Retarded potentials

(502) | |||

(503) |

The solutions to these equations are easily found using the Green's function for Poisson's equation (480):

The time-dependent Maxwell equations reduce to

(506) | |||

(507) |

We can solve these equations using the time-dependent Green's function (499). From Eq. (486) we find that

(508) |

These are the general solutions to Maxwell's equations. Note that the time-dependent solutions, (509) and (510), are the same as the steady-state solutions, (504) and (505), apart from the weird way in which time appears in the former. According to Eqs. (509) and (510), if we want to work out the potentials at position and time then we have to perform integrals of the charge density and current density over all space (just like in the steady-state situation). However, when we calculate the contribution of charges and currents at position to these integrals we do not use the values at time , instead we use the values at some earlier time . What is this earlier time? It is simply the latest time at which a light signal emitted from position would be received at position before time . This is called the

(511) |

The time dependence in the above equations is taken as read.

We are now in a position to understand electromagnetism at its most fundamental level.
A charge distribution
can thought of as built up
out of a collection, or series, of charges which
instantaneously come into existence, at some point and some time , and
then disappear again. Mathematically, this is written

(514) |

(515) |

(516) |

(517) |

Let us compare the steady-state law

(518) |

These two formulae look very similar indeed, but there is an important difference. We can imagine (rather pictorially) that every charge in the Universe is continuously performing the integral (519), and is also performing a similar integral to find the vector potential. After evaluating both potentials, the charge can calculate the fields, and, using the Lorentz force law, it can then work out its equation of motion. The problem is that the information the charge receives from the rest of the Universe is carried by our spherical waves, and is always slightly out of date (because the waves travel at a finite speed). As the charge considers more and more distant charges or currents, its information gets more and more out of date. (Similarly, when astronomers look out to more and more distant galaxies in the Universe, they are also looking backwards in time. In fact, the light we receive from the most distant observable galaxies was emitted when the Universe was only about one third of its present age.) So, what does our electron do? It simply uses the most up to date information about distant charges and currents which it possesses. So, instead of incorporating the charge density in its integral, the electron uses the

Consider a thought experiment in which a charge appears at position at time
, persists for a while, and then disappears at time . What is the electric field
generated by such a charge? Using Eq. (519), we find that

(520) |

Now, (since there are no currents, and therefore no vector potential is generated), so

(521) |

This solution is shown pictorially in Fig. 37. We can see that the charge effectively emits a Coulomb electric field which propagates radially away from the charge at the speed of light. Likewise, it is easy to show that a current carrying wire effectively emits an Ampèrian magnetic field at the speed of light.

We can now appreciate the essential difference between time-dependent electromagnetism and
the action at a distance laws of Coulomb and Biot & Savart. In the latter
theories, the field-lines act
rather like rigid
wires attached to charges (or circulating around currents). If the charges (or currents) move then
so do the field-lines, leading inevitably to unphysical action at a distance type behaviour.
In the time-dependent theory, charges act rather like water sprinklers: *i.e.*, they spray out the
Coulomb field in all directions at the speed of light. Similarly,
current carrying wires throw out magnetic field
loops at the speed of light. If we move a charge (or current) then field-lines emitted beforehand
are not affected, so the field at a distant charge (or current) only responds to the change
in position
after a time delay sufficient for the field to propagate between the two charges (or currents) at
the speed of light.

In Coulomb's law and the Biot-Savart law, it is not entirely obvious that the electric and magnetic fields
have a real existence. After all, the only measurable quantities are the forces acting between charges and
currents. We can describe the force acting on a given charge or current, due to the other charges
and currents in the Universe,
in terms of the local electric and magnetic fields, but we have no way of knowing whether these
fields persist when the charge or current is not present (*i.e.*, we could argue that electric and
magnetic fields are just a convenient way of calculating forces, but, in reality, the forces
are transmitted directly between charges and currents by some form of magic).
However, it is patently obvious that electric and magnetic fields have a real existence
in the time-dependent theory. Consider the following thought experiment.
Suppose that a charge comes into existence for a period of time, emits a Coulomb
field, and then disappears. Suppose that a distant charge interacts with this field,
but is sufficiently far from the first charge that by the time the field arrives the
first charge has already disappeared. The force exerted on the second charge is only ascribable
to the electric field: it cannot be ascribed to the first charge, because this charge no longer exists
by the time the force is exerted. The electric field clearly transmits energy and momentum
between the two charges. Anything which possesses energy and momentum is ``real'' in a physical
sense. Later on in this course, we shall demonstrate that electric and magnetic fields conserve
energy and momentum.

Let us now consider a moving charge. Such a charge is continually emitting spherical waves in the scalar potential, and the resulting wavefront pattern is sketched in Fig. 38. Clearly, the wavefronts are more closely spaced in front of the charge than they are behind it, suggesting that the electric field in front is larger than the field behind. In a medium, such as water or air, where waves travel at a finite speed, (say), it is possible to get a very interesting effect if the wave source travels at some velocity which exceeds the wave speed. This is illustrated in Fig. 39.

The locus of the outermost wave front is now a cone instead of a sphere. The wave intensity on the cone is extremely large: this is a