# Displacement Field

## Main Question or Discussion Point

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 $$\epsilon_{0}$$?

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Ben Niehoff
Gold Member
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.

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.

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)?

Andy Resnick