This is a relatively simple, and hence very standard problem.

Now, we have no desire to ``reinvent the sphere''^{16.1} but it is important that you understand where our results come
from. First of all, let us introduce dimensionless, scaled versions of
the *relative* permeability and permittivity (a step that Jackson
apparently performs in J10 but does not document or explain):
&epsi#epsilon;_r & = & &epsi#epsilon;(&omega#omega;)/&epsi#epsilon;_0

&mu#mu;_r & = & &mu#mu;(&omega#omega;)/&mu#mu;_0 &ap#approx;1
where we assume that we are not at a resonance so that the spheres have
normal dispersion and that these numbers are basically real. The latter
is a good approximation for non-magnetic, non-conducting scatterers e.g.
oxygen or nitrogen molecules.

If you refer back to J4.4, equation J4.56 and the surrounding text, you will see that the induced dipole moment in a dielectric sphere in terms of the relative permittivity is: = 4&pi#pi;&epsi#epsilon;_0 ( &epsi#epsilon;_r - 1&epsi#epsilon;_r + 2 ) a^3 _inc

To recapitulate the derivation (useful since this is a *common*
question on qualifiers and the like) we note that the sphere has
azimuthal symmetry around the direction of
, so we can express the
scalar potential inside and outside the sphere as
&phis#phi;_in & = & &sum#sum;_&ell#ell;A_&ell#ell;r^&ell#ell;P_&ell#ell;(&thetas#theta;)

&phis#phi;_out & = & &sum#sum;_&ell#ell;{ B_&ell#ell;r^&ell#ell;+ C_&ell#ell;
1r^&ell#ell;+ 1 } P_&ell#ell;(&thetas#theta;) .

We need to evaluate this. At infinity we know that the field should be (to
lowest order) undisturbed, so the potential must asymptotically go over to
_ r &rarr#to;&infin#infty; &phis#phi;_out = -E_0 z = -E_0 r &thetas#theta;= -E_0
r P_1(&thetas#theta;)
so we conclude that
and all other
. To proceed
further, we must use the **matching conditions** of the tangential and
normal fields at the surface of the sphere:
- 1a . &phis#phi;_in&thetas#theta; |_r=a = -
1a . &phis#phi;_out&thetas#theta; |_r=a
(tangential component) and
- &epsi#epsilon;. &phis#phi;_inr |_r=a = - .
&epsi#epsilon;_0 &phis#phi;_outr |_r=a
(normal
onto
).

Since this is the surface of a sphere (!) we can project out each spherical
component if we wish and cause these equations to be satisfied term by term.
From the first (tangential) equation we just match
itself:
1a (A_&ell#ell;a^&ell#ell;) = 1a ( B_&ell#ell;a^&ell#ell;+ C_&ell#ell;
1a^&ell#ell;+1 )
or (using our knowledge of
)
A_1 & = & -E_0 + C_1a^3 &ell#ell;= 1

A_&ell#ell;& = & C_&ell#ell;a^2&ell#ell;+1 else

From the second (normal) equation we get
&epsi#epsilon;_r A_1 & = & -E_0 - 2C_1a^3 &ell#ell;= 1

&epsi#epsilon;_r A_&ell#ell;& = & - (&ell#ell;+1) C_&ell#ell;a^2&ell#ell;+1
else.

The second equation of each pair are incompatible and have only the trivial
A_&ell#ell;= C_&ell#ell;= 0 &ell#ell;&ne#ne;1 .
Only the
term survives. With a little work one can show that
A_1 & = & -3 E_02 + &epsi#epsilon;_r

C_1 & = & (&epsi#epsilon;_r - 1&epsi#epsilon;_r+2 ) a^3 E_0
so that
&phis#phi;_in & = & - ( 3&epsi#epsilon;_r+2 ) E_0 r
&thetas#theta;

&phis#phi;_out & = & - E_0 r &thetas#theta;+ ( &epsi#epsilon;_r -
1&epsi#epsilon;_r + 2 ) E_0 a^3r^2 &thetas#theta;.

When we identify the second term of the external field with the dipole potential and compare with the expansion of the dipole potential &phis#phi;() = 14&pi#pi;&epsi#epsilon;_0 ·r^3 we conclude that the induced dipole moment is: = 4&pi#pi;&epsi#epsilon;_0 ( &epsi#epsilon;_r - 1&epsi#epsilon;_r + 2 ) a^3 E_0 . as given above.

There is no magnetic dipole moment, because and therefore the sphere behaves like a ``dipole antenna''. Thus and there is no magnetic scattering of radiation from this system. This one equation, therefore, (together with our original definitions of the fields) is sufficient to determine the differential cross-section: d &sigma#sigma;d &Omega#Omega; = k^4 a^6 &epsi#epsilon;_r - 1&epsi#epsilon;_r + 2^2 ^&ast#ast;·_0^2 where remember that (for dispersion) and hopefully everybody notes the difference between dielectric and polarization (sigh - we need more symbols). This equation can be used to find the explicit differential cross-sections given , as desired.

However, the light incident on the sphere will generally be unpolarized. Then the question naturally arises of whether the various independent polarizations of the incident light beam will be scattered identically. Or, to put it another way, what is the angular distribution function of radiation with a definite polarization? To answer this, we need to consider a suitable decomposition of the possible polarization directions.

This decomposition is apparent from considering the following picture of the general geometry:

Let
define the plane of scattering. We have to fix
and
relative to this scattering plane and
average over the polarizations in the incident light,
and
(also fixed relative to this plane). We can always choose
the directions of polarization such that
is
perpendicular to the scattering plane and
are
in it, and perpendicular to the directions
and
respectively. The dot products are thus
^(1) &ast#ast; ·_0^(1) & = & ·_0 = &thetas#theta;

^(2) &ast#ast; ·_0^(2) & = & 1 .

We need the average of the squares of these quantities. This is essentially
averaging
and
over
.
Alternatively, we can meditate upon symmetry and conclude that the average is
just
. Thus (for the polarization in the plane (
) and
perpendicular to the plane (
) of scattering, respectively) we have:
d &sigma#sigma;_&par#parallel;d &Omega#Omega; & = & k^4 a^6
&epsi#epsilon;_r - 1&epsi#epsilon;_r + 2^2 ^2 &thetas#theta;2

d &sigma#sigma;_&perp#perp;d &Omega#Omega; & = & k^4 a^6 &epsi#epsilon;_r -
1 &epsi#epsilon;_r + 2^2 12
We see that light polarized perpendicular to the plane of scattering has no
dependence, while light polarized in that plane is not scattered
parallel to the direction of propagation at all (along
or
).
We will invert this statement in a moment so that it makes more sense. See
the diagram below.

Unfortunately, everything thus far is expressed with respect to the plane of
scattering, which varies with the direction of the scattered light. If we
define the **polarization**
of the scattered radiation to be
&Pi#Pi;(&thetas#theta;) = d&sigma#sigma;_&perp#perp;d&Omega#Omega; -
d&sigma#sigma;_&par#parallel;d&Omega#Omega;d&sigma#sigma;_&perp#perp;d&Omega#Omega; +
&sigma#sigma;_&par#parallel;d&Omega#Omega; = ^2 &thetas#theta;1 + ^2 &thetas#theta;
then we obtain a quantity that is in accord with our intuition.
is maximum at
. The radiation scattered through an angle of
90 degrees is completely polarized in a plane perpendicular to the plane of
scattering.

Finally, we can add the two pieces of the differential cross-section
together:
d&sigma#sigma;d&Omega#Omega; = k^4a^6 ( &epsi#epsilon;- 1&epsi#epsilon;+ 2
)^2 12(1 + ^2 &thetas#theta;)
which is strongly and symmetrically peaked forward and backward. Finally,
this is easy to integrate to obtain the **total cross-section**:
&sigma#sigma;= 8&pi#pi;3 k^4 a^6 ( &epsi#epsilon;_r - 1&epsi#epsilon;_r
+ 2 )^2 .

At last, we can put it all together. Molecules in the atmosphere behave, far
from resonance, like itty-bitty dielectric spheres to a remarkable
approximation. Since blue light is scattered more strongly than red, light
seen away from its direction of incidence (the sky and not the sun) is shifted
in color from white to blue. When Mr. Sun is examined directly through a
thick layer of atmosphere (at sunset) the blue is all scattered out and the
remaining light looks red. Finally, light from directly overhead at sunup or
sundown is polarized in a north-south direction; at noon the light from the
horizon is polarized parallel to the horizon (and hence is filtered by
vertical transmission axis polarized sunglasses. You should verify this
at your next opportunity outdoors with a pair of polarized sunglasses,
as this *whole discussion* is taught in *elementary* terms in
second semester introductory physics courses.

Don't say I never taught you anything^{16.2}.

The last remarks I would make concern the total cross-section. Note that if we factor out a we get the ``area'' of the sphere times a pure (dimensionless) number associated with the relative size of the sphere radius and the wavelength and a second pure number involving only the dielectric properties of the medium: &sigma#sigma;= (4&pi#pi;a^2) (ka)^4 {23 ( &epsi#epsilon;_r - 1&epsi#epsilon;_r + 2 )^2 }. This expression isn't any more useful than the one above, but it does make the role of the different terms that contribute to the total scattering cross-section more clear.