Question

A sphere of homogenous linear dielectric material is placed in an otherwise uniform elctric fild E not . find the elctric field inside the sphere?

Answer 1

Element surface

$$dS=R^2sin(\theta)d\theta d\alpha$$

Electric field

$E_{add}=k\frac{\sigma dS}{R^2}=k\frac{Pcos(\theta)R^2sin(\theta)d\theta d\alpha}{R^2}=kPcos(\theta)sin(\theta)d\theta d\alpha$

$E_{||}=dE_{add}cos(\theta)=kPcos^2(\theta)sin(\theta)d\theta d\alpha$

Integration

$E_{add}=kP\int cos^2(\theta)sin(\theta)d\theta\int d\alpha=\frac{1}{3}\frac{P}{\varepsilon_0}$

Then the electric field in center of the sphere

$$\vec{E}=\vec{E_0}+\frac{1}{3}\frac{\vec{P}}{\varepsilon_0}$$