# Determining the Resistivity of a Metal ## Experimental Aims - The primary objective is to determine the resistivity ($\rho$) of a conducting wire by measuring how resistance changes with wire length - This experiment uses a constantan wire (or similar alloy) approximately 2 metres in length - By plotting resistance against length, the resistivity can be extracted from the gradient of the graph ### Variables **Independent variable:** - Length of wire, $L$ (m) **Dependent variable:** - Current through the wire, $I$ (A) **Control variables:** - Potential difference across the wire (kept constant throughout) - Material composition of the wire - Temperature of the wire (maintained by switching off between readings) - Cross-sectional area of the wire ## Equipment and Apparatus | Equipment | Purpose | |-----------|---------| | **Ammeter** | Measures current flowing through the wire | | **Voltmeter** | Measures potential difference across the wire section | | **Power supply** | Provides constant voltage (typically 3-6 V) | | **2.0 m test wire** | The conductor whose resistivity is being determined (e.g. constantan, nichrome) | | **Crocodile clip lead** | Allows electrical connection at different positions along the wire | | **Metre ruler** | Measures the length of wire in the circuit | | **Micrometer screw gauge** | Measures the diameter of the wire with high precision | | **Connecting wires** | Complete the circuit connections | ### Measurement Resolution - Metre ruler: **1 mm** - Micrometer screw gauge: **0.01 mm** - Voltmeter: **0.1 V** - Ammeter: **0.01 A**

A micrometer screw gauge provides much better precision for diameter measurements than vernier calipers, crucial since area depends on diameter squared

## Experimental Method ### Setup and Procedure 1. **Measure the wire diameter:** - Use the micrometer screw gauge to measure the diameter at 5-10 random positions along the wire - Take care to close the micrometer gently to avoid deforming the wire - Calculate the mean diameter from all measurements - This accounts for any manufacturing variations in wire thickness 2. **Construct the circuit:** - Secure the test wire alongside a metre ruler using tape or clamps - Attach one end of the wire firmly to the circuit at the 0 cm mark - Connect the ammeter in series with the wire - Connect the voltmeter in parallel across the wire segment being tested - Use a crocodile clip as a movable connection point 3. **Initial measurement:** - Position the movable clip at 25 cm from the fixed end - Set the power supply to provide 3.0 V (or another suitable constant voltage) - Switch on and immediately record the ammeter reading - Switch off the circuit immediately after recording to prevent heating 4. **Systematic data collection:** - Repeat the current measurement three times at this length position - Move the crocodile clip to the next position (e.g. 50 cm, 75 cm, 100 cm...) - Continue in regular intervals (typically 25 cm steps) up to the full 2.0 m length - Aim for 8-10 different length measurements total - Take three current readings at each position 5. **Calculate average current:** - For each wire length, compute the mean current from the three trials - This reduces the impact of random fluctuations in readings 6. **Determine resistance values:** - For each average current, calculate resistance using: $$R = \frac{V}{I}$$ - Where $V$ is the constant voltage and $I$ is the average current for that length ### Example Data Table | Length $L$ / m | Current₁ / A | Current₂ / A | Current₃ / A | Average Current / A | Resistance $R$ / Ω | |----------------|--------------|--------------|--------------|---------------------|-------------------| | 0.25 | | | | | | | 0.50 | | | | | | | 0.75 | | | | | | | 1.00 | | | | | | | 1.25 | | | | | | | 1.50 | | | | | | | 1.75 | | | | | | | 2.00 | | | | | | ## Analysis of Results ### The Resistivity Equation The relationship between resistance and the physical properties of a wire is given by: $$R = \frac{\rho L}{A}$$ Where: - $R$ = resistance (Ω) - $\rho$ = resistivity (Ω m) - $L$ = length of conductor (m) - $A$ = cross-sectional area (m²) ### Extracting Resistivity from the Graph **Step 1: Calculate cross-sectional area** For a circular wire with mean diameter $d$: $$A = \pi r^2 = \frac{\pi d^2}{4}$$ Remember to convert diameter from millimetres to metres before calculation.

Resistivity: $\rho = \frac{RA}{L}$, measured in ohm-metres (Ω m), describes how strongly a material opposes current flow

**Step 2: Plot resistance against length** - Place length $L$ on the x-axis - Place resistance $R$ on the y-axis - Plot all data points and draw a line of best fit - The line should pass through or very near the origin The expected relationship is linear, as shown in this typical result:

**Step 3: Determine the gradient** The resistivity equation can be rearranged: $$R = \frac{\rho}{A} \times L$$ This matches the straight-line equation $y = mx$, where: - Gradient $m = \frac{\rho}{A}$ Calculate the gradient using: $$\text{gradient} = \frac{\Delta R}{\Delta L}$$ **Step 4: Calculate resistivity** $$\rho = \text{gradient} \times A$$ The resistivity equals the gradient multiplied by the cross-sectional area calculated in Step 1. ## Evaluating the Experiment ### Systematic Errors **Zero error in length measurement:** - Ensure the fixed connection point is exactly at the 0 cm mark on the ruler - If the starting position is offset, all subsequent length measurements will be systematically wrong - Check alignment before beginning data collection ### Random Errors **Temperature variation:** - As current flows, the wire heats up, increasing its resistance independent of length - This is why the circuit must be switched off between readings - Even brief current flow can cause measurable temperature rise in thin wires - Keep currents small (under 0.5 A if possible) **Diameter measurement uncertainty:** - The diameter may vary along the wire due to manufacturing imperfections - Taking 5-10 measurements at random positions and averaging reduces this error - The micrometer must be zeroed before use to avoid systematic offset

Since $A \propto d^2$, small errors in diameter measurement are squared when calculating area, significantly affecting the final resistivity value

**Meter resolution:** - Multiple readings at each length position help average out reading uncertainties - Digital meters may fluctuate by ±1 in the last digit ### Improving Accuracy - Use longer wire lengths to increase the range of resistance values - Ensure voltmeter and ammeter are properly calibrated - Take readings quickly to minimize heating effects - Verify the power supply maintains constant voltage throughout ## Safety Considerations **Electrical heating hazards:** - Thin wires carrying current become hot very rapidly - Never touch the test wire while the circuit is energized - Even low voltages (3-6 V) can heat thin wires enough to cause burns **Preventing wire damage:** - Switch off immediately if you smell burning insulation - Excessive current can melt the wire or damage surrounding materials - Keep the circuit off between measurements **General laboratory safety:** - Keep liquids away from all electrical apparatus - Ensure connections are secure to avoid short circuits - Check the power supply voltage setting before switching on

To minimize heating: use the lowest practical voltage, take readings swiftly, and allow cooling time between measurements

## Worked Example A student investigates a 2.0 m nichrome wire and obtains the following data: | Length / m | Current₁ / A | Current₂ / A | Current₃ / A | Average Current / A | Resistance / Ω | |------------|--------------|--------------|--------------|---------------------|----------------| | 0.30 | 0.62 | 0.61 | 0.62 | 0.62 | 8.06 | | 0.60 | 0.41 | 0.42 | 0.41 | 0.41 | 12.20 | | 0.90 | 0.31 | 0.31 | 0.30 | 0.31 | 16.13 | | 1.20 | 0.25 | 0.26 | 0.25 | 0.25 | 20.00 | | 1.50 | 0.21 | 0.21 | 0.20 | 0.21 | 23.81 | | 1.80 | 0.18 | 0.18 | 0.18 | 0.18 | 27.78 | The wire diameter measurements (mm): 0.26, 0.25, 0.26, 0.27, 0.25, 0.26 Voltage maintained at: 5.0 V Calculate the resistivity of nichrome. ### Solution **Step 1: Calculate average diameter and cross-sectional area** Mean diameter = $\frac{0.26 + 0.25 + 0.26 + 0.27 + 0.25 + 0.26}{6} = 0.258$ mm Converting to metres: $d = 0.258 \times 10^{-3}$ m Cross-sectional area: $$A = \frac{\pi d^2}{4} = \frac{\pi (0.258 \times 10^{-3})^2}{4} = 5.23 \times 10^{-8} \text{ m}^2$$ **Step 2: Plot the graph and find gradient** Plotting $R$ against $L$ gives a straight line through the origin:

Using two points from the line of best fit: - At $L = 0.30$ m, $R = 8.00$ Ω - At $L = 1.80$ m, $R = 28.00$ Ω $$\text{Gradient} = \frac{\Delta R}{\Delta L} = \frac{28.00 - 8.00}{1.80 - 0.30} = \frac{20.00}{1.50} = 13.33 \text{ Ω m}^{-1}$$ **Step 3: Calculate resistivity** $$\rho = \text{gradient} \times A = 13.33 \times (5.23 \times 10^{-8})$$ $$\rho = 6.97 \times 10^{-7} \text{ Ω m}$$ **Result:** The resistivity of the nichrome wire is approximately $7.0 \times 10^{-7}$ Ω m. This value is reasonably close to the accepted resistivity of nichrome (around $1.0 - 1.5 \times 10^{-6}$ Ω m depending on composition), with variations due to experimental uncertainties.