# Sommerfeld radiation conditions for an electromagnetic field

There is some confusion in the definition of Sommerfeld radiation conditions for an electromagnetic field, which are related to the asymptotic behaviour of the field for a distance $r \to \infty$ (from the origin of a spherical coordinates system) in order to assure the unicity of the field itself as a solution of the Maxwell equations.

These are known also as Silver-Muller conditions and they may be presented as

$$\displaystyle \lim_{r \to \infty} r|\mathbf{E}| < q_1$$

$$\displaystyle \lim_{r \to \infty} r|\mathbf{H}| < q_2$$

$$\displaystyle \lim_{r \to \infty} r(\mathbf{E} - \eta \mathbf{H} \times \mathbf{\hat{u}}_r) = 0$$

$$\displaystyle \lim_{r \to \infty} r \left( \mathbf{H} - \frac{\mathbf{\hat{u}}_r \times \mathbf{E}}{\eta} \right) = 0$$

where $\mathbf{\hat{u}}_r$ is the radial unit vector in a spherical coordinate system with center in the origin (where are the sources of the field) and $q_1, q_2$ are two arbitrary real constants.

(The sources of the fields can be considered all "near" the origin, if their distances to the origin is finite, because now $r \to \infty$).

In this form the first conditions state that the asymptotic behaviour of the fields (with any sources) is like $1 / r$ for $r \to \infty$. This is consistent with the field of an electrical (hertzian) dipole, whose far field components are proportional to $1 / r$.

But sometimes the conditions are presented like the following ones:

$$\displaystyle \lim_{r \to \infty} r|\mathbf{E}| = 0$$

$$\displaystyle \lim_{r \to \infty} r|\mathbf{H}| = 0$$

Does it makes sense? It would not be compatible with the field of a hertzian dipole as before. However, if these latter conditions were the true ones, it would be simpler to prove the unicity of the solution of an electromagnetic field: if we considered an infinite spherical volume of radius $r \to \infty$ and its spherical boundary surface $S$, the Poynting vector would always be zero in all the points of this surface.

This question deals with the unicity of the solution.

So, which are the correct conditions? Those with $q_1, q_2$ or those with $0$?

Thank you anyway!

• Where is this "boundary of the space" ? Your second set of conditions seems too restrictive; it effectively precludes any presence of radiation at arbitrary large distances and allows there only static electromagnetic field of static charges. This would mean that in sufficiently distant past and before the charges were static, which is a very special and ungrounded assumption. – Ján Lalinský Apr 9 '14 at 19:36
• I agree with you (and maybe you can understand it from my post): the latter conditions are too restrictive and unreal. With "boundary of the space" I mean the surface $S$ enclosing a volume $V$ which extends to infinity, that is, which tends to be include all the free space. – BowPark Apr 10 '14 at 8:32
• But with the former (and more realistic) conditions it seems impossible up to now to demonstrate the unicity of the solution of the Maxwell equation! (as showed in the link in the post, that is this link). – BowPark Apr 10 '14 at 8:41