 10.8 × 1.02 × 10⁻⁷ = 1.10 × 10⁻⁶ Ω m, the right value for nichrome.

Evaluating the result

Idealised readings with one clip left dirty. The crocodile clip at 0.600 m was closed over a smear of oxide, adding about 0.45 Ω that has nothing to do with the wire. The third column divides R by L, which the model says is ρ/A and the same at every length.

Idealised illustrative data, chosen so the working is easy to follow. Real readings scatter about the line rather than sitting on it, and your own graph will have points either side of the best fit.

Three rows return 10.8 Ω m⁻¹ and the spoiled one returns 11.6. On the R against L graph it sits above the line while its neighbours sit on it, and above is the direction a contact fault pushes a reading in this series arrangement, because bad contact adds resistance. A high point suggests a contact fault without proving one, so the test is the repeat: reclip, scrape the wire if you must, and read again, and if the point drops back to the line the clip was the culprit. A reading you can explain and repeat is a corrected reading, while one you delete for spoiling a line is something else.

Where the uncertainty comes from

• Diameter: The dominant term: it is squared in the area, doubling its percentage uncertainty. Multiple readings in two directions catch an oval or tapering wire.
• Temperature: Current heats the wire and raises its resistance mid-experiment; low current, brief readings.
• Contact position: The crocodile clip has width; clip to the same edge each time and read the length at that edge.
• Meter resolution: At short lengths the resistance is small, so the meter's last digit is a larger percentage of it. A least-squares line applies no weighting; what the long lengths give is a smaller percentage uncertainty in R and a longer lever on the gradient, so run the readings out to the full metre.

What earns the marks

Safety