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A capacitor is made of two circular plat...

A capacitor is made of two circular plates of radius R each, separated by a distacne `d lt lt R.` The capacitor is connected to a constant voltage. A thin conducting disc of radius ` r lt lt R`and thickness `t lt lt 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.

Text Solution

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Assuming initially the disc is in touch with the bottom plate, so the entire plate is a equipotential the electric field on the disc, when potential difference V is applied across it, given by ltBrgt `E=(V)/(d)`
Let charge q' is transferred to the disc during the process,
therefore by gauss' theorem
`thereforeq'=-epsi_(0)(V)/(d)pir^(2)`
Since, gauss theorem states that `phi=(q)/(epsi_(0))` or `q=(epsi_(0))/(phi)`
`=epsiEA=(epsi_(0)V)/(d)A`
The force acting on the disc is
`-(V)/(d)xxq'=epsi_(0)(V^(2))/(d^(2))pir^(2)`
If the disc is to be lifted then,
`epsi_(0)(V^(2))/(d^(2))pir^(2)=mgimpliesV=sqrt((mgd^(2))/(piepsi_(0)r^(2)))`
this is the required expression.
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