Question

Two metallic wires A and B are connected in parallel. Wire A has length l and radius r, wire B has a length 2l and radius 2r. Compute the ratio of the total resistance of parallel combination and the resistance of wire A.

Answer 1

The resistance of a wire is directly proportional to its length and inversely proportional to the area of cross-section of the wire.
⇒ $R\propto \frac{l}{A}\propto \frac{l}{r^2}$
Wire A has length l and radius r and wire B has a length 2l and radius 2r. So, we can say,
$\frac{R_A}{R_B}=\frac{\frac{l}{r^2}}{\frac{2l}{4r^2}}=\frac{2}{1}$ .....(1)
Net resistance in parallel combination = $R_{rq}=\frac{R_A{R}_B}{R_A+R_B}=\frac{R_A\times \frac{R_A}{2}}{R_A+\frac{R_A}{2}}=\frac{1}{3}$...........(2)
The ratio of the total resistance of parallel combination and the resistance of wire A = $\frac{R_{eq}}{R_A}=\frac{1}{3}$.