Question

A capacitor is made of two circular plates of radius R each, separated by a distance d << R . The capacitor is connected to a constant voltage. A thin conducting disc of radius r<< R and thickness t <<r is placed at the centre of the bottom plate. Find the minimum voltage required to lift the disc if the mass of the disc is m.

• A.
• B.
• C.
• D.

Answer 1

Answer: (C)

Answer 2

:

As shown in figure, the disc is in touch with the botton plate. The electric field on the disc is therefore, charge transferred to the disc

$\mathrm { q } ^ { \prime } = \mathrm { CV } = \left( \frac { \varepsilon _ { 0 } \mathrm {~A} } { \mathrm {~d} } \right) \mathrm { V } = \varepsilon _ { 0 } \frac { \mathrm {~V} } { \mathrm {~d} } \pi \mathrm { r } ^ { 2 }$

Force acting on the disc,

$\mathrm { F } = \varepsilon _ { 0 } \frac { \mathrm { V } ^ { 2 } } { \mathrm {~d} ^ { 2 } } \pi \mathrm { r } ^ { 2 }$

If the disc is to be lifted,

i.e., $\varepsilon _ { 0 } \frac { \mathrm {~V} ^ { 2 } } { \mathrm {~d} ^ { 2 } } \pi \mathrm { r } ^ { 2 } = \mathrm { mg }$ (for minimum V)

$\therefore \mathrm { V } = \sqrt { \frac { \mathrm { mgd } ^ { 2 } } { \pi \varepsilon _ { 0 } \mathrm { r } ^ { 2 } } }$