A non-conducting sphere of radius R has a spherical cavity of radius `R//2` as shown. The solid part of the sphere has a uniform volume charge density `rho`. Find the magnitude and direction of electric field at point (a) O and (b) A.

Explanation

For a non-conducting sphere of Radius `R` having volume charge density `rho`.
(i) `r lt R ("inside") ,E = (rho r)/(3 in_(0))`
(ii) `r gt R ("outside") , E = (rho R^(3))/(3 in_(0) r^(2))`
(iii) ` r = R ("surface") , E = (rho R)/(3 in_(0))`
At `O`:
`E_(1) = 0 , E_(2) = (rhoR//2)/(3 in_(0)) = (rho R)/(6 in_(0))` , towards left
`E_(0) = (rho R)/(6 in_(0))` , towards left
(b) At `A` :
`E_(1) = (rho R)/(3 in_(0))` , towards right
`E_(2) = (rho (R//2)^(3))/(3 in_(0) (3R//2)^(2)) = (rho R)/(54 in_(0))` , towards left
`E_(A) = E_(1) - E_(2) = (rho R)/(in_(0)) [(1)/(3) - (1)/(54)]`
`= (17 rho R)/(54 in_(0))` , towards right