In the "Before" scenario, there is conductive plate in electrostatic equilibrium with uniform surface charge density +η (the plate has some thickness but the width and length are significantly larger). In the "After" scenario, an infinite plane of negative charge with fixed uniform surface charge density -η is placed parallel below the plate. The plate is allowed to re-establish electrostatic equilibrium.
a) Find the electric field value at three points, (1) above the plate, (2) between the plate and plane, (3) below the plane.
b) The surface charge densities of the two sides of the plate.
Only η and ε0 are known values.
Econductor.surface = η/ε0
Eplane = η/2ε0
Eplane = η/2ε0
The Attempt at a Solution
I'm assuming, since the plate has a finite length, is in electrostatic equilibrium, and we aren't given width and length, that we use the electric field for a conductive surface.
I originally thought of the plate having positive on both the top and the bottom, but then I started thinking of the positive charge in the plate being attracted towards the plane leaving the top negative, or would it just be neutral? Honestly, I've spent so long on this problem, I've started to confuse myself.
Currently, assuming that both the top and bottom of the plate are positive, I have for part a:
(1) Etotal = η/ε0 (UP) + η/2ε0 (DOWN) = η/2ε0 (UP)
(2) Etotal = η/ε0 (DOWN) + η/2ε0 (DOWN) = 3η/2ε0 (DOWN)
(3) Etotal = η/ε0 (DOWN) + η/2ε0 (UP) = η/2ε0 (DOWN)
But when I started looking at part b, I started second guessing myself and I have found absolutely nothing anywhere to help clarify my confusion.
Originally, I was thinking that for part b:
ηTop = Etotal.1ε0
ηBottom = Etotal.2ε0
But that didn't get me anywhere, since I was still in terms of η