Parallel Plates: Part 1

Take a look at the plane $x=0$ . Clearly the the voltage everywhere on this plane is zero. We can imagine replacing the plane $x=0$ by another grounded plate, and the problem will be equivalent!

Let's do that. Consider the problem of a charge $Q=-92 \space \mu C$ located at $(L/2,0,0)$ surrounded by two parallel, grounded, conducting plates located at $x=0$ and $x=L$ . By symmetry, the induced charge on these plates should be the same. Furthermore, the combined induced charge on both of the plates should be $92 \space \mu C$ . Now we can see that the induced charge on the plate at $x=L$ should be $\frac{92}{2} \space \mu C = 46 \space \mu C$ . But remember that this problem is equivalent to our original problem! Therefore the answer is $\boxed{46 \space \mu C}$ .