Electric field below conducting plane

Question

In Griffiths introduction to Electrodynamics, the classic image problem is presented: There is a charge $q$, above a grounded conducting plane. Griffith says that the electric field below the plane is zero. My question is why is the electric field below the plane zero? Does not the induced charge on the plane create an electric field?

Answer 1

Does not the induced charge on the plane create an electric field?

Yes, of course it does. But don't forget that there is also the electric field of the point charge $q$ above the plane to consider. Remarkably, since this is the electrostatic case, it must be that the electric field of the induced charge precisely cancels the electric field of the point charge $q$ in the region below the plane.

This must be the case since (1) there is no charge below the plane and (2) the electric potential must therefore satisfy Laplace's equation in that region.

Assuming the potential is set to zero at $\pm \infty$, the fact that the grounded conducting plane is, by definition, at zero potential implies that the potential is zero in the region below the plane.

If this were not the case, there would necessarily be a maximum or minimum of the potential in the region below the plane but then the potential would not satisfy Laplace's equation. From Wolfram Mathworld:

A function $\psi$ which satisfies Laplace's equation is said to be harmonic. A solution to Laplace's equation has the property that the average value over a spherical surface is equal to the value at the center of the sphere (Gauss's harmonic function theorem). Solutions have no local maxima or minima

(emphasis is mine)

Since the potential is constant in the region below the plane, the electric field is necessarily zero.

Answer 2

Think of it this way: The potential distribution below the surface obeys the Laplace equation, and the boundary conditions associated with it is that the potential should be zero at the conductor's boundary. Now a trivial solution for this boundary value problem is an identically zero potential everywhere below the conductor. And by the uniqueness theorem of the Laplace equation, it is also the only solution. Hence, the potential, and thus the electric field beneath the conductor, are identical to zero.

About your statement on the induced charge; it is true that some charge gets induced on the conductor. However, the potential of the point charge above the surface cancels the potential created by the induced charges, leaving a zero potential in that region.