Answer to Question #141618 in Electric Circuits for Kashf Noor

Question

When a potential difference V is applied between the ends of a wire of diameter d and length l , the current in the wire is I. What is the current when a potential difference of 2V is applied between the ends of a wire of the same material of diameter 2d and the length 2l ? Assume that the temperature of the wire remains constant. 

Answer 1

resistance of wire:

$R = \rho\times\frac{l}{S}$

where $\rho$ - electrical resistivity of material, l - length of wire, S - cross sectional area of wire.

current:

$I = \frac{V}{R}$

then:

$R = \frac{V}{I}$

this implies:

$\frac{V}{I} = \rho\times\frac{l}{S}$

Cross-sectional area for wire:

$S=\pi \times \frac{d^2}{4}$

this implies:

$\frac{V}{I} = \rho\times\frac{4\times l}{\pi\times d^2}$

expressing I:

$I=\frac{V\times \pi \times d^2}{4\times \rho \times l}$

expressing current for second case:

$I_{2}=\frac{2\times V\times \pi \times 4 \times d^2}{4\times \rho \times 2 \times l}$

simplifying

$I_{2}=\frac{V\times \pi \times d^2}{\rho \times l}$