And the fundamental proposition in potentials, viz. that, if n be the
external normal at any point of a closed surface, the integral
[int][int](dV/dn)dS, taken over the whole surface, has the
value--4[pi][mu]0, where [mu]0 is the sum of the values of [mu] for
each source lying within the surface, follows almost intuitively from
the mere consideration of what it means as regards light. For every
source external to the closed surface sends in light which goes out
again. But the light from an internal source goes wholly out; and the
amount per second from each unit source is 4[pi], the total area of
the unit sphere surrounding the source.
It is well to observe, however, that the analogy is not quite
complete. To make it so, all the sources must lie on the same side of
the surface whose illumination we are dealing with. This is due to the
fact that, in order that a surface may be illuminated at all, it must
be capable of scattering light, i.e. it must be to some extent opaque.
Hence the illumination depends mainly upon those sources which are on
the same side as that from which it is regarded.
Though this process bears some resemblance to the heat analogy
employed by Lord Kelvin (Sir W. Thomson) for investigations in
statical electricity and to Clerk Maxwell's device of an
incompressible fluid without mass, it is by no means identical with
them. Each method deals with a substance, real or imaginary, which
flows in conical streams from a source so that the same amount of it
passes per second through every section of the cone. But in the
present process the velocity is constant and the density variable,
while in the others the density is virtually constant and the velocity
variable. There is a curious reciprocity in formulae such as we have
just given. For instance, it is easily seen that the light received
from a uniformly illuminated surface is represented by
[int][int]r^(-2) cos [theta] dS.
As we have seen that this integral vanishes for a closed surface which
has no source inside, its value is the same for all shells of equal
uniform brightness whose edges lie on the same cone.
Public-domain text, read in full here on John Shaqi.
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