Question

Two metallic wires $A$ and $B$ of same material are connectedin parallel. Wire $A$ has length $l$ and radius $r$ and wire $B$ has length $2l$. Compute the ratio of the total resistance of parallel combination and the resistance of wire $A$?

Answer 1

The electrical resistance of a uniform conductor is given in terms of resistivity by:

$R=\frac{\rho l}{a}$

where $l$ is the length of the conductor in SI units of meters,

$a$ is the cross-sectional area (for a round wire $a=\pi r^2$ if $r$ is radius) in units of meters squared, and $\rho$ is the resistivity in units of ohm·meters.

$\therefore R_A=\frac{\rho l}{\pi r^2}$ and $R_B=\frac{2l\rho }{\pi r^2}$

Resistance in parallel will be

$\frac{1}{R_{eq}}=\frac{1}{R_A}+\frac{1}{R_B}=\frac{\pi r^2}{\rho l}+\frac{\pi r^2}{2l\rho }=\frac{3\pi r^2}{\rho 2l}$

$\therefore R_{eq}=\frac{2\rho l}{3\pi r^2}$

Now the ratio between total resistance and reistance of wire $A$ is

$\frac{R_{eq}}{R_A}=\frac{\frac{2\rho l}{3\pi r^2}}{\frac{\rho l}{\pi r^2}}=\frac{2}{3}$