Question

A copper wire is stretched to make 0.5 $\%$ longer. The percentage change in its electrical resistance if its volume remains unchanged is:
A. 0.25$\%$
B. 0.5 $\%$
C. 1.0 $\%$
D. 2.0 $\%$

Answer 1

In this question we are asked to find the change in electric resistance under the given conditions that the copper wire is stretched 0.5$\%$ and the volume is unchanged. Therefore, we will be using the formula of resistance which deals with the length and volume of the wire. We will also be using log functions to calculate the change in copper wire.

Formula used:
$R=\rho \dfrac{l}{A}$
Where,
$\rho$ is the resistivity of the specific resistance.
R is the specific resistance
l is the length of the given wire
A is the area of the given wire

Complete step-by-step answer:
We have been given the change in length and condition that volume remains constant.
Therefore,
Using the formula,
$R=\rho \dfrac{l}{A}$
Now, since we are given a condition on volume, we will multiply and divide by l.
We get,
$R=\rho \dfrac{l}{A}\times \dfrac{l}{l}$
Therefore,
$R=\rho \dfrac{{{l}^{2}}}{V}$ ……………… (since volume is product of length and area)
Now, taking log on both sides
We get,
$\log R=\log \rho +2\log l-\log V$
After differentiating above equation for small change we get,
$\dfrac{\Delta R}{R}=\dfrac{\Delta \rho }{\rho }+2\dfrac{\Delta l}{l}-\dfrac{\Delta V}{V}$
Now, it is said that the volume remains constant and we know that there is no change in resistivity
Therefore,
$\dfrac{\Delta R}{R}=2\dfrac{\Delta l}{l}$
From above equation we can say that change in resistance is directly proportional to twice the change in length
Therefore,
$R\propto 2l$
Therefore,
From the above equation we can say that if the length is increased by 0.5%, the resistance will increase by 1.0%.

So, the correct answer is “Option C”.

Note: Resistance is the property of a material to oppose the flow of current. If resistance in a material is more the material is a bad conductor and vice versa. The resistance can be calculated using Ohm’s law which deals with potential difference across a conductor and the current flowing through it. Resistance can be given by potential difference across the conductor over the current flowing through it. The unit of resistance is Ohms $\Omega$.