Describe an experiment to determine resistance using a voltmeter and an ammeter and do the appropriate calculations

Objective (AO1‑AO3)

To determine the resistance of an unknown component by measuring the voltage across it and the current through it with a voltmeter and an ammeter, to record the data, carry out the required calculations (including uncertainties) and to evaluate the result against the Cambridge IGCSE 0625 (Core 4.2.4) syllabus.

Syllabus links

Theory

1. Ohm’s law – definition of resistance (AO1)

For an ohmic conductor the voltage, current and resistance are related by

\$V = I R\$

Re‑arranged, this gives the definition of resistance:

\$R = \frac{V}{I}\$

Resistance is a property of the material and its geometry; it is measured in ohms (Ω).

2. Resistance and geometry (AO1)

For a uniform conductor

\$R \propto \frac{L}{A}\$

where L is the length and A the cross‑sectional area. Consequently:

• If the length is doubled, the resistance doubles.
• If the area is doubled, the resistance is halved.

These relationships are part of the core syllabus and are therefore presented in the main text rather than as an optional extension.

3. Temperature effect (AO2)

For most metals the resistivity increases with temperature, approximately linearly over a small range:

\$\rho(T) \approx \rho0[1+\alpha (T-T0)]\$

where α is the temperature coefficient. In practice the resistor may warm as current flows, causing the measured resistance to drift. A simple observation of this drift is included in the procedure.

4. V–I relationship for an ideal resistor (AO2)

An ideal (ohmic) resistor gives a straight‑line graph of voltage (V) against current (I) that passes through the origin. The gradient of the line equals the resistance.

Safety (AO3)

• Always switch off the power supply before making or breaking connections.
• Use insulated crocodile clips and keep wires away from the body.
• Do not touch live terminals while the supply is on.
• Do not exceed the voltage/current ratings of the ammeter, voltmeter or the unknown component.
• If the resistor becomes noticeably hot, allow it to cool before continuing.

Apparatus

• Unknown resistor (or any component whose resistance is to be measured)
• Digital voltmeter – high internal resistance (connect in parallel)
• Digital ammeter – low internal resistance (connect in series)
• Adjustable DC power supply
• Connecting wires with crocodile clips
• Switch (optional, for safety)
• Ruler or caliper (optional, for geometry extension)
• Thermometer or infrared thermometer (optional, to record temperature rise)

Experimental circuit

Pre‑lab checklist (AO2)

1. Set the ammeter to a range that will comfortably accommodate the expected current (e.g. 0–0.5 A).
2. Set the voltmeter to a range that covers the expected voltage drop (e.g. 0–12 V).
3. Zero the instruments if they have a “zero” or “tare” function.
4. Record the nominal (set) voltage of the power supply and, if possible, measure the actual output with a multimeter.
5. Inspect all leads and clips for damage; ensure good contact.

Procedure (AO2)

1. Switch off the power supply and assemble the circuit as shown in Figure 2.
2. Close the switch (if used) and set the power‑supply to its lowest voltage (≈ 2 V). Allow the circuit to stabilise for about 2 s.
3. Record:

   • Voltage across the resistor, V (V)
   • Current through the circuit, I (A)
   • Temperature of the resistor (optional, °C)

4. Increase the supply voltage in equal steps (e.g. 2 V, 4 V, 6 V, 8 V, 10 V). For each step repeat the measurements of V, I and temperature.
5. After the final reading switch the supply off and disconnect the circuit.
6. (Optional geometry extension) Measure the length L and diameter d of the resistor wire, calculate the cross‑sectional area A = πd²/4 and the theoretical resistance R = ρL/A using the appropriate resistivity ρ.

Data table

Calculations (AO2)

1. Resistance for each reading

   \$Rk = \frac{Vk}{I_k}\$

2. Mean resistance

   \$\overline{R} = \frac{\displaystyle\sum{k=1}^{n} Rk}{n}\$

   where \$n\$ is the number of measurements (normally 5).

3. Standard deviation (random error)

   \$\sigma = \sqrt{\frac{\displaystyle\sum{k=1}^{n}(Rk-\overline{R})^{2}}{n-1}}\$

4. Propagation of uncertainties (if the instrument uncertainties are \$\Delta V\$ and \$\Delta I\$):

   \$\frac{\Delta R}{R}= \sqrt{\left(\frac{\Delta V}{V}\right)^{2}+\left(\frac{\Delta I}{I}\right)^{2}}\$

   Hence

   \$\Delta R = R\;\sqrt{\left(\frac{\Delta V}{V}\right)^{2}+\left(\frac{\Delta I}{I}\right)^{2}}\$

5. Final result – present as \$\overline{R}\pm\sigma\$ Ω or \$\overline{R}\pm\Delta R\$ Ω, and state the percentage uncertainty.

Example calculation

For the 4 V setting the recorded values are:

\$V = 3.96\ \text{V},\qquad I = 0.020\ \text{A},\qquad \Delta V = \pm0.01\ \text{V},\qquad \Delta I = \pm0.0005\ \text{A}\$

Resistance:

\$R = \frac{3.96}{0.020}=198\ \Omega\$

Uncertainty:

\$\frac{\Delta R}{R}= \sqrt{\left(\frac{0.01}{3.96}\right)^{2}+\left(\frac{0.0005}{0.020}\right)^{2}}=0.025\$

\$\Delta R = 198 \times 0.025 \approx 5\ \Omega\$

If the five calculated resistances are 198 Ω, 200 Ω, 202 Ω, 199 Ω and 201 Ω, then

\$\overline{R}=200\ \Omega,\qquad \sigma\approx1.6\ \Omega\$

Result (rounded to one significant figure in the uncertainty):

\$\boxed{R = 200 \pm 2\ \Omega}\$

V–I Graph (AO2)

Plot the measured voltage (vertical axis, V) against the corresponding current (horizontal axis, A). Use a straight‑line fit (or the “line of best fit” tool on a graphing calculator). The gradient of this line is the resistance. A sample graph is shown in Figure 1.

Extension – Non‑ohmic devices (optional, AO2)

• Filament lamp – the V–I curve starts shallow and becomes steeper as the filament heats, illustrating the temperature‑dependence of resistance.
• Diode – a “knee” voltage is observed; current is essentially zero for reverse bias, showing that Ohm’s law does not apply.

Sources of error (AO2/3)

• Instrument internal resistance – The voltmeter draws a tiny current; the ammeter adds a small voltage drop. Both cause systematic error.
• Contact resistance – Poor connections at crocodile clips add extra resistance, most noticeable at low currents.
• Temperature rise of the resistor – As the resistor warms, its resistance increases, leading to a drift in the measured \$R\$.
• Resolution and reading error – Digital displays are limited to a certain number of significant figures; rounding introduces random error.
• Power‑supply stability – If the output voltage varies during the experiment, the assumed \$V_{\text{set}}\$ will not be accurate.
• Uncertainty propagation – Failure to combine the uncertainties of \$V\$ and \$I\$ correctly can underestimate the overall error.

Conclusion (AO3)

By measuring the voltage across an unknown component and the current through it, the resistance can be calculated using the definition \$R = V/I\$. Repeating the measurement at several supply voltages and averaging the results reduces random errors, while the calculation of uncertainties and the discussion of systematic errors provide a robust estimate of the true resistance. The experiment also reinforces key syllabus concepts: the correct placement of ammeters and voltmeters, the geometric and temperature dependence of resistance, and the linear V–I relationship that characterises an ideal (ohmic) resistor.