Question

The length of a certain wire is kept same while its radius is doubled. what is the change in the resistance of this wire?

Answer 1

The resistance of a wire is given by
$R= \frac{\rho L}{A}$
where $\rho$ is the resistivity of the material, L the length of the wire and A its cross-sectional area. 

In the problem, $\rho$ and L remain the same, while A changes because the radius changes. The area is given by:
$A=\pi r^2$
This means that if we double the radius (2r), the area becomes
$A_{new}= \pi (2r)^2 = 4 \pi r^2 =4A$
And therefore, the new value of the resistance is
$R_{new} = \frac{\rho L}{4 A}= \frac{1}{4}R$

So, when the radius is doubled, the resistance becomes $\frac{1}{4}$ of its original value.