An electric current passes through a circuit containing two wires of the same material connected in parallel. If the lengths of the wires are in the ratio of 4/3 and radius of the wires are in the ratio of 2/3, then the ratio of the currents passing through the wires will be

Given,

The lengths of the wires are in the ratio $=\frac{4}{3}$

 Radius of the wires are in the ratio $=\frac{2}{3}$

The potential difference across the wires will be the same since wires are connected in parallel.

${\mathrm{I}}_1=\frac{\mathrm{V}}{{\mathrm{R}}_1}$ and ${\mathrm{I}}_2=\frac{\mathrm{V}}{{\mathrm{R}}_2}$

$\Rightarrow \frac{{\mathrm{I}}_1}{{\mathrm{I}}_2}=\frac{{\mathrm{R}}_2}{{\mathrm{R}}_1}$

Therefore,

$\begin{array}{r}{R}_1=\rho \frac{I_1}{A_1}=\rho \frac{I_1}{\pi r_1^2}\end{array}$and

$R_2=\rho \frac{I_2}{\pi r_2^2}$

$\Rightarrow \frac{{\mathrm{l}}_1}{{\mathrm{I}}_2}=\frac{\rho \frac{{\mathrm{l}}_2}{\pi {\mathrm{r}}_2^2}}{\rho \frac{{\mathrm{I}}_1}{\pi {\mathrm{r}}_1^2}}=\frac{{\mathrm{I}}_2{\mathrm{r}}_1^2}{{\mathrm{l}}_1{\mathrm{r}}_2^2}$

$\Rightarrow \frac{{\mathrm{l}}_1}{{\mathrm{l}}_2}=\frac{3}{4}\times {\left(\frac{2}{3}\right)}^2=\frac{1}{3}$