A point charge, Q, is located @ (0,0,d) above infinite conducting plane that lies in xy plane and is maintained at ground potential. Find:
a.) surface charge density as a function of x and y on conducting plane, and
b.) total charge induced on conducting plane.
The Attempt at a Solution
Well, I know that the E field is upward directed in the z-direction because the E-field is always perpindicular to a conducting surface. The answer is in the back of the book, and it is:
a.) ρs = (-2Qd)/(4*pi*[ρ2 + d2](3/2)
b.) Qsurface = -Q
In part a, I do not see how the author got [ρ2 + d2](3/2) in the denominator of this formula. This equation shows up in the E-field equation for a uniform disc of charge, and how can we assume this?! If I were to follow this implementation, assuming I can follow the assumptions for an uniform disc of charge, r = daz, and r' = xax + yay. (r - r') = daz - xax - yay. The magnitude of this term to the third power (still following equations for uniform disc of charge) = sqrt(d2 + x2 + y2) = sqrt(d2 + ρ2).
I get stuck here. If someone could let me know if I should continue to follow this method, and if so, how to continue. If I should try another way of thinking, please offer some tips to get me going in the right direction. Thank you for all help, it's most appreciated!