Let q be the point charge of this point. The image of this charge with respect to the grounded sphere is shown in red. It has a charge of q′=−qR/p and lies on a line connecting the center of the sphere and the inner charge at vector position . It can be seen that the potential at a point specified by radius vector due to both charges alone is given by the sum of the potentials: Multiplying through on the rightmost expression yields: and it can be seen that on the surface of the sphere (i.e. when r=R), the potential vanishes. The potential inside the sphere is thus given by the above expression for the potential of the two charges. This potential will NOT be valid outside the sphere, since the image charge does not actually exist, but is rather "standing in" for the surface charge densities induced on the sphere by the inner charge at . The potential outside the grounded sphere will be determined only by the distribution of charge outside the sphere and will be independent of the charge distribution inside the sphere. If we assume for simplicity (without loss of generality) that the inner charge lies on the z-axis, then the induced charge density will be simply a function of the polar angle θ and is given by: The total charge on the sphere may be found by integrating over all angles: Note that the reciprocal problem is also solved by this method. If we have a charge q at vector position outside of a grounded sphere of radius R, the potential outside of the sphere is given by the sum of the potentials of the charge and its image charge inside the sphere. Just as in the first case, the image charge will have charge −qR/p and will be located at vector position . The potential inside the sphere will be dependent only upon the true charge distribution inside the sphere. Unlike the first case the integral will be of value −qR/p. Electric dipole moments The image of an electric point dipole is a bit more complicated. If the dipole is pictured as two large charges separated by a small distance, then the image of the dipole will not only have the charges modified by the above procedure, but the distance between them will be modified as well. Following the above procedure, it is found