A small ball of mass `m=21 g` suspended by a insulating thread at a height `h=12 cm` from a large horizontal conducting plane performs small oscillations (figure) . After a charge `q` had been imparted to the ball, the oscillation period changed `eta=2.0` times. Find `q`.

Explanation

In the absence of the charge, the oscillation period of the ball
`T=pi sqrt(l//g)`
when we impart the charge `q` to the ball, it will be influenced by the induced charges on the conducting plane. From the electric image method the electric force on the ball by the plane equals `(q^(2))/(4pivarepsilon_(0)(2h)^(2))` and is directed downward. Thus in this case the effective acceleration of the ball
`g^(')=g+(q^(2))/(16pi epsilon_(0)mh^(2))`
and the corresponding time period
`T^(')=2pi sqrt ((l)/(g^(')))=2pisqrt((l)/(g+(q^(2))/(16pi epsilon_(0)m h^(2))))`
From the condition of the problem
`T=etaT^(')`
So, `T^(2)=eta^(2)T'^(2)` or ` (1)/(g)=eta^(2)((1)/(g+(q^(2))/(16pi epsilon_(0)mh^(2))))`
Thus on solving
`q=4h sqrt(pi epsilon_(0)mg(eta^(2)-1))=2 mu C`