A positive charge Q is uniformly distributed throughout the volume of a dielectric sphere of radius R. A point mass having charge +q and mass m is fired toward the center of the sphere with velocity v from a point at distance `r (r gt R)` from the center of the sphere. Find the minimum velocity v so that it can penetrate `(R//2)` distance of the sphere. Neglect any resistance other than electric interaction. Charge on the small mass remains constant throughout the motion.

Explanation

Using conservation of mechanical energy `Delta K + Delta U = 0`,
`(0 - (1)/(2) m v^2) + q(V_f - V_i) = 0`
or `(1)/(2)m v^2 = q (V_f - V_i)` ….(i)
`V_i = Q/(4 pi epsilon_0 r) and V_f = q/(8 pi epsilon_0 R)[3 - (r^2)/(R^2)]`
where `r = (R)/(2) , here, V_f = (11 Q)/(32 pi epsilon_0 R)`
Putting the values of `V_i` and `V_f` in Eq. (i)
`(1)/(2) m v^2 = (11 q Q)/(32 pi epsilon_0 R) - (q Q)/(4 pi epsilon_0 r)`
or `m v^2 = (11q Q)/(16 pi epsilon_0 R) - (q Q)/(2 pi epsilon_0 r)`
=`(q Q)/(2 pi epsilon_0) [(11)/(8 R) - (1)/(r)]`
or `v^2 = (q Q)/(2 m pi epsilon_0 R)[ (11)/(8) - (R)/(r)]`
Hence `V = sqrt((q Q)/(2 m pi epsilon_0 R)[(11)/(8) - (R)/(r)])`.