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.

Answer

` [ ( 2K Qq) /( mR) ( r-R)/r + 3/8 ) ]^(1/2)`

Explanation

`V_A = (Kq)/r`
`V_("inside") = (KQ)/(2K^3) (3 R^2 -r^2)`

`V_B = (KQ)/(2R^3) [ 3 R^2 - (R/2)^2] rArr V-B = (11)/8 (KQ)/R`
From conservation of energy , Gain of `PE = Loss ` of ` KE`
`q[V_B - V-A ] = 1/2 mv^2`
`rArr q [ (11)/ 8 (KQ)/R - (Kq)/r]= 1/2 mv^2`
`:. v = sqrt ((2 KQq)/m ((11)/(8R) - 1/r))= sqrt ((2 KQq)/(mR) (r-R)/r + 3/8)`.