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Retarded fields
We know the solution to Maxwell's equations in terms of retarded potentials. Let us now
construct the associated electric and magnetic fields using
It is helpful to write

(535) 
where
. The retarded time becomes , and a general retarded
quantity is written
. Thus, we can write the retarded
potential solutions of Maxwell's equations in the especially compact form:
where
.
It is easily seen that
where use has been made of

(539) 
Likewise,
Equations (533), (534), (538), and (540) can be combined to give

(541) 
which is the timedependent generalization of Coulomb's law,
and

(542) 
which is the timedependent generalization of the BiotSavart law.
Suppose that the typical variation timescale of our charges and currents is . Let
us define , which is the distance a light ray travels in time . We
can evaluate Eqs. (541) and (542) in two asymptotic limits: the near field
region , and the far field region . In the near field region

(543) 
so the difference between retarded time and standard time is relatively small. This
allows us to expand retarded quantities in a Taylor series. Thus,

(544) 
giving

(545) 
Expansion of the retarded quantities in the near field region yields
In Eq. (546), the first term on the righthand side corresponds to Coulomb's law, the second
term is the correction due to retardation effects, and the third term corresponds to Faraday
induction. In Eq. (547), the first term on the righthand side is the BiotSavart law,
and the second term is the correction due to retardation effects. Note that the retardation
corrections are only of order . We might suppose, from looking at Eqs. (541) and
(542), that the corrections should be of order . However, all of the order
terms canceled out in the previous expansion. Suppose, then, that we have a d.c. circuit
sitting on a laboratory benchtop. Let the currents in the circuit change on a
typical timescale of
one tenth of a second. In this time, light can travel about meters, so
kilometers. The lengthscale of the experiment is about one meter, so
meter. Thus, the retardation corrections are of order
. It is clear that we are fairly safe just using Coulomb's law, Faraday's law,
and the BiotSavart law to analyze the fields generated by this type of circuit.
In the far field region, , Eqs. (541) and (542) are dominated by the terms which
vary like , so
where

(550) 
Here, use has been made of
and
.
Suppose that our charges and currents are localized to some region in the vicinity of
. Let
, with
.
Suppose that the extent of the current and charge containing region is much less than .
It follows that retarded quantities can be written

(551) 
etc. Thus, the electric field reduces to

(552) 
whereas the magnetic field is given by

(553) 
Note that

(554) 
and

(555) 
This configuration of electric and magnetic fields is characteristic of an electromagnetic wave
(see Sect. 4.7).
Thus, Eqs. (552) and (553) describe an electromagnetic wave propagating radially
away from the
charge and current containing region. Note that the wave is driven by timevarying electric
currents. Now, charges moving with a constant velocity constitute a steady current, so a
nonsteady current is associated with accelerating charges. We conclude that accelerating
electric charges emit electromagnetic waves. The wave fields, (552) and (553), fall off
like the inverse of the distance from the wave source. This behaviour should be contrasted with
that of Coulomb or BiotSavart fields, which fall off like the inverse square of
the distance from the source. The fact that wave fields attenuate fairly gently with increasing
distance from the source is what makes astronomy possible. If wave fields obeyed an inverse square
law then no appreciable radiation would reach us from the rest of the Universe.
In conclusion, electric and magnetic fields look simple in the near field region (they are
just Coulomb fields, etc.) and also in the far field region (they are just electromagnetic
waves). Only in the intermediate region, , do things start getting really complicated
(so we generally do not look in this region!).
Next: Summary
Up: Timedependent Maxwell's equations
Previous: Advanced potentials?
Richard Fitzpatrick
20060202