Radiating Systems, Multipole Fields and Radiation — John David Jackson
Physics — E&M · aperture-dipole-moments
Problem
The aperture or apertures in a perfectly conducting plane screen can be viewed as the location of effective sources that produce radiation (the diffracted fields). An aperture whose dimensions are small compared with a wavelength acts as a source of dipole radiation with the contributions of other multipoles being negligible.
(a) Beginning with (10.101) show that the effective electric and magnetic dipole moments can be expressed in terms of integrals of the tangential electric field in the aperture as follows:
p=ϵn∫(x⋅Etan)da(9.72)m=iωμ2∫(n×Etan)da
where Etan is the exact tangential electric field in the aperture, n is the normal to the plane screen, directed into the region of interest, and the integration is over the area of the openings.
(b) Show that the expression for the magnetic moment can be transformed into
m=μ2∫x(n⋅B)da(9.74)
Be careful about possible contributions from the edge of the aperture where some components of the fields are singular if the screen is infinitesimally thick.