Electric field inside a polarized sphere

[Fabio010]

Homework Statement

A sphere of radius R carries a polarization
$\vec{P}$= k$\vec{r}$,

where k is a constant and $\vec{r}$ is the vector from the center.


Find the field inside and outside the sphere.


In solution, the field outside sphere is 0.

I interpreted that as the field produced by the polarization charges.

Their sum is 0. So, the outside field is zero.

My problem is inside sphere.

Can i consider always the electric field inside a sphere as -$\frac{1}{3εo}$*$\vec{P}$
??


Because the formula -$\frac{1}{3εo}$*$\vec{P}$ is obtained by a uniformly polarized sphere.

In the exercise $\vec{P}$ is not uniform.

 

[rude man]

Since there is no free charge anywhere inside the sphere, I would think the E field, by Gauss, would be zero everywhere inside the sphere.

 

[TSny]

Can i consider always the electric field inside a sphere as -$\frac{1}{3εo}$*$\vec{P}$
??

Not sure where you're getting that expression. The polarization produces a bound charge density inside the sphere: $\rho_b = -\vec{\nabla}\cdot \vec{P}$

From the bound charge you can get the field using Gauss' law.

 

[Fabio010]

So you are telling me that $\vec{E}$(A) = $\frac{∫-∇.\vec{P}dv}{εo}$

 

[TSny]

So you are telling me that $\vec{E}$(A) = $\frac{∫-∇.\vec{P}dv}{εo}$

Yes (although I'm not real clear on your notation in this equation). See what you get for the bound charge density as a function of $r$ and then use it in Gauss' law to find the magnitude of the field inside, $E(r)$, as a function of $r$.