Question

A resistive wire is stretched till its length is increased by 100% due to the consequent decrease in diameter, the change in the resistance of a stretched wire will be _____.
A. $300\% $
B. $200\% $
C. $100\% $
D. $50\% $

Answer 1

Hint: To answer this question students should know what is ohm’s law and equation of ohm’s law. Ohm’s law gives the relation between current passing and voltage applied across any circuit. Then what is resistance? How is it related to ohm’s law?

Complete step by step answer:
Ohm's law helps us in determining the values of resistance, the current flowing through a circuit, and the voltage applied. Hence with the help of these values, we can find the values of other factors like drift speed, resistivity and many more. It also allows us to calculate the value of power consumption.
$I\propto V$ or $I\propto V$
$ \Rightarrow V = IR$
Where R is a constant called resistance of the conductor. The value of this constant depends on the nature, length, area of cross section and temperature of the conductor.

As per Ohm's law we have,
$V = IR$
Were, $R = \rho \dfrac{l}{a} - - - - (1)$
Here, $\rho $ is the resistivity ‘l’ is the length of material and ‘a’ is the area of the material.
As the volume is constant,
Initial volume = final volume
$la = 2la' - - - - (2)$
(Here initially the length was l and it was increased by $100\% $ so the final length will become $2l$ )
$a' = \dfrac{a}{2}$

Now, substituting the value in the equation of resistance,
$ \Rightarrow R' = \rho \dfrac{{2l}}{{\dfrac{a}{2}}} - - - - (3)$
$ \Rightarrow R' = \rho \dfrac{{2l}}{a} \times 2 - - - - (4)$
Further simplifying we get
$ \Rightarrow R' = 4\rho \dfrac{l}{a}$
$ \Rightarrow R' = 4R$

Therefore, change in the resistance of stretched wire will be
$percentage{\text{ new }}R' = \dfrac{{final{\text{ resistance - initial resistance}}}}{{initial{\text{ resistance}}}} \times 100$
Substituting the given values
$ \Rightarrow {{\% }}R' = \dfrac{{4R - R}}{R} \times 100$
$ \Rightarrow \dfrac{{3R}}{R} \times 100 = 300\% $
$\therefore R' = 300\% $

Hence, the correct answer is option A.

Note: For ohm’s law it is a necessary condition that the temperature must be constant. It’s true only for linear electrical elements. For non-linear electrical elements with parameters like capacitance, resistance etc, the voltage and current won’t be constant with respect to time making it difficult to use Ohm’s law. Ohm’s law is not applicable for unilateral electrical elements like diodes and transistors as they allow the current to flow through in one direction only.