I have two earthed metal plates, separated by a distance d with a plane of charg
ID: 1375545 • Letter: I
Question
I have two earthed metal plates, separated by a distance d with a plane of charge density ? placed a distance a from the lower plate. I want to derive expressions for the strength of the electric field in the regions between the plane and the top plate and the plane and the bottom plate.
I'm not sure how to apply Gauss's Law to the given situation. I think the best Gaussian surface to use would be a cylinder(?), and then somehow integrate over the two regions. But this would suggest that the electric field between the charged plane and the bottom surface was independent of the total separation, d of the two earthed plates, which I don't think is correct. I don't yet fully understand Gauss's Law, and need more practice with it.
Please can someone point me in the correct direction, and give me a hint as to what I should be looking to integrate?
With very many thanks,
Explanation / Answer
Your idea of a cylinder/pillbox enclosing a portion of the charged plane looks good to me.
Assume the plates and plane are much, much larger in extent than d. Then the field is parallel with the cylindrical "side" of the pillbox (and no field "leaks" out the side), and the field is indeed independent of d (and constant, as well).
Finally, to solve the problem: 1) Gauss' law applied to the pillbox gives you a relation between the fields above and below the plane. 2) A second relation is obtained by equating the potential at the plane calculated by a) integrating "up" from the bottom plate and b) integrating "down" from the upper plate. Together, the two relations determine the fields uniquely.