A simple pendulum of length `l` and bob mass `m` is hanging in front of a large nonconducting sheet having surface charge density `sigma`. If suddenly a charge `+q` is given to the bob `&` it is released from the position shown in figure. Find the maximum angle through which the string is deflected from vertical .
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Explanation

The electric field intensity due to nonconducting sheet is
`E = (sigma)/(2epsilon_0)` … (i)
According to work energy theorem, `W_("net") = Delta k = 0`
Since initial and final velocity of bob is zero,
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Therefore, `W_E + W_g = 0` (ii)
`W_E = q E l sin theta`
`W_g = -mg (l - l cos theta)`
Putting the values in Eq. (ii)
`q E l sin theta - m g l (1 - cos theta) = 0`
or `q E l sin theta = mg l (1 - cos theta)`
or `(q E l)/(m g l)= (1 - cos theta)/(sin theta)` or `(q E)/(m g) = (2 "sin"^2 theta/2)/(2 "sin" theta/2"cos" theta/2 )`
or `(q E)/(m g) = "tan" theta/2` or `theta/2= "tan" ^-1((q E)/(m g))`
or `theta = 2 tan^-1 ((qE)/(mg))`
Putting the value of E from Eq. (i),
`theta=2 tan^(-1)((q sigma)/(2 epsilon_0 mg))`.