# Displacement Field

1. Jul 24, 2008

### Apteronotus

Hi,

Suppose we apply an electric field E to a dielectric material. It is my understanding that the actual field that is formed as a result of our applied field is given by the displacement field D.

D=$$\epsilon_{0}$$E+P

I know that the field P is due to the polarization of the atoms withing the dielectric.

1. what is the physical meaning behind $$\epsilon_{0}$$E?
2. Specifically, why is the contribution of our applied field E being scaled by the permittivity of free space $$\epsilon_{0}$$?

Last edited: Jul 24, 2008
2. Jul 24, 2008

### Ben Niehoff

The $\epsilon_0$ is there to make the units work out correctly. The polarization field is measured in coulombs per meter squared (perpendicular to the vector, just as current density is measured). The D field uses the same units as P.

This is just a convention; one could just as easily define

$$\vec D_{new} = \vec E + \frac 1{\epsilon_0} \vec P$$

and you could work out the same equations, only with different constants. In fact, in some systems of units, $\epsilon_0 \equiv 1$, so this question becomes moot.

3. Jul 24, 2008

### Andy Resnick

What you wrote is a highly restricted form of a constitutive relation- you wrote it for a linear material which is not moving.

There's lots of equivalent ways to write what you wrote (D = $\epsilon$E = $\epsilon_{r}\epsilon_{0}$E = (1+4$\pi\chi$)$\epsilon_{0}$E =...)

The idea is that the displacement field in regions of matter is composed of the "matter-free" field and an additional contribution from the matter.

4. Jul 24, 2008

### Apteronotus

Fantastic!! Thank you both very much.

On a related item...

I now that the equation is valid only for Linear, Homogenous Isotropic materials.
The material that I'm concerned with is water (which I believe to be isotropic --Encyclopedia Britannica).
And my applied electric field is dynamic (sinusoidal).

Off the top of your heads... is there a great leap between the equation
$$\vec D = \epsilon_0 \vec E + \vec P$$
and one which would apply in my case?

Could you direct me to any resources where the above equation is given for a more general case (ie. not so restrictive)?

5. Jul 25, 2008

### Andy Resnick

Sinusoidal fields are about as basic as they come, and application of the field to a material is independent of the material response to the field.

For water, your constitutive relation is fine, as long as the frequency of oscillation doesn't go to high and the field amplitude isn't too large. In those cases, you need a more general, nonlinear, constitutive relation:

D ~ aE +bE^2 +cE^3+...

If it's moving, there's components from the magnetization that also figure in.