# F2-3: The Temperature Coefficient of Resistance of Copper¶

## Apparatus¶

Metre bridge board; clamp and stand; $$1.5\text{V}$$ cell; galvanometer; jockey; standard resistor $$10\Omega$$; copper wire & thermometer in a test tube; $$1\text{L}$$ beaker of water; bunsen burner (or other heat source); 8 connecting leads (5 long, 3 short); 1 sheet of graph paper. ## Procedure¶

1. Set up the apparatus as above, connecting the battery last. Check carefully that all connections are secure. Do not begin heating yet. Find the balance point length $$L$$ where the galvanometer reads zero. Disconnect the battery. Read the temperature $$\Theta$$.
2. Begin heating the water. At temperatures approximately $$30, 35, 40, 45,... \text{ up to } 90\text{°C}$$, reconnect the battery, find $$L$$, and read $$\Theta$$ (to the nearest $$0.1\text{°C}$$). Disconnect the battery between readings.
3. Tabulate the readings of $$L$$ and $$\Theta$$.

## Theory¶

1. Resistivity (the Greek $$\rho$$ ) at a given temperature is:

$\rho = \text{Resistance} \left( \frac{ \text{Area}}{ \text{length}} \right)$

Therefore the resistance of a given sample varies with temperature. This is given by:

$R_{\Theta} = R_0 (1+ \alpha \Theta + \beta \Theta^2)$

where:

$\begin{split}R_{\Theta} &= R \quad \text{ at } \Theta \text{°C} \\ R_0 &= R \quad \text{ at } 0 \text{°C} \\ \alpha \text{ } & \text{and } \beta \text{ are constants}\end{split}$

$$\beta$$ is very small, and is usually neglected. In this experiment, assume that $$\beta =0$$.

2. The circuit is a Wheatstone Bridge: $\frac{R_1}{R_2} = \frac{R_3}{R_4} \label{eqn1} \tag{equation 1}$

And therefore in this experiment when the units of L are cm:

$R_{\Theta} = 10 \left( \frac{L}{100-L} \right) \label{eqn2} \tag{equation 2}$

## Analysis¶

1. For each value of $$L$$, calculate a value of $$R_\Theta$$ using the above formula $$(\ref{eqn2})$$.

2. Plot a graph of $$R_\Theta$$ vs. $$\Theta$$°C. Find the gradient and the y-intercept.

NB: it is not necessary for the $$R_\Theta$$ axis to extend down to zero.

3. Use the formula given in 1 of the theory, together with the gradient and y-intercept only, to calculate $$\alpha$$, the temperature coefficient of resistance of copper.

## Questions¶

1. Use $$\ref{eqn1}$$ above to prove $$\ref{eqn2}$$.
2. Calculate the expected resistance of the copper wire when its temperature is $$300\text{°C}$$.
3. If copper has a resistivity of $$1.7 \times 10^{-8}\Omega\text{m}$$ at 293K, find the resistance of a sample of copper length $$5\text{cm}$$, uniform cross-sectional area $$10^{-6} \text{m}^2$$, at:
1. $$0\text{°C}$$
2. $$100\text{°C}$$