Question

Two metal spheres, one of radius R and the other of radius 2R both have same surface charge density $\sigma$. They are brought in contact and separated. The surface charge density of bigger sphere becomes

• A. $2\sigma$
• B. $\frac{5}{3}\sigma$
• C. $\frac{5}{6}\sigma$
• D. $\frac{4}{3}\sigma$

Answer 1

Answer: (C)

$Q_1=4\pi R^2\sigma$, $Q_2=16\pi R^2\sigma$
$Q_1+Q_2=20\pi R^2\sigma$
After contact, potential is same.
$\frac{Q_1^{{}^{\prime }}}{4\pi {\epsilon }_{o}R}=\frac{Q_{2^{{}^{\prime }}}}{4\pi {\epsilon }_o\left(2R\right)}$
$\Rightarrow Q_2^{{}^{\prime }}=2Q_1^{{}^{\prime }}$
$\Rightarrow Q_2^{{}^{\prime }}=2(20\pi R^2\sigma -Q_2^{{}^{\prime }})$
$\Rightarrow 3Q_2^{{}^{\prime }}=(40\pi R^2\sigma )$
$\Rightarrow Q_2^{{}^{\prime }}=\frac{40\pi R^2\sigma }{3}$
$Q_2^{\prime }=\frac{40\pi R^2\sigma }{3\times 4\pi \times 4R^2}=\frac{5}{6}\sigma$