A square plate of side-length L, charged with uniform surface charge density η.
It is centred at x = y = z = 0 and orientated in the z = 0 plane.
The task is to determine the z-component of the electric field at the point r(x, y, z) = r(x, 0, d), offset along the x-axis and a height d above the plane of the plate.
A strategy is to break the plate into a set of thin strips, with the i th strip having a thickness ∆xi and position xi along the x-axis (see figure).
The electric fields of the strips can then be individually determined, and summed up via an integral to obtain the total electric field.
(a) Draw two sketches of the system in the y = 0 plane (side view), the first showing the electric field lines from the plate, and the second showing the total electric field vector at point r as well as its z-component.
(b) Using your field vector sketch, find the electric field in the z direction (the z component)
In the y = 0 plane bisecting the plate, the electric field magnitude of strip i is
E(i) = ηLΔx(i)/(4πεr(i)√(r(i)2+L2/4))
where η=surface charge density
r(i) = distance from the centre of the strip to the point r
Δx(i) = distance of strip along x-axis
The Attempt at a Solution
I have drawn multiple diagrams but I'm not confident in my calculations. I'm pretty sure that the field lines from the plate act perpendicular to the surface. However, from this I'm not sure how to get to the z component of the electric field...