Answer
$\sqrt2$
Work Step by Step
Use equation 18–3 to find the resistances. The cross-sectional area $A=\pi r^2=\frac{\pi d^2}{4}$.
$$R=\rho\frac{\mathcal{l}}{A}=\rho\frac{4 \mathcal{l}}{\pi d^2}$$
The wires have the same resistance and the same resistivity.
$$R_{long}=R_{short}$$
$$\rho\frac{4 \mathcal{l}_{long}}{\pi d_{long}^2}=\rho\frac{4 \mathcal{l}_{short}}{\pi d_{short}^2}$$
The long wire is twice as long as the short one.
$$\rho\frac{4(2 \mathcal{l}_{short})}{\pi d_{long}^2}=\rho\frac{4 \mathcal{l}_{short}}{\pi d_{short}^2}$$
$$\frac{d_{long}}{d_{short}}=\sqrt2$$
