An instructive paradox in the method of images

The standard method-of-images problem of a point charge next to an infinite flat conducting surface is examined, and it is shown that the dipole moment has two different values depending on whether it is deduced from the form of the field or from the actual charge distribution. The resolution to the paradox is also presented, and provides yet another instance of subtle flaws that can lie hidden in pedagogically useful idealizations of real physical setups in electromagnetism.