Gauss's Law in Matter: What is It?

[Apteronotus]

Hi,

The Poisson equation (or Gauss Law) in a vacuum is given by
\nabla^2\phi=-\frac{\rho}{\epsilon_0}
where \rho \mbox{ and } \epsilon_0 are the charge density and vacuum permittivity or (electric constant of space).

My question is what is the Gauss's Law in a dielectric material? Do we simply replace the vacuum permittivity by the permittivity of the material?
ie.
\nabla^2\phi=\frac{\rho}{\epsilon_r}
where above \epsilon_r is the relative permittivity (or permittivity of the dielectric material).


Thanks,

 

[conquest]

Hi,

I would check some books for follow up.

How is it Gauss's law in vacuum? What is then carrying the charge?

Beyond that I think it matters whether you're talking linear dielectric or non-linear dielectric.

For a linear dielectric if you state it using the fields use the displacement field instead of the electric field and then it accounts for the free charge with the new permittivity.

 

[Apteronotus]

From what I've read, Gauss's law is usually given in its integral form, saying that "the amount of electric flux through a surface is proportional to the amount of charge within that closed surface".
Then the differential form is derived using divergence theorem relating the divergence of the electric field to the surface charge density \rho.

So in short everything seems to be in a vacuum. But how do these equations compare if say we have a linear, homogeneous, isotropic material?

 

[pabloenigma]

Poissons equation is bit different from Gauss Law(In addition to it,Poissons eqn requires that the field is derivable from a scalar potential)

Coming to the point,In matter, Gauss Law is expressed as the surface Integral of the the displacement vecotr being equal to the the total enclosed free charge.
If the matter is linear and Homogeneous,it can be written in form of Poissons equation with source Free charge divided by epsilon.
The equation you proposed is dimensionally incorrect,as you can see.

 

[jtbell]

http://farside.ph.utexas.edu/teaching/em/lectures/node69.html

In particular, equations 807 and 809 are the differential and integral forms of Gauss's law in a dielectric medium.

 

[Apteronotus]

Thank you all for your responses. jtbell the resource you linked to is very informative, thank you.

Following equation 813, would it be correct to deduce
\nabla^2\phi=-\frac{\rho_{f}}{\epsilon \epsilon_0}

 

[pabloenigma]

Yes,for Linear and Homogeneous dielectrics

 

[Meir Achuz]

and isotropic

 

[Dickfore]

In a dielectric material, Gauss's Law holds for the electric displacement field \mathbf{D}:
<br /> \nabla \cdot \mathbf{D} = \rho_f<br />
where \rho_f is the free charge density.

The dielectric is modeled by a relative dielectric constant, which meaning (for an isotropic medium) is:
<br /> \mathbf{D} = \varepsilon_0 \, \varepsilon_r \, \mathbf{E}<br />
The electric field, being irrotational, is still derivable from the electrostatic potential:
<br /> \mathbf{E} = -\nabla \, \phi<br />