Question

The resistance of a $20cm$ long wire is $5ohms$ The wire is stretched to form a uniform wire of $40cm$ length. The resistance now will be?

Answer 1

We are asked to find the resistance of a wire when it is stretched to a different length. We can start to attempt this question by finding the volume of the wire, even with stretching the volume of the wire doesn’t change. Once we find the volume of the wire, we can find the new cross-sectional area and with that value, we are able to find the new value of resistance thus leading us to the required solution.

Formulas used: The equation of resistance is given by, $R = \dfrac{{\rho l}}{A}$
Where $\rho$ is the resistivity of the material
$l$ is the length of the wire
$A$ is the cross sectional area of the wire
The volume of the wire can be found using the formula $V = lA$

Complete step by step answer:
Let us start by writing the given data down
The length of the wire is initially $l = 20cm$
The resistance in the initial case is $R = 5\Omega$
The new length of the wire after stretching is $l' = 40cm$
We can now move onto finding how much the area of the wire changes using the formula of volume $V = lA$
Bringing area to one side, we have $A = \dfrac{V}{l}$
Substituting this value in the formula to find the resistance, we have $R = \dfrac{{\rho l}}{A} \Rightarrow R \propto \dfrac{l}{{\left( {\dfrac{V}{l}} \right)}}$
This means that the resistance is proportional to the square of the length of the wire
The length of the wire changes to doubles and the resistance is changing to the square of two which is four.
In conclusion, the new resistance will be $R' = 4 \times 5 = 20\Omega$

Note: Resistance is defined as the opposition force offered by a material to the flow of current through the circuit. Resistance depends on the area of cross-section, and the length of the material. From this relation, we arrive at the formula for resistance as $R = \dfrac{{\rho l}}{A}$
Resistivity is constant for a material as it is the resistance offered by the material when the area of cross-section and the length of the wire are constant.