Helmholtz's theorem

for electric fields, and

for magnetic fields. There are no other field equations. This strongly suggests that if we know the divergence and the curl of a vector field then we know everything there is to know about the field. In fact, this is the case. There is a mathematical theorem which sums this up. It is called

Let us start with scalar fields. Field equations are a type of differential equation:
*i.e.*, they deal with the infinitesimal differences in quantities between neighbouring
points. The question is, what
differential equation completely specifies a
scalar field? This is easy. Suppose that we have a scalar field
and a field equation which tells us the gradient of this field at all points:
something like

(283) |

(284) |

Suppose that we have a vector field . How many differential equations
do we need to completely specify this field? Hopefully, we only need two: one
giving the divergence of the field, and one giving its curl. Let us
test this hypothesis. Suppose that we have two field equations:

where is a scalar field and is a vector field. For self-consistency, we need

(287) |

In other words, we are saying that a general field is the sum of a conservative field, , and a solenoidal field, . This sounds plausible, but it remains to be proved. Let us start by taking the divergence of the above equation, and making use of Eq. (285). We get

Note that the vector field does not figure in this equation, because the divergence of a curl is automatically zero. Let us now take the curl of Eq. (288):

(290) |

So, we have transformed our problem into four differential equations, Eq. (289) and Eqs. (291)-(293), which we need to solve. Let us look at these equations. We immediately notice that they all have exactly the same form. In fact, they are all versions of Poisson's equation. We can now make use of a principle made famous by Richard P. Feynman: ``the same equations have the same solutions.'' Recall that earlier on we came across the following equation:

where is the electrostatic potential and is the charge density. We proved that the solution to this equation, with the boundary condition that goes to zero at infinity, is

Well, if the same equations have the same solutions, and Eq. (295) is the solution to Eq. (294), then we can immediately write down the solutions to Eq. (289) and Eqs. (291)-(293). We get

and

(297) | |||

(298) | |||

(299) |

The last three equations can be combined to form a single vector equation:

We assumed earlier that
. Let us check to
see if this is true. Note that

(302) |

Now

(304) |

(305) |

(306) |

from Eq. (303), provided is bounded as . However, we have already shown that from self-consistency arguments, so the above equation implies that , which is the desired result.

We have constructed a vector field which satisfies Eqs. (285) and (286)
and behaves sensibly at infinity: *i.e.*,
as
. But, is our solution the only possible solution
of Eqs. (285) and (286) with sensible boundary conditions at infinity? Another way of posing
this question is to ask whether there are any solutions of

(309) |

These solutions certainly satisfy Laplace's equation, and are well-behaved at infinity. Because the solutions to Laplace's equations are unique, we know that Eqs. (310) are the only solutions to Eqs. (308). This means that there is no vector field which satisfies physical boundary equations at infinity and has zero divergence and zero curl. In other words, our solution to Eqs. (285) and (286) is the

We have just proved a number of very useful, and also very important, points.
First, according to Eq. (288), a general vector field can be written as the
sum of a conservative field and a solenoidal field. Thus, we ought to be able
to write electric and magnetic fields in this form. Second, a general vector
field which is zero at infinity is completely specified once its divergence
and its curl are given. Thus, we can guess that the laws of electromagnetism
can be written as four field equations,

(311) | |||

(312) | |||

(313) | |||

(314) |

without knowing the first thing about electromagnetism (other than the fact that it deals with two vector fields). Of course, Eqs. (279)-(282) are of exactly this form. We also know that there are only four field equations, since the above equations are sufficient to completely reconstruct both and . Furthermore, we know that we can solve the field equations without even knowing what the right-hand sides look like. After all, we solved Eqs. (285)-(286) for completely general right-hand sides. [Actually, the right-hand sides have to go to zero at infinity, otherwise integrals like Eq. (296) blow up.] We also know that any solutions we find are unique. In other words, there is only one possible steady electric and magnetic field which can be generated by a given set of stationary charges and steady currents. The third thing which we proved was that if the right-hand sides of the above field equations are all zero then the only physical solution is . This implies that steady electric and magnetic fields cannot generate themselves. Instead, they have to be generated by stationary charges and steady currents. So, if we come across a steady electric field we know that if we trace the field-lines back we shall eventually find a charge. Likewise, a steady magnetic field implies that there is a steady current flowing somewhere. All of these results follow from vector field theory (