CP7 – Electrical Resistivity | Edexcel A Level Physics

Theory — Electrical Resistivity

Understanding how resistance depends on material, length and cross-sectional area

Objectives

Measure resistance R at different lengths L using the voltmeter-ammeter method, plot R vs L, find ρ from gradient
Measure R for wires of different diameters d (same material, same length), plot R vs 1/A, find ρ from gradient
Use a micrometer to measure wire diameter and calculate cross-sectional area A = πd²/4

The Resistivity Equation

R = ρL / A → ρ = RA / L

R = resistance (Ω) · ρ = resistivity (Ω m) · L = length (m) · A = cross-sectional area (m²)
For circular wire: A = π(d/2)² = πd²/4

Linearised Graphs

Vary L (fixed A): R = (ρ/A)·L → plot R vs L, gradient = ρ/A → ρ = gradient × A Vary d (fixed L): R = (ρL)·(1/A) → plot R vs 1/A, gradient = ρL → ρ = gradient / L

Material Resistivities

Material	ρ / Ω m	Typical use
Nichrome	1.10 × 10⁻⁶	Heating elements, resistance wire
Constantan	4.90 × 10⁻⁷	Precision resistors, thermocouples
Manganin	4.40 × 10⁻⁷	Standard resistors, shunts

Voltmeter-Ammeter Method

Connect a voltmeter directly across the wire and an ammeter in series. Adjust the rheostat to set a low, steady current. Read V and I, then calculate R = V/I. Keep current low to prevent the wire heating — resistivity increases with temperature for metals.

R = V / I

In the simulation you drag the rheostat slider to set current. The voltmeter and ammeter both show analogue needle deflection and a digital readout.

Procedure

Two investigations — select the one you are performing.

Equipment

Resistance wire (nichrome, constantan or manganin) · Metre rule · Crocodile clips × 2 · Power supply (0–6 V DC) · Variable resistor (rheostat) · Ammeter · Voltmeter · Micrometer screw gauge · Connecting leads

Investigation 1 — Varying Length L

Keep the same wire (same material, same diameter) throughout. Move the crocodile clip to change the length in the circuit. Measure R at each length and plot R vs L. Gradient = ρ/A → ρ = gradient × A.

1

Measure the wire diameter with a micrometer

Before starting, measure d at three positions, correcting for any zero error. Calculate the mean d, then A = πd²/4. This is the fixed cross-sectional area used throughout Investigation 1.

💡 A small error in d causes a large error in A = πd²/4 — measure carefully. 0.01 mm error in d gives ~4% error in A for a 0.5 mm wire.

2

Set up the circuit

Stretch the wire along a metre rule. Connect one clip at the 0 cm end. Circuit: power supply → rheostat → ammeter → (clip 1) → wire → (clip 2) → back to supply. Voltmeter connects directly between clip 1 and clip 2.

💡 The voltmeter must be directly across the wire between the clips — not across the ammeter leads — to avoid including ammeter resistance in R.

3

Set L and adjust the rheostat

Place clip 2 at L = 20 cm. Adjust the rheostat to give a steady, low current (e.g. 0.3 A). Read V from the voltmeter and I from the ammeter. Calculate R = V/I.

💡 Keep current low and read quickly — wire heating increases resistance. If R drifts upward as you watch, reduce the current.

4

Repeat for at least 8 different lengths

Move clip 2 to new positions (e.g. 20, 30, 40, 50, 60, 70, 80, 90 cm). Keep the rheostat setting constant so the same current flows each time.

💡 Wider range of L gives a more reliable gradient. Aim for at least a 4:1 ratio between shortest and longest length.

5

Plot R vs L — find ρ

Gradient = ρ/A. Calculate ρ = gradient × A (A in m², L in m → ρ in Ω m). The line should pass through the origin — a positive y-intercept indicates contact resistance at the clips.

💡 Contact resistance is typically 0.01–0.1 Ω and is independent of L. To remove its effect, measure the slope of the graph rather than using individual R values.

Investigation 2 — Varying Cross-sectional Area A

Fix L = 60 cm. Use the same material but different wire diameters. Measure d with a micrometer for each wire, calculate A = πd²/4 and 1/A. Plot R vs 1/A. Gradient = ρL → ρ = gradient / L.

1

Set and measure fixed length L

Place the crocodile clips at exactly L = 60.0 cm for every wire. Re-measure L when you swap wires — the clips can shift slightly.

💡 Marking the clip positions on the metre rule with a pencil dot ensures L is truly constant between swaps.

2

Measure diameter of first wire

Use a micrometer to measure d at three positions along the wire. Correct for zero error. Calculate mean d, A = πd²/4, and 1/A.

💡 Rotate the wire 90° between readings to detect any ovality. Use the mean of all readings as d.

3

Measure R using voltmeter-ammeter method

Apply a low current and record V and I. Calculate R = V/I. Take 3 readings at different currents and use the mean R to reduce random error.

💡 Multiple V/I readings and taking the mean is equivalent to finding the gradient of a V vs I graph — both reduce uncertainty.

4

Swap wire and repeat

Replace with a wire of the same material but different diameter. Maintain the same fixed length L. Measure d and R for each new wire.

💡 Use the same material throughout — different materials have different ρ, so the R vs 1/A graph would not be linear if materials are mixed.

5

Plot R vs 1/A — find ρ

Gradient = ρL. Calculate ρ = gradient / L (with L in m, A in m² → ρ in Ω m). Line must pass through origin.

💡 A small error in d has a large effect on A = πd²/4 and therefore 1/A — this is the main source of uncertainty in Investigation 2.

Material

Wire Diameter (fixed)

Diameter d 0.45 mm

0.28 mm0.71 mm

Variable — Length L

Clip position 0.20 m

0.10 m1.00 m

Rheostat — set current

Resistance dial mid

max currentmin current

R = V / I

— Ω

adjust rheostat to set current

Length L

—

Resistance R

—

Area A

—

ρ = RA/L

—

Readings: 0/8 minimum

Material

Fixed Length = 0.60 m

L = 0.60 m (fixed for all wires)

Variable — Wire Gauge

Rheostat — set current

Resistance dial mid

max currentmin current

R = V / I

— Ω

adjust rheostat to set current

Diameter d

—

Area A

—

1/A

—

R

—

Readings: 0/5 minimum

Results Data Table

Investigation 1 — Varying Length

#	Material	d / mm	A / mm²	L / m	V / V	I / A	R = V/I / Ω	ρ = RA/L / Ω m
No data yet.

Investigation 2 — Varying Diameter

#	Material	SWG	d / mm	A / mm²	1/A / mm⁻²	V / V	I / A	R / Ω
No data yet.

Graph & Analysis

Resistivity from R vs L

—Collect ≥3 readings to plot

Parameters

Gradient (ρ/A)—

R²—

Expected ρ—

% difference—

Interpretation

Record readings to see analysis.

Resistivity from R vs 1/A

—Collect ≥3 readings to plot

Parameters

Gradient (ρL)—

R²—

Expected ρ—

% difference—

Interpretation

Record readings to see analysis.

Discussion Questions

Write your answers and reveal model answers when ready.

Q1

Explain why the R vs L graph should pass through the origin, and what a positive y-intercept indicates in practice.

From R = ρL/A, when L = 0 there is no wire and R = 0. The relationship R ∝ L means the graph is directly proportional — a straight line through the origin. A positive y-intercept indicates contact resistance at the crocodile clip connections, which is independent of L. This is why the voltmeter must be connected directly across the wire at the clip positions to exclude any resistance in the connecting leads or clips.

Q2

A nichrome wire has diameter 0.46 mm, length 0.800 m and resistance 12.3 Ω. Calculate the resistivity.

d = 0.46 mm = 4.6 × 10⁻⁴ m. A = π(d/2)² = π × (2.3 × 10⁻⁴)² = π × 5.29 × 10⁻⁸ = 1.66 × 10⁻⁷ m². ρ = RA/L = 12.3 × 1.66 × 10⁻⁷ / 0.800 = 2.04 × 10⁻⁶ / 0.800 = 1.06 × 10⁻⁶ Ω m. The accepted value is 1.10 × 10⁻⁶ Ω m — a difference of about 4%, within typical experimental uncertainty.

Q3

Explain why it is important to use a low current, and how you would detect if the wire had heated significantly during the experiment.

Resistivity increases with temperature for metals. If the current is too large, P = I²R heats the wire, increasing ρ and therefore R — giving a measured value higher than the true room-temperature value. To detect heating: take readings quickly and watch whether R drifts upward during a single reading. Alternatively, reduce the current and check whether R decreases — if it does, the wire was self-heating at the higher current. Use the lowest practical current that gives a readable voltage.

Q4

In Investigation 2, explain why all wires must be the same material, and why the line must pass through the origin.

All wires must be the same material because ρ must be constant for R = (ρL)(1/A) to give a straight line — if ρ varies between wires, the gradient changes and the points don't fall on a single line. The line passes through the origin because as A → ∞ (infinite cross-section), R → 0. When 1/A = 0, R = ρL × 0 = 0. A positive y-intercept would indicate contact resistance independent of wire geometry.

Q5

A small error in measuring diameter d leads to a large percentage error in resistivity ρ. Explain why, using the equation A = πd²/4.

Because A = πd²/4, the percentage error in A is approximately twice the percentage error in d (since A ∝ d²). For example, a 2% error in d gives approximately 4% error in A. Since ρ = RA/L, the percentage error in ρ ≈ percentage error in R + percentage error in A. So a 1% error in d contributes ~2% to the final uncertainty in ρ. This is why micrometer readings must be taken at multiple positions and orientations — to get the most accurate mean d possible.