Welcome to Core Practical 7: Electrical Resistivity!

In this practical, we are moving beyond just looking at how much a specific component resists current. We are looking at resistivity—a fundamental property of the material itself. Whether you have a tiny scrap of copper or a massive copper beam, the resistivity remains the same, even though their resistances are very different. This experiment is a classic in the IAS Physics curriculum and frequently appears in Unit 3 exam questions.

1. The Science Behind the Experiment

Resistance (\(R\)) depends on the shape and size of an object. However, resistivity (\(\rho\)) is a constant for a material at a specific temperature. The relationship is given by the formula:

\(R = \frac{\rho l}{A}\)

Where:
\(R\) = Resistance (measured in ohms, \(\Omega\))
\(\rho\) = Resistivity (measured in ohm-metres, \(\Omega \text{m}\))
\(l\) = Length of the wire (measured in metres, \(\text{m}\))
\(A\) = Cross-sectional area of the wire (measured in \(\text{m}^2\))

The Goal: By measuring the resistance of different lengths of a wire, we can determine the resistivity of the metal it is made from.

2. The Experimental Setup

To find \(\rho\), we need to measure three main things: Resistance, Length, and Area.

Measuring Diameter (to find Area \(A\))

The wire is very thin, so we use a micrometer screw gauge.
Pro tip: The micrometer has a resolution of 0.01 mm. This is much more precise than a standard ruler!

  • Measure the diameter \(d\) at several points along the wire and at different orientations (rotations).
  • Calculate the average diameter to account for any non-uniformity in the wire.
  • Calculate the cross-sectional area using: \(A = \frac{\pi d^2}{4}\) (or \(A = \pi r^2\) where \(r = d/2\)).

Measuring Resistance (\(R\)) and Length (\(l\))

1. Attach the wire to a meter rule using tape.

2. Set up a circuit with a power supply, an ammeter (in series), and a voltmeter (in parallel across the test wire). Alternatively, an ohmmeter can be used for a direct reading.

3. Use crocodile clips or a "sliding contact" (like a jockey) to change the length \(l\) of the wire being tested.

4. For each length (e.g., 10 cm, 20 cm, up to 100 cm), record the current \(I\) and potential difference \(V\).

5. Calculate resistance for each length using Ohm’s Law: \(R = \frac{V}{I}\).

Quick Review: Key Variables

Independent Variable: Length of the wire (\(l\)).
Dependent Variable: Resistance (\(R\)).
Control Variables: Temperature of the wire (keep current low!) and the material of the wire.

3. Data Analysis: The Graph

To find the resistivity accurately, we don't just use one reading; we plot a graph.
If we rearrange our formula to \(R = (\frac{\rho}{A})l\), it looks exactly like the equation for a straight line: \(y = mx + c\).

  • y-axis: Resistance (\(R\)) in \(\Omega\)
  • x-axis: Length (\(l\)) in \(\text{m}\)
  • Gradient (\(m\)): \(\frac{\rho}{A}\)

To find resistivity, simply calculate the gradient of your best-fit line (using a large triangle on your graph) and multiply it by the cross-sectional area:

\(\rho = \text{gradient} \times A\)

4. Dealing with Uncertainties and Errors

Physics exams love to ask how you can make your results better. Here are the "must-know" points for Core Practical 7:

Systematic Errors

Zero Errors: Check the micrometer screw gauge when it is fully closed. If it doesn't read \(0.00 \text{ mm}\), you must add or subtract this "zero error" from all your diameter readings.

Contact Resistance: The crocodile clips might have their own resistance. If your graph does not go through the origin \((0,0)\), it might be due to this extra resistance where the clips meet the wire.

Random Errors

Heating Effect: This is the most common mistake! As current flows, the wire gets hot. Increased temperature increases resistance in metals.
Solution: Use a low current and switch off the power supply between readings to keep the temperature constant.

Parallax Error: When measuring the length with the meter rule, look directly down over the marking to avoid misreading the scale.

Uncertainty Calculations

Recall that for repeat readings, the uncertainty is half the range: \(\frac{\text{max} - \text{min}}{2}\).
For a single reading, the uncertainty is half the resolution of the instrument (e.g., \(\pm 0.005 \text{ mm}\) for a micrometer).

5. Summary Checklist

Key Takeaways:

  • Resistivity is a material property: \(\rho = \frac{RA}{l}\).
  • Use a micrometer for diameter (resolution 0.01 mm) and take readings in multiple places.
  • The gradient of an \(R\) vs \(l\) graph is \(\frac{\rho}{A}\).
  • Keep current low to prevent temperature changes from ruining your data.
  • Area must be in \(\text{m}^2\)—don't forget to convert from \(\text{mm}^2\) if necessary! (\(1 \text{ mm}^2 = 1 \times 10^{-6} \text{ m}^2\)).

Don't worry if the math seems a bit heavy! Just remember the graph: Resistance on the side, Length on the bottom. The steeper the line, the higher the resistivity (for a constant area)!

Cross-reference: If you want to see how resistivity fits into broader circuit theory, check out the Electric Circuits section of Unit 2.