Answer to the Question 04/02
SCATTERING DIPOLES
The question was:
A system consists of two ideal dipoles placed at positions
(0,0,0) and (0,0,a).
Dipole moment p of each dipole is related to electric field
E on that dipole via relation p=gE.
(g is so small that you can neglect the interaction between
the dipoles, i.e. electric field created by one dipole at the
position of the other dipole is negligible.) An incoming
electromagnetic wave of wavelength L=2a is scattered
by the system. Consider two cases: (a) the incoming wave is
in x-direction, and (b) the incoming wave is
in z-direction. Estimate the ratio between the total
scattering cross sections between those two cases. (You may neglect
dimensionless prefactors of order unity.)
(2/2003) We did not receive any solutions of the problem, and
decided to publish the following solution and comments.
Solution:
(Comment: All expressions are presented in Gaussian electromagnetic units.)
When electromagnetic field propagates x direction, both dipoles will
will feel the same field at the same time, and the total electric dipole
of the system will be p=2gE. Since the dipoles are not close
to each other, the system will also have higher moments (e.g. quadrupole),
and its radiation will not be pure dipole radiation. However, the order of
magnitude estimate of dipole radiation will suffice in an estimate of the
cross section. We note the the power radiated by a dipole must be proportional
to p2, since the electric field must be proportional to
charge (and thus to dipole moment), and the power is proportional to
squared field. From dimensional considerations, the radiated power
must be of order cp2/L4, where c is
the speed of light. Substituting, expression for p, and dividing the
result
by the flux of the incoming wave, we find that the cross section is
of order g2/L4. (This expression could be obtained
directly from dimensional analysis.)
When electromagnetic field propagates z direction, both dipoles will
feel field pointing in opposite directions, and consequently the total dipole
moment will vanish. However, one can easily convince himself, that
non-vanishing elements of quadrupole moment are of order Q=pa=gaE.
From dimensional considerations, we can also establish that the radiation
of quadrupole is of order cQ2/L6.
Substituting, expression for Q, and dividing the result
by the flux of the incoming wave, we find that the cross section is
of order g2a2/L6. Since L=2a,
this expression is of the same order as cross section obtained in the i
previous ("dipole radiation").
Those results really should not be surprising: Once we established that cross
section must be proportional to squared charges (and thus to g2,
and keeping in mind that the units of g are [length]3,
we must divide g2 by something with dimensions
[length]4 to obtain a cross section (that has units
[length]2). Since a and L are of the same
order, we can use one of them to get the correct result
g2/L4.
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