Question

Two isolated metallic solid spheres of radii $R$ and $2R$ are charged such that both have same charge density $\sigma$. The spheres are then connected by a thin conducting wire. If the new charge density of the bigger sphere is ${\sigma }^{\prime }$. The ratio $\frac{{\sigma }^{\prime }}{\sigma }$ is :

• A. $\frac{5}{3}$
• B. $\frac{9}{4}$
• C. $\frac{5}{6}$
• D. $\frac{4}{3}$

Answer 1

Answer: (C)

$\frac{Q_1^{\prime }}{4\pi {\epsilon }_{0}R}=\frac{Q_2^{\prime }}{4\pi {\epsilon }_0(2R)}$
$\therefore Q_2^{\prime }=2Q_1^{\prime }$
$Q_1^{\prime }+Q_2^{\prime }=Q_1+Q_2$
$\therefore \frac{Q_2^{\prime }}{2}+Q_2^{\prime }=20\pi R^2\sigma$
$\frac{3}{2}{Q}_2^{\prime }=20\pi R^2\sigma$
$\therefore \frac{Q_2^{\prime }}{4\pi (2R{)}^2}=\frac{2}{3}\cdot \frac{20\pi R^2\sigma }{16\pi R^2}$
$\therefore \frac{{\sigma }^{\prime }}{\sigma }=\frac{5}{6}$