Question

If a copper wire is stretched to make it 0.1% longer, the percentage change in its resistance is
(a). 0.2% increase
(b). 0.2% decrease
(c). 0.1% increase
(d). 0.1% decrease

Answer 1

- Hint: In this question remember to use the formula and also remember that increasing the length of wire by stretching it doesn’t change the volume of wire i.e. V initial = V final here “V” stands for volume of wire, use this information to approach towards the solution.

Complete step-by-step answer:
According to the given information wire is stretched to it 0.1%
So let L be the initial length of wire
And L’ be the final length of wire since the wire is stretched to make 0.1% longer
So the final length of wire will be L’ = L + 01%L
$ \Rightarrow $L’ = L + $\dfrac{{0.1}}{{100}} \times L$
$ \Rightarrow $L’ =$\dfrac{{100L + 0.1L}}{{100}}$
$ \Rightarrow $L’ =$\dfrac{{100.1L}}{{100}}$ = 1.001 L
Now we know that when a length of a wire is increased by stretching it the volume of wire doesn’t changes whereas the area of the wire changes
So volume of final wire will be equal to initial volume of wire i.e. V’ = V
Since we know that $Volume = length \times Area$
So \[A' \times L' = A \times L\]
Substituting the given value in the above equation
\[A' \times \left( {1.001} \right)L = A \times L\]
So the final area of wire will be $A' = \dfrac{A}{{1.001}}$
To find the new resistance let’s use the formula i.e. $R = \rho \dfrac{L}{A}$
Substituting the new dimensions of wire in the above formula we get
$R' = \rho \dfrac{{L'}}{{A'}}$
Now substituting the values in the above equation
$R' = \rho \dfrac{{1.001L}}{A} \times 1.001$
$ \Rightarrow $$R' = \rho \dfrac{L}{A}{\left( {1.001} \right)^2}$
Since we know that the initial value of resistance of wire is $R = \rho \dfrac{L}{A}$ therefore
$R' = R{\left( {1.001} \right)^2}$
R’ = 1.002 R
So the new resistance of new wire will be (1.002) R
So the percentage of resistance change will be $\dfrac{{R' - R}}{R} \times 100$
Substituting the value in the above equation we get
$\dfrac{{1.002R - R}}{R} \times 100$
$ \Rightarrow $$\dfrac{{0.002R}}{R} \times 100$
$ \Rightarrow $$0.002 \times 100$
$ \Rightarrow $0.2 %
Therefore the new resistance will increase by 0.2%

Hence option A is the correct one.

Note: In the above solution we used a term “resistance” which can be explained as the property of a material to resist the flow of current the resistance of material is different from each other also resistance is the reason of the conductors dissipate heat when electric current is flowed through it let's explain this concept with the help of an example suppose you have a wire through in which you supplied the current of 1A then you will notice that the output of the current will not same as the input of current this is because of resistance of wire and the loss in current losses in the form of heat.