Question concerning gauss's law

[mmblind10]

Homework Statement

Find the Electric Field everywhere in space for an infinitely long slab in the x-y plane with thickness (t) and uniform volume charge density (rho).

Find the electric field if the sheet becomes a conductor.

The Attempt at a Solution

I know the E field above and below the sheet only corresponds to z but stays constant along z. I also believe that E = (rho)(t)/[2(epsilon)] for both above and below the sheet. However I am confused as to what to do for inside the slab. I believe it involves an integral from -t/2 to t/2 but i am not quite sure what I would be integrating. Any help would be appreciated.

For the second part I know the E field inside the slab would be zero. So using Gauss's law I got that E = (rho)(t)/(epsilon) for above and below the slab. Any feedback would be a great help. Thanks.

 

[anathema]

Hello,

For the first part of the problem, you are correct in your understanding that the electric field above and below the slab will only have a component in the z-direction and will be constant along z. However, for the electric field inside the slab, you will need to take into account the contribution from each individual charge element within the slab. This can be done by setting up an integral over the entire thickness of the slab, similar to what you mentioned. The integral would be over the variable z, from -t/2 to t/2, and the integrand would be the electric field due to a small charge element at a distance z from the center of the slab. This can be calculated using Coulomb's law and then integrated to get the total electric field inside the slab.

For the second part of the problem, you are correct in using Gauss's law to determine the electric field inside the conductor. Since the conductor is now a perfect conductor, the electric field inside will be zero. However, outside the conductor, the electric field will still follow the same formula as before, E = (rho)(t)/(2(epsilon)).

I hope this helps clarify the problem for you. Let me know if you have any further questions. Good luck with your solution!

 

[Moetasim]

Your understanding of the electric field above and below the sheet is correct. To find the electric field inside the slab, you will need to use Gauss's law and consider a Gaussian surface that encloses a portion of the slab. The charge enclosed by this surface will be the volume charge density (rho) multiplied by the volume of the portion of the slab enclosed by the surface. The electric field inside the slab will be constant and perpendicular to the surface, making the integral of the electric field over the surface equal to the electric field multiplied by the surface area. This will give you the equation E = (rho)(t)/[2(epsilon)] for the electric field inside the slab.

For the second part, you are correct that the electric field inside the slab will be zero since it is now a conductor. This is because in a conductor, the charges will rearrange themselves in such a way that the electric field inside the conductor is zero. Using Gauss's law, you can find the electric field above and below the conductor to be E = (rho)(t)/(epsilon), as you have correctly calculated. Good job!