Question

Find the temperature at which the resistance of a wire made of silver will be twice its resistance at $20^\circ$C. Take $20^\circ$C as the reference temperature and temperature coefficient of resistance of silver at $20^\circ$C = $4.0 \times 10^{-3\ \text{K}^{-1}$.}

Hint:

For temperature dependence of resistance, always use the relation $R = R_0 (1 + \alpha \Delta T)$ where $\Delta T$ is the change in temperature from the reference temperature.

Answer 1

Step 1: Write the relation between resistance and temperature.
The resistance of a conductor at temperature $T$ is given by the relation:
\[ R = R_0 \left(1 + \alpha (T - T_0)\right) \]
where $R_0$ is the resistance at reference temperature $T_0$, and $\alpha$ is the temperature coefficient of resistance.

Step 2: Substitute the given condition.
It is given that the resistance becomes twice the resistance at $20^\circ$C.
Therefore,
\[ R = 2R_0 \] Substituting in the formula:
\[ 2R_0 = R_0 \left(1 + \alpha (T - 20)\right) \] Dividing both sides by $R_0$:
\[ 2 = 1 + \alpha (T - 20) \]

Step 3: Substitute the value of $\alpha$.
\[ 2 = 1 + (4.0 \times 10^{-3})(T - 20) \] \[ 1 = (4.0 \times 10^{-3})(T - 20) \]

Step 4: Solve for temperature.
\[ T - 20 = \frac{1}{4.0 \times 10^{-3}} \] \[ T - 20 = 250 \] \[ T = 270^\circ C \]

Step 5: Final answer.
The resistance of the silver wire becomes twice its value at
\[ T = 270^\circ C \]