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.
$\frac{Q^{'}_1}{4\pi\varepsilon_{o}R}=\frac{Q_{2^{'}}}{4\pi\varepsilon_{o}\left(2R\right)}$
$\Rightarrow Q^{'}_2 = 2 Q^{'}_1$
$\Rightarrow Q^{'}_2 = 2(20 \pi R^2\sigma - Q_2^{'})$
$\Rightarrow 3Q_2^{'} = (40 \pi R^2\sigma)$
$\Rightarrow \,Q _2^{'} = \frac {40\pi R^2\sigma}{3}$
$Q'_2\,=\, \frac{40\pi R^2 \sigma}{3\times 4\pi \times 4R^2} = \frac {5}{6} \sigma $