[NOTES/EM-04009] The Method of Images

The method of images makes use of uniqueness theorem. Consider a volume \(V\) containing  a  set of conductors having boundary surfaces \(S_1, S_2, \ldots\) as shown in figure. Let \(S\) be  the surface enclosing the volume \(V\). Suppose we are interested in finding the electric potential \(\phi\) in region \(R\) bounded by the surfaces \(S\) and \(S_1, S_2, \ldots\). The electric potential \(\Phi\) in the region of interest is uniquely determined by  the charge density in \(R\) and the values of potential on the boundary surfaces  \(S, S_1, S_2, \ldots\).

Example 

Suppose we want  to find the potential \(\Phi\) in region \(R\), which is the volume inside the surface \(S\) and outside  the surfaces \(S_1, S_2, S_3\). It is given that

• The region \(R\) has charge density \(\rho\).
• The potential on the boundary \(S\) has a value \(\phi_0\).
• The conductors are held at constant potentials so that potentials on the surfaces \(S_1, S_2, S_3,...\) have values \(\phi_1, \phi_2, \phi_3,...\) respectively.

The uniqueness theorem guarantees that the solution of the above problem  coincides with the solution of a second problem differing from the original problem as  follows.

• The conductors are removed, only their boundaries are marked.
• Auxiliary charges, called image charges are placed outside region \(R\) in such a manner the potentials on the boundary \(S\) and on the boundaries of the conductors, \(S_1, S_2, S_3,...\) remains as in original problem.
• The only constraint on location of the image charges is that they should be placed outside \(R\). This ensures that the charge density in region of interest remains the same as in the original problem. The desired solution is then given by the potential of the charge distribution \(\rho\) and the image charges.