It's tempting to work with potential fields, as we did earlier, but the potential becomes infinite. So we must work with the actual force field. However, by the symmetry of the plane, the pull to the left has to equal the pull to the right. Only the vertical force matters.
In polar coordinates, the distance is the square root of z2+r2. Take the inverse square, extract the z component, and multiply by r for polar integration, giving the following integrand.
dzr × (z2+r2)-3/2
Integrate with respect to r and get dz (constant has been adjusted) over sqrt(z2+r2). Evaluate at 0 and infinity, and get dz/sqrt(z2), or d. Integrating with respect to θ merely changes the constant, so don't worry about that.
The force field does not depend on z. You can be sitting on the plane, or a million miles above it.
In the real world, there aren't any infinite planes, but there are capacitors, which often consist of parallel plates, whose area is large compared to their distance apart. If you are an electron between the plates, and away from the edges, the foregoing model is reasonably accurate. The electric field inside the capacitor is constant.