A particle of mass `1 kg` and charge `1//3 mu C` is projected toward a nonconducing fixed spherical shell of radius `r = 1 mm` having the same charge uniformly distributed on its surface. Find th minimum initial velocity of projection requires if the particle just grazes the shell.
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Explanation

Applying conservation principle of angular momentum, we get
`m v_0 d = m v r`
or `v = (v_0 d)/( r) = (v_0)/(2)(because d = r/(2) = 0.5 mm)`
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Applying mechanical energy conservation principle, we get
`U_i + T_ i = U_f + T_f`
or `0 + (1)/(2) m v_0^2 = (q^2)/(4 pi epsilon_0 r) + (1)/(2) m v^2`
or `(1)/(2) m v_0^2 = (q^2)/(4 pi epsilon_0 r) + (1)/(2) m v^2`
or `(1)/(2) m v_0^2 = (q^2)/(4 pi epsilon_0 r) + (1)/(2) m ((v_0)/(2))^2`
or `(1)/(2) m v_0^2 - (m v_0^2)/(8) = (q^2)/(4 pi epsilon_0 r)`
or `(1)/(2) mv _0^2 xx (3)/(4) = (q^2)/(4 pi epsilon_0 r)`
or `(1)/(2) mv_0^2 = (q^2)/(3 pi epsilon_0 r)`
or `v_0^2 = (8 q^2)/(3 xx 4 pi epsilon _0 m r)= (8)/(3)`
or `v_0 = sqrt ((8)/(3)) = 2 sqrt((2)/(3)) ms^-1`.