Show that the field inside a spherical cavity cut in a uniform dielectric medium is uniform and of magnitude
Ecav = 3erEm/(2Er+1),
where er is the relative permittivity of the medium and Em is the uniform field in the dielectric at a point distant from the cavity.
As usual we introduce the "electric displacement field" D:
D = er E (1)
where er is the relative permittivity .
Then, as explained in your textbook,
div D = rho_free/epsilon_0 (2)
where rho_free is the charge density due to free charges, not the induced polarization charges. So, the polarization effects of the medium do not appear in D and D behaves like the electric field would if there were no medium.
To solve problems like this one where you have different regions with constant relative permittivities, you reason as follows. In each of the regions equation (1) is valid, albeit it with a different er for each region. If we introduce the electric potential V defined by E = -nabla V, then it follows from (1) that in each region:
D = -er nabla V (3)
Insert this in (2) to obtain:
nabla^2 V = -rho_free/(er epsilon_0) (4)
Or if there are no free charges, such as in this problem, we have:
nabla^2 V = 0 (5)
So, we just need to solve the Laplace equation. To do that we need to know the boundary conditions. Let's consider the boundary of two dielectrics. It follows from the Maxwell equation nabla times E = 0 (which allows you to write E = nabla V in the first place), via Stokes' Theorem that:
A detailed explanation is given.