Solved: Gauss's Law Problem w/Infinite Slab of Charge

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In summary, the electric field outside the slab is 2EA while the electric field inside the slab is 1/2EA.
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[SOLVED] Gauss's Law problem

Homework Statement

An infinite slab of charge parallel to the yz-plane has density p for -b < x < b, 0 otherwise. Find the electric field at all points.

The Attempt at a Solution

I am able to do the electric field outside the slab. But I am off by a factor of 1/2 for the electric field inside. I made a Gaussian cylinder that starts at the axis to x. I'll call the area of the circular faces $A = \pi r^2$. Since there are two of these faces, net flux is $2EA$. The charge enclosed in the surface is $\rho A x$.

$$2 E A = \frac{\rho A x}{\epsilon_0} => E = \frac{\rho x}{2 \epsilon_0}$$

My book says the solution is just $\rho x / \epsilon_0$. So, I know the error probably came from saying the flux was 2EA. But, for the electric field outside the slab, I used 2EA for the flux, and since the charge enclosed in a cylinder running from -x to x, for |x| > b, was 2b, the factors of 2's cancel out. Here they don't. I can't figure out what I'm doing wrong.

I suppose I could just say the flux was one EA because the field at x = 0 is 0. But, for the next problem, I have to find the electric field for a slab whose density is $\rho(x) = \rho_0 e^{-|x/b|}$, and I still have that extra 1/2 factor in there, without the field at x = 0 being 0.

Last edited:
By having your Gaussian surface extend from the origin to a distance x from the origin you lose the symmetry needed to apply Gauss's law. You must have your Gaussian volume centered on the origin--that way you know that the electric field is the same at each end. Have it go from -x to +x.

Ah, that makes sense. Thanks.

1. What is Gauss's Law?

Gauss's Law is a fundamental law in electromagnetism that relates the electric field to the distribution of electric charge in a given space.

2. How do you solve a Gauss's Law problem with an infinite slab of charge?

To solve a Gauss's Law problem with an infinite slab of charge, you must first determine the direction of the electric field at a given point outside the slab. Then, you can use the formula E = σ/2ε0 to calculate the magnitude of the electric field.

3. What is an infinite slab of charge?

An infinite slab of charge is a hypothetical object with an infinite amount of charge distributed evenly throughout its surface. It is often used in Gauss's Law problems as an idealized scenario to simplify calculations.

4. How does the electric field vary in an infinite slab of charge?

In an infinite slab of charge, the electric field is constant and perpendicular to the surface of the slab. This means that the magnitude of the electric field remains the same at all points outside the slab, and its direction is always pointing away from the slab.

5. Can Gauss's Law be used to solve problems with other charge distributions?

Yes, Gauss's Law can be used to solve problems with various charge distributions, including point charges, line charges, and spherical shells. However, the shape and symmetry of the charge distribution may affect the approach and calculations used to solve the problem.

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