Proving identical electric fields

Question

This is a discussion question from the book University Physics with Modern Physics: 14th Edition.

CHAPTER 23: ELECTRIC POTENTIAL, DISCUSSION QUESTION #22

A positive point charge is placed near a "very large" conducting plane . A professor of physics asserted that the field caused by this configuration is the same as would be obtained by removing the plane and placing a negative point charge of equal magnitude in the mirror-image position behind the initial position of the plane. Is this correct? Why or why not?

I found that the plane (where the conductor is located) is an equipotential surface of the field generated by the two charges that are equal in magnitude and opposite in sign, since it is halfway between the charges and therefore perpendicular to the field lines (the y- and z- components of the electric field cancel out at points on the plane). However, I'm stuck there and I can't determine whether the fields generated by the two scenarios are identical. Also, can you explain it to me using only the concepts of potential and without referring to "method of images" or any advanced concepts?

Answer 1

In fact, you don't need to show that the fields are identical. It is guaranteed by uniqueness of the solution of the boundary value problem. If you write down a solution of the Poisson equation that satisfies the boundary conditions — in this case, the potential on the conducting surface and the surface at infinity — and has the same charge distribution in the volume of interest, then you have found the only solution.