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Earlier on in this lecture course, we had to solve Poisson's equation

(475) 
where is denoted the source function. The potential
satisfies the boundary condition

(476) 
provided that the source function
is reasonably localized. The solutions to Poisson's equation
are superposable (because the equation is linear). This property is exploited in the Green's
function method of solving this equation. The Green's function
is the
potential, which satisfies the appropriate boundary conditions,
generated by a unit amplitude
point source located at .
Thus,

(477) 
Any source function can be represented as a weighted sum of point
sources

(478) 
It follows from superposability that the potential generated by the source
can be written as the weighted sum of point source driven
potentials (i.e., Green's functions)

(479) 
We found earlier that the Green's function for Poisson's equation is

(480) 
It follows that the general solution to Eq. (475) is written

(481) 
Note that the point source driven potential (480) is perfectly sensible. It is spherically symmetric
about the source, and falls off smoothly with increasing distance from the source.
We now need to solve the wave equation

(482) 
where is a timevarying source function. The potential
satisfies the boundary conditions

(483) 
The solutions to Eq. (482) are superposable (since the equation is linear), so
a Green's function method of solution is again appropriate. The Green's function
is the potential generated by a point impulse located at
position and applied at time . Thus,

(484) 
Of course, the Green's function must satisfy the correct boundary conditions. A general
source can be built up from a weighted sum of point impulses

(485) 
It follows that the potential generated by can be written as the weighted
sum of point impulse driven potentials

(486) 
So, how do we find the Green's function?
Consider

(487) 
where is a general scalar function. Let us try to prove the following theorem:

(488) 
At a general point,
, the above expression reduces to

(489) 
So, we basically have to show that is a valid solution of the free space
wave equation.
We can easily show that

(490) 
It follows by simple differentiation that
where
. We can derive analogous equations for
and
.
Thus,

(492) 
giving

(493) 
which is the desired result. Consider, now, the region around
. It is
clear from Eq. (491) that the dominant term on the righthand
side as
is
the first one, which is essentially
.
It is also clear that
is negligible compared to this term.
Thus, as
we find
that

(494) 
However, according to Eqs. (477) and (480)

(495) 
We conclude that

(496) 
which is the desired result.
Let us now make the special choice

(497) 
It follows from Eq. (496) that

(498) 
Thus,

(499) 
is the Green's function for the driven wave equation (482).
The timedependent Green's function (499) is the same as the steadystate Green's function
(480), apart from the deltafunction appearing in the former. What does this deltafunction do?
Well, consider an observer at point . Because of the deltafunction, our observer
only measures a nonzero potential at one particular time

(500) 
It is clear that this is the time the impulse was applied at position (i.e., )
plus the time taken for a light signal to travel between points and
. At time , the locus of all the points at which the potential is nonzero
is

(501) 
In other words, it is a sphere centred on whose radius is the distance traveled by
light in the time interval since the impulse was applied at position .
Thus, the Green's function (499) describes a spherical wave which emanates from position
at time and propagates at the speed of light. The amplitude of the wave
is inversely proportional to the distance from the source.
Next: Retarded potentials
Up: Timedependent Maxwell's equations
Previous: Electromagnetic waves
Richard Fitzpatrick
20060202