The resistivity of a wire is given by:
[tex]R=\rho\frac{l}{\pi r^2}[/tex]
Where [tex]\rho[/tex] is the electrical resistivity of the wire, [tex]l[/tex] is the length of the wire and r is the radius os the wire, Recall that [tex]r=\frac{d}{2}[/tex]. So:
[tex]R=\rho\frac{l}{\pi(\frac{d}{2})^2}\\R=4\rho\frac{l}{\pi d^2}[/tex]
We have [tex]l'=4l, \rho'=\rho, R'=R[/tex]. Thus:
[tex]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[/tex]
The diameter of the new wire must be twice of the original wire.
