A positive point charge of 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. [2025]

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 $\frac{1}{4 π ε_{0}} = 9 \times 10^{9} {Nm}^{2} / C^{2}$ (where $ε_{0}$ is the permittivity of free space), which of the following statements is/are correct? [2025]

Smart Achievers · Accepted Answer

(1, 2, 3)

Before grounding,

${(V_{sphere})}_{_{net}} = {(V_{c})}_{q} + {(V_{c})}_{_{ind}}$

$V_{sphere} = \frac{k q}{ℓ} + 0 = \frac{9 \times 10^{9} \times 10^{- 8}}{0.2} = \frac{90}{0.2} = \frac{900}{2} = 450 volt$

So, option (1) is correct.

$\frac{k Q}{ℓ} + \frac{k q_{s}}{R} = V'_{sphere} = 0$

$∴$ $q_{s} = - \frac{R}{ℓ} q = - \frac{0.1}{0.2} \times 10^{- 8} = - 5 \times 10^{- 9} C$

$q_{s} = - 5 \times 10^{- 9} C$

Charge flowing from sphere to ground $= 5 \times 10^{- 9} C$

After grounding is removed, the charge on the sphere $= - 5 \times 10^{- 9} C$

So option (2) and (3) are correct.

Final electrostatic potential of the sphere

${(V_{sphere})}_{final} = \frac{k q}{ℓ} + \frac{k q_{s}}{R}$

$= \frac{9 \times 10^{9} \times 10^{2} \times 10^{- 8}}{30} - \frac{9 \times 10^{9} \times 5 \times 10^{- 9} \times 10^{2}}{10}$

$= \frac{9 \times 1000}{30} - 450 = 300 volt - 450 volt = - 150 volt$

So option (4) is incorrect.

Solution

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

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

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

4 The final electrostatic potential of the sphere is 300 V.

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