Question

How does resistance (R) of a wire depend on its radius r?

Answer 1

Resistance can be defined as the opposition to the flow of current. It depends on the length and cross-sectional area of the wire or the conductor. Obtain the dependence of the resistance on the above two quantities. Transform the in terms of the radius of the wire and then we can find our answer.

Complete answer: Resistance of a material can be defined as the opposition offered by the material to the flow of current through the material. The resistance of a conductor depends on the length of the conductor, the cross-sectional area of the conductor and the material of the conductor.
The relation between the resistance and the length and cross-sectional area of the wire is given by the resistivity formula.
The resistance of a conductor is directly proportional to the length of the conductor and inversely proportional to the cross-sectional area of the conductor. We can write,
$R\propto \dfrac{L}{A}$
We can equate the above equation by introducing a proportionality constant.
$R=\rho \dfrac{L}{A}$
Where, $\rho$ is the resistivity of the conductor.
Where R is the resistance of the conductor, L is the length of the conductor and A is the cross-sectional area of the conductor.

Now, let r be the radius of the conductor. Then we can define the cross-sectional area of the conductor as,

$A=\pi {{r}^{2}}$
Putting this value on the above equation, we get that,
$R=\rho \dfrac{L}{\pi {{r}^{2}}}$
So, we can say that the resistance of a wire is inversely proportional to the square of the radius of the wire.
$R\propto \dfrac{1}{{{r}^{2}}}$

Note:
The unit of resistance is Ohm. The resistivity of the conductor or material can be defined as the resistance offered by the material of specific size and specific material. For different material with the same size, the resistivity is also different.