Question

Two isolated metallic solid spheres of radii $R$ and $2 R$ 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 \varepsilon_0 R}=\frac{Q_2^{\prime}}{4 \pi \varepsilon_0(2 R)} $
$ \therefore Q_2^{\prime}=2 Q_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(2 R)^2}=\frac{2}{3} \cdot \frac{20 \pi R^2 \sigma}{16 \pi R^2} $
$ \therefore \frac{\sigma^{\prime}}{\sigma}=\frac{5}{6}$