A sphere of radius R has a uniform volume density `rho`. A spherical cavity of radius b, whose center lies at `veca`, is removed from the sphere.
i. Find the electric field at any point inside the spherical cavity.
ii. Find the electric field outside the cavity
(a) at points inside the large sphere but outside the cavity and
(b) at points outside the large sphere.

Explanation

The electric field within the cavity or outside is the
superposition of the electric field due to the original
uncut sphere, plus the electric field due to a sphere of
the size of the cavity but with a uniform negative charge
density. The effective charge distribution is composed
of a uniformly charged sphere of radius R, charge
density `rho`, superposed on it is a charge density - `rho` filling
the cavity. An electric field `vecE_1` is caused by the charge
distribution `+rho` at a point `vecr` inside the spherical cavity.

`vecE_1 = (rhor)/(3epsilon_0) r^2 = (rhovecr)/(3epsilon_0)`
where `hatr` is a unit vector in radial direction. Similarly, the electric field `vecE_2` formed by the charge density `-rho`
inside the cavity is
`vecE_2 = (rho(-vecs))/(3epsilon_0)`
Here, `vecs` is the radius vector from the cavity centre to
the point P. From vector triangle
`vecr = veca + vecs or vecs = vecr - veca`
`:. vecE_2 = (-rho(vecr-veca))/(3epsilon_0)`
The resultant electric field inside the cavity is, therefore,
given by the superposition of `vecE_1 and vecE_2`. So
`vecE= vecE_1 + vecE_2 = (rhovecr)/(3epsilon_0) + [(-rho(vecr-veca))/(3epsilon_0)] = + (rhoveca)/(3epsilon_0)` = constant.
`:. vecE = (rhoveca)/(3epsilon_0)`
ii. (a) Electric field at points inside the large sphere but
outside the cavity :
`vecE_1 = (rhovecr)/(3epsilon_0)`
and `vecE_2 = 1/(4piepsilon_0) (q(-vecs))/(s^3) = -((4/3pirhob^3)(vecr - veca))/(4epsilon_0|(vecr -veca)|)`
The resultant electric field is
`vecE = vecE_1 + vecE_2 = rho/(3epsilon_0)[vecr - (b/(vecr -veca))^3 (vecr-veca)]`
(b) Electric field at points outside the large sphere,
`vecE_1 = Q_t otal)/(4piepsilon_0r^3) vecr = (4/3piR^3rho)/(4piepsilon_0r^3) vecr = (R^3rho)/(3epsilon_0r^3) vecr `
`vecE_2 = Q_(t otal)/(4piepsilon_0s^3) vecs = (-(4/3pib^3rho))/(4piepsilon_0(|vecr -veca|)^3) (vecr - veca)`
`= (-rhob^3)/(3epsilon_0(|vecr-veca|)^3 (vecr-veca)`
The resultant electric field is
`vecE = vecE_1 + vecE_2 = (rho)/(3epsilon_0) [(R/r)^3 vecr - (b/(vecr-veca))^3 (vecr-veca)]`
`:. E (vecr) = {((rhoveca)/(3epsilon_0)), (rho/(3epsilon_0)[vecr-(b/|vecr-veca|)^3 (vecr-veca)]), (rho/(3epsilon_0)[(R/r)^3vecr - (b/|vecr-veca|)^3 (vecr-veca)]`:}`
Electric field inside the cavity
Electric field outside the cavity but inside the large cavity
electric field outside the large sphere .