Energy of a material with permanent polarization

[Rafael]

In electrostatics, for what I understand the when I have an electric field, the density of the energy stored in it is given by the following formula:
$$W = \frac{1 }{2} E \cdot D$$But when there is some material permantent polarization the above formula fails to work.
Is this correct?
How can the energy be calculated?

 

[Dale]

But when there is some material permantent polarization the above formula fails to work.

Why does it fail?

 

[Rafael]

Why does it fail?

is derived from:

But ρ is the free charge:
$$ρ = \nabla \cdot D$$

In a system with just a electrical polarizated material there isn`t free charge (ρ = 0), so the formula above should predict that the total energy is 0.

 

[Dale]

Hmm, I am not sure where you got that derivation, but it is not the only way. From Maxwell’s macroscopic equations you can easily get $\partial_t W_E=E\cdot \partial_t D$. Then if we assume a linear dispersionless medium $D=\epsilon E$ then we get $\partial_t W_E=\partial_t (\frac{1}{2}E\cdot D)$ so therefore $W_E=\frac{1}{2}E\cdot D$