Question

A silver wire has a resistance of $2.1\Omega $ at ${{27.5}^{\circ }}C$ , and a resistance of $2.7\Omega $ at ${{100}^{\circ }}C$. Determine the temperature coefficient of resistivity of silver.

Answer 1

Hint: At first we need to look for all the points or values given in the question. Now we have to find a formula or a process by which we can solve the question. When we figure out the formula we write that formula and then we place the values that are given in the question for resistance and temperature to get the required result.
Formula Used:
${{R}_{{{t}'}}}={{R}_{t}}\left( 1+\alpha \Delta T \right)$

Complete answer:
We know that the resistance of silver wire at ${{27.5}^{\circ }}C$is $2.1\Omega $.
And the resistance of that same silver wire at ${{100}^{\circ }}C$ is$2.7\Omega $.
Now, according to the question we are to determine the temperature coefficient of the resistivity of silver,
We know that ,
${{R}_{{{t}'}}}={{R}_{t}}\left( 1+\alpha \Delta T \right)$ ,
Now considering the cases,
${{R}_{{{t}'}}}=2.7\Omega $, and \[{t}'={{100}^{\circ }}C\]
${{R}_{t}}=2.1\Omega $, and $t={{27.5}^{\circ }}C$
Now, we are substituting the values in the equation,
 $2.7=2.1\left( 1+\alpha \left( 100-27.5 \right) \right)$,
On simplifying the equation we get,
$\alpha =\dfrac{2.7-2.1}{2.1(100-27.5)}$
On solving the above equation we get ,
$\alpha ={{0.0039}^{\circ }}{{C}^{-1}}$
This is the required result, so the temperature coefficient of the silver wire is ${{0.0039}^{\circ }}{{C}^{-1}}$.

Additional Information:
The relative change in a physical property of a body that is associated with change in temperature is known as temperature coefficient.

Note:
In the formula ${{R}_{{{t}'}}}={{R}_{t}}\left( 1+\alpha \Delta T \right)$, ${{R}_{{{t}'}}}$is the resistance of the silver wire at a higher temperature , ${{R}_{t}}$ is the resistance of the temperature at a lower temperature, $\Delta T$ is the change in temperature of the silver wire and $\alpha $ is the temperature coefficient for resistivity of silver. Students must at first check that all the values are available or not to find the answer.