Question 197: Electrostats & Capacitance

Question

A positive point charge of $10^{-8}$ C is kept at a distance of 20 cm from the center of a neutral conducting sphere of radius 10 cm. The sphere is then grounded and the charge on the sphere is measured. The grounding is then removed and subsequently the point charge is moved by a distance of 10 cm further away from the center of the sphere along the radial direction. Taking $\frac{1}{4\pi\epsilon_0} = 9 	imes 10^9$ Nm$^2$/C$^2$ (where $\epsilon_0$ is the permittivity of free space), which of the following statements is/are correct:

Accepted Answer

Before grounding
$$ \begin{aligned} & \mathrm{V}_{	ext {sphere }}=\left(\mathrm{V}_{\mathrm{C}}ight)_{	ext {net }}=\left(\mathrm{V}_{\mathrm{C}}ight)_{\mathrm{q}}+\left(\mathrm{V}_{\mathrm{C}}ight)_{	ext {ind }} \ & \mathrm{V}_{	ext {sphere }}=\frac{\mathrm{kq}}{\ell}+0=\frac{9 	imes 10^9 	imes 10^{-8}}{0.2}=\frac{90}{0.2}=\frac{900}{2}=450 	ext { volt } \end{aligned} $$
After grounding
$$ \begin{aligned} & \frac{\mathrm{kQ}}{\ell}+\frac{\mathrm{kq}_{\mathrm{s}}}{\mathrm{R}}=\mathrm{V}_{	ext {sphere }}^{\prime}=0 \ & \mathrm{q}_{\mathrm{s}}=-\frac{\mathrm{R}}{\ell} \mathrm{q}=-\frac{1}{2} 	imes 10^{-8}=-5 	imes 10^{-9} \ & \mathrm{q}_{\mathrm{s}}=-5 	imes 10^{-9} 	ext { Coulomb } \end{aligned} $$
Charge flower from sphere to ground $=5 	imes 10^{-9}$ Coulomb
After grounding is removed
$$ \begin{aligned} & \left(\mathrm{V}_{	ext {sphere }}ight)_{	ext {final }}=\frac{\mathrm{kq}}{\ell^{\prime}}+\frac{\mathrm{kq}_{\mathrm{s}}}{\mathrm{R}} \ & =\frac{9 	imes 10^9 	imes 10^2 	imes 10^{-8}}{30}-\frac{9 	imes 10^9 	imes 5 	imes 10^{-9} 	imes 10^2}{10} \ & =\frac{9 	imes 1000}{30}-450=300 \mathrm{volt}-450 \mathrm{volt}=-150 \mathrm{volt} \end{aligned} $$

Answer

Before the grounding, the electrostatic potential of the sphere is 450 V.

Answer

Charge flowing from the sphere to the ground because of grounding is $5 	imes 10^{-9}$ C.

Answer

After the grounding is removed, the charge on the sphere is $-5 	imes 10^{-9}$ C.

Answer

The final electrostatic potential of the sphere is 300 V.

Electrostats & Capacitance

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