MAGNETIC FIELDS IN MATTER — Edward Purcell & David Morin
Physics — E&M · field-center-polarized-sphere
Problem
If you don't trust the E=−P/3ϵ0 result we obtained in Section 10.9 for the field inside a uniformly polarized sphere, you will find it more believable if you check it in a special case. By direct integration of the contributions from the σ=Pcosθ surface charge density, show that the field at the center is directed downward (assuming P points upward) with magnitude P/3ϵ0.