Question

A wire of resistivity rhorho must be replaced in a circuit by a wire of the same material but 4 times as long. If, however, the resistance of the new wire is to be the same as the resistance of the original wire, the diameter of the new wire must be

Answer 1

The resistivity of a wire is given by:

$R=\rho\frac{l}{\pi r^2}$

Where $\rho$ is the electrical resistivity of the wire, $l$ is the length of the wire and r is the radius os the wire, Recall that $r=\frac{d}{2}$. So:

$R=\rho\frac{l}{\pi(\frac{d}{2})^2}\\R=4\rho\frac{l}{\pi d^2}$

We have $l'=4l, \rho'=\rho, R'=R$. Thus:

$R'=4\rho'\frac{l'}{\pi d'^2}\\R=4\rho\frac{4l}{\pi d'^2}\\4\rho\frac{l}{\pi d^2}=4\rho\frac{4l}{\pi d'^2}\\\frac{1}{d^2}=\frac{4}{d'^2}\\d'^2=4d^2\\d'=2d$

The diameter of the new wire must be twice of the original wire.