40, 46; CP2, CP3Electric CircuitsCore

Core Practicals 2–3: Resistivity and Internal Resistance

A graph can separate a wire’s resistance from contact offsets and a cell’s emf from its internal voltage drop. The experiment must keep heating and measurement loading under control.

40 min 3.2 Circuit Analysis

What you'll be able to do

Measure resistivity with micrometer, ammeter and voltmeter
Infer emf and internal resistance from a terminal-pd graph
Explain circuit design, heating controls and gradient units

CP2: resistivity of a wire

Connect an ammeter in series and a voltmeter across a measured length of wire; use a current-limiting variable resistor and switch. Measure diameter with a zero-checked micrometer in several places and orientations. Vary the test length using a reliable contact and record V and I for each length.

Calculate R=V/I and plot R against L. Gradient is ρ/A, so multiply by the cross-sectional area to obtain resistivity. Keep current low and switch off between readings to reduce heating, since resistance depends on temperature. Contact resistance can give a non-zero intercept; the gradient is less affected by a fixed series offset than a single R/L value.

R=\frac VI=\frac{\rho L}{A},\qquad \rho=A\frac{\Delta R}{\Delta L}

A=πd²/4; resistivity has SI units Ω m.

CP3: emf and internal resistance

Connect a variable load and ammeter in series with the cell and measure terminal pd with a high-resistance voltmeter. Vary load resistance to obtain a range of current and terminal-pd pairs. Avoid a short circuit; large currents heat the cell and change the model parameters.

Plot terminal pd V vertically against current I horizontally. Extrapolating the fitted line to I=0 gives emf ε. The slope is −r, so internal resistance is the magnitude of the negative gradient. An open-circuit voltmeter gives an approximate emf if its current draw is negligible.

V=\varepsilon-Ir

Treat ε and r as approximately constant over the short experiment.

Evaluate the measurement

Place the voltmeter across the actual test component, and check the meaning of each meter reading against the circuit. An ammeter has low resistance; a voltmeter has high resistance. Finite meter resistances can matter when the component resistance is extreme.

Repeat a reference reading at the end: a shifted value may indicate warming, contact drift or a changing cell. When estimating gradient uncertainty, use lines consistent with the scatter or error bars and report units. A negative fitted slope does not mean a negative physical internal resistance.

Work through the reasoning

Try each problem before opening the worked solution. State the physical model, show substitutions with consistent units, and explain what your answer means.

Example 1A wire has diameter 0.40 mm and R–L gradient 4.0 Ω m⁻¹. Calculate resistivity.

A=\pi(0.40\times10^{-3})^2/4=1.26\times10^{-7}

m².

\rho=A(4.0)=5.03\times10^{-7}

Ω m.

Answer

5.0\times10^{-7}

Ω m.

Example 2A cell’s fitted V–I line passes through (0.20 A, 1.40 V) and (0.60 A, 1.20 V). Find r and ε.

1Gradient

=(1.20-1.40)/(0.60-0.20)=-0.50

Ω.

r=0.50

Ω.

\varepsilon=V+Ir=1.40+0.20(0.50)=1.50

Answerr=0.50 Ω and emf=1.50 V.

Equation recap

R=\frac VI=\frac{\rho L}{A},\qquad \rho=A\frac{\Delta R}{\Delta L}

A=πd²/4; resistivity has SI units Ω m.

V=\varepsilon-Ir

Treat ε and r as approximately constant over the short experiment.

Common mistakes to avoid

Reporting internal resistance as the negative slope value.

Use the positive magnitude r.

Placing the voltmeter in series with the wire.

It should measure pd across the test length.

Key takeaways

Use a length graph and measured area to obtain resistivity.
A cell’s V–I line gives emf and internal resistance.
Heating, contact resistance and meter loading can bias electrical measurements.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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