Question

How does the resistance of a wire vary with its cross sectional area?

Answer 1

Hint: This problem can be solved by recalling the formula for the resistance of a body at a specific temperature in terms of the specific resistance or resistivity of the material at that temperature, the length of the body and the cross sectional area of the body.

Formula used: $R=\rho \dfrac{l}{A}$

Complete step by step answer:
The resistance of a wire is a measure of the opposition that it offers to the flow of electric current to it. Greater the resistance of a wire, greater is the opposition offered by the wire to the flow of current across it.
The resistance of a wire can be mathematically written in terms of its dimensions, that is, length and cross sectional area, and the resistivity of the material of which the wire has been made.
The resistance $R$ of a body of length $l$ and cross sectional area $A$ is given by
$R=\rho \dfrac{l}{A}$ --(1)
Where $\rho $ is the specific resistance or the resistivity of the material of which the wire is made.
Therefore, from (1) we can see for a wire, the resistance can be written as inversely proportional to the cross sectional area.
$\therefore R\propto \dfrac{1}{A}$
Therefore, the resistance of a wire varies inversely with the cross sectional area.

Therefore, the resistance of a wire varies inversely with the cross sectional area.

Note: Since the resistance of a wire varies inversely with the cross sectional area, it means that a thicker wire has a smaller resistance and vice versa. This is also the reason why wires that are used for transmission of electric power over long distances are very thick so that they do not offer much resistance and offer an easy path for the flow of electric current and power.