A unifrom electric field of strength `E = 100 V//m` is generated inside a ball made of uniform istropic dielectric with permitivity `epsilon = 5.00`. The radius of the ball is `R = 3.0 cm`. Find the maximum surface density of the bound chagres and the total bound charge of one sign.

Explanation

Within the ball the electric field can be resolved into normal and tangential components.
`E_(n) = E cos theta, E_(r ) = E sin theta`
Then, `D_(n) = epsilon epsilon_(0) E cos theta`
and `P_(n) = (epsilon - 1) epsilon_(0) E cos theta`
or, `sigma' = (epsilon - 1) epsilon_(0) E cos theta`
so, `sigma_(max) = (epsilon - 1) epsilon_(0) E`,
and total of one sign,
`q' = int_(0)^(1) (epsilon - 1) epsilon_(0) E cos theta 2 pi R^(2) d ( cos theta) = pi R^(2) epsilon_(0) (epsilon - 1) E`
(Since we are intersted in the total charge of one sign we must intergative `cos theta` from 0 to 1 only).