The boundary conditions are that the fields are zero at infinity, assuming that the generating charges and currents are localized to some region in space. According to Helmholtz's theorem, the above field equations, plus the boundary conditions, are sufficient to

The field equations can be integrated to give:

Here, is a closed surface enclosing a volume . Also, is a closed loop, and is some surface attached to this loop. The field equations (346)-(349) can be deduced from Eqs. (350)-(353) using Gauss' theorem and Stokes' theorem. Equation (350) is called Gauss' law, and says that the flux of the electric field out of a closed surface is proportional to the enclosed electric charge. Equation (352) has no particular name, and says that there is no such things as a magnetic monopole. Equation (353) is called Ampère's circuital law, and says that the line integral of the magnetic field around any closed loop is proportional to the flux of the current through the loop. Finally. Eqs. (351) and (353) are incomplete: each acquires an extra term on the right-hand side in time-dependent situations.

The field equation (347) is automatically satisfied if we write

Here, is the electric scalar potential, and is the magnetic vector potential. The electric field is clearly unchanged if we add a constant to the scalar potential:

(356) |

(357) |

(358) |

(359) |

Taking the
divergence of Eq. (354) and the curl of Eq. (355), and making use of the
Coulomb gauge, we find that the four field equations (346)-(349) can be reduced to
Poisson's equation written four times over:

Poisson's equation is just about the simplest

Poisson's equation

(363) |

The function is called the Green's function. The Green's function for Poisson's equation is

Note that this Green's function is proportional to the scalar potential of a point charge located at : this is hardly surprising, given the definition of a Green's function.

According to Eqs. (360), (361), (362), (364), and (365), the
scalar and vector potentials generated by
a set of stationary charges and steady currents take the form

Making use of Eqs. (354), (355), (366), and (367), we obtain the fundamental force laws for electric and magnetic fields. Coulomb's law,

(368) |

Of course, both of these laws are examples of action at a distance laws, and, therefore, violate the theory of relativity. However, this is not a problem as long as we restrict ourselves to fields generated by

The question, now, is by how much is this scheme which we have just worked out going to be disrupted when we take time variation into account. The answer, somewhat surprisingly, is by very little indeed. So, in Eqs. (346)-(369) we can already discern the basic outline of classical electromagnetism. Let us continue our investigation.