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Question

A thin metallic spherical shell of radius $$R$$ carries a charge $$Q$$ on its surface. A point charge $$Q/2$$ is placed at the centre $$C$$ and another charge $$+2Q$$ is placed outside the shell at $$A$$ at a distance $$x$$ from the centre as shown in the figure.
(i) Find the electric flux through the shell.
State the law used.
(ii) Find the force on the charges at the centre $$C$$ of the shell and at the point $$A$$.

Solution
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(i) Electric flux through a Gaussian surface, $$\varphi = \dfrac {Total\ enclosed\ charge}{\epsilon_{0}}$$
Net charge enclosed inside the shell, $$q = 0$$
$$\therefore$$ Electric flux through the shell $$\dfrac {q}{\epsilon_{0}} - 0$$

(ii) Gauss's Law: Electric flux through a Gaussian surface is $$\dfrac {1}{\epsilon_{0}}$$ times the net charge enclosed within it.
Mathematically, $$\oint \vec {E} . \vec {ds} = \dfrac {1}{\epsilon_{0}} \times q$$
(ii) We know that electric field or net charge inside the spherical conducting shell is zero. Hence, the force on charge $$Q/2$$ is zero.
Force on charge at $$A, F_{A} = \dfrac {1}{4\pi \epsilon_{0}} \dfrac {2Q\left (Q + \dfrac {Q}{2}\right )}{x^{2}} = \dfrac {1}{4\pi \epsilon_{0}} \dfrac {3Q^{2}}{x^{2}}$$.

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