# Gauss' Law between infinite plates

• flintbox
In summary, the problem involves calculating the electric field and potential difference between two infinite plates with a uniform charge density. The solution can be found by applying Gauss' Law and taking advantage of symmetry. The direction of the electric field can be determined by considering different points between the plates and using the symmetry of the problem. Care must be taken with the Gaussian surfaces, as they must enclose the charge between the plates. The solution can also be found by solving Poisson's equation.

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

Doc Al said:
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.

flintbox said:
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).

flintbox said:
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?

nrqed said:
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.

Doc Al said:
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

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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flintbox said:
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.

flintbox said:
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 ...