Question

Figure shows two conducting spheres separated by large distance and of radius 2cm and 3cm containing charges 10$\mu$C arc 20$\mu$C respectively. When the spheres are connected by a conducting wire then find out following :
i)Ratio of the final charge.
ii)Final charge on each sphere.
iii)Ratio of final charge densities.

Answer 1

$\left(i\right)$When they are connected, their potential becomes same.

Hence, $\frac{q_1}{r_1}=\frac{q_2}{r_2}$

$⟹\frac{q_1}{q_2}=\frac{2}{3}=2:3$

$\left(ii\right)$ $q_1+q_2=30\mu C$

$⟹q_2+\frac{2}{3}{q}_2=30$

$⟹q_2=18\mu C$

and $q_1=\frac{2}{3}{q}_2=12\mu C$

$\left(iii\right)$

Charge densities $\sigma \propto \frac{q}{r^2}$

$⟹\frac{{\sigma }_1}{{\sigma }_2}=\frac{q_1}{q_2}{\left(\frac{r_2}{r_1}\right)}^2$

$=\frac{2}{3}\times \frac{3^2}{2^2}$

$⟹{\sigma }_1:{\sigma }_2=3:2$