SCATTERING DIPOLES

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.)

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.