Electric displacement of a uniformly polarized sphere

[tanzhongm]

Homework Statement

Suppose we have a uniformly polarized sphere of radius R. The polarization vector is $$\mathbf{P}$$.

Goal: To find the electric field everywhere.

Homework Equations

I know the actual solution already. It should be $$\mathbf{E} = \frac{-1}{3 \epsilon_0} \mathbf{P}$$

The Attempt at a Solution

However, if we use Gauss' Law $$\oint \mathbf{D} \cdot d\mathbf{A} = Q_{enc,free}$$.

We get $$Q_{enc,free}=0$$

Hence, $$\mathbf{D}=0$$ and $$\epsilon_0 \mathbf{E}=\mathbf{P}$$

Therefore, $$\mathbf{E} = \frac{\mathbf{P}}{\epsilon_0}$$.

But these two values are different.

 

[kloptok]

You can't use Gauss' law for the D-field if you don't have a symmetrical problem.In other words, since the polarisation is not spherically symmetric Gauss' law for the D-field with the free charge on the right hand side is not valid. Remember that curl of D is not zero, but equal to minus the curl of P if i remember correctly.

To find the E-field everywhere you can set up a boundary value problem for the electric potential with appropriate boundary conditions at the boundary of the sphere.