# Gauss' Law between infinite plates

## 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

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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.

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

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
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Gold Member
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
Mentor
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?

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.

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
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Gold Member
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.)

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.

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Doc Al
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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
Homework Helper
Gold Member
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 ...