Gauss' Law between infinite plates

[flintbox]

Homework Statement

The volume between two infinite plates located at x=L and x=-L respectively is filled with a uniform charge density ##\rho##. Calculate the electric field in the regions above, between and below the plates. Calculate the potential difference between the points x=-L and x=L.

Homework Equations

& attempt[/B]
I want to apply Gauss' Law, but I don't know how to. To me it seems that inside the plates, the charge enclosed is that of any surface, but I wouldn't know the flux of the electric field. I tried searching literature, but they all consider charged plates, whereas here, the plates are just boundaries.

 

[Doc Al]

Hint: Take advantage of symmetry. Imagine a Gaussian surface in the shape of a cube centered at x = 0.

 

[flintbox]

Hint: Take advantage of symmetry. Imagine a Gaussian surface in the shape of a cube centered at x = 0.

But I am confused about the direction of the electric field inside the plates, since there is a charge density everywhere.

 

[nrqed]

But I am confused about the direction of the electric field inside the plates, since there is a charge density everywhere.

Other hint: consider any point exactly midway between the two plates, what can you say about the E field there?

Now, consider another point between the two plates but not exactly midway this time. You should be able to tell what the direction of the E field is, there. Using only the symmetry of the problem (consider the plates to be infinite).

 

[Doc Al]

But I am confused about the direction of the electric field inside the plates, since there is a charge density everywhere.

Take ##\rho## as positive. All that matters in the charge within your Gaussian surface. If the charge enclosed is positive, which way must the field point?

 

[flintbox]

Other hint: consider any point exactly midway between the two plates, what can you say about the E field there?

Now, consider another point between the two plates but not exactly midway this time. You should be able to tell what the direction of the E field is, there. Using only the symmetry of the problem (consider the plates to be infinite).

The E field just above the center points upward and the E field below downwards. Thank you! I think I can do it now.

 

[flintbox]

Take ##\rho## as positive. All that matters in the charge within your Gaussian surface. If the charge enclosed is positive, which way must the field point?

Then the field points outwards! Thanks

 

[rude man]

Careful with the Gaussian surfaces! In addition to the volume charges there are also induced surface charges!

(This problem is also easily solved by solving Poisson's equation.)

 

[flintbox]

I've used Gauss to determine the Electric field inside to be ##2\pi \rho x## (CGS units), but what about outside? I don't know how to apply Gauss since there is no charge enclosed.

 

[Doc Al]

I don't know how to apply Gauss since there is no charge enclosed.

If a Gaussian surface extends beyond the plates, then it encloses the charge between them.

 

[rude man]

I've used Gauss to determine the Electric field inside to be $2\pi \rho x$ (CGS units), but what about outside? I don't know how to apply Gauss since there is no charge enclosed.

I can't read your post and I'd have to convert to SI.
Run a surface from inside one of the plates to any outside region. Remember what I said about surface charges ...