Radiation in the far field for a current carrying loop

[Mr. Rho]

Hi people, I have a problem with some integral here.

I have a loop of radius a, with a current I = I_oe^-iωt' and trying to calculate the radiating fields in the far zone, my procedement is:

Current density: J(r',t') = I_oδ(r'-a)δ(θ'-π/2)e^-iωt'/2πa² φ (φ direction)

Here t' = t - |r-r'|/c (retarded time)

I evaluate the vector potential: A(r,t) = ∫_vJ(r',t')dV/|r-r'|

I approximate 1/|r-r'| ≈ 1/r because in the far zone this term does not affect too much compared with the exponential, and |r-r'| ≈ r - aSinθCos(φ-φ') for the exponential, that plays a major role in the far zone.

The problem is that I reach integrals that I can't solve:

∫ sinφ' e^{-ikaSinθCos(φ-φ')} dφ' for x direction

and ∫ sinφ' e^{-ikaSinθCos(φ-φ')} dφ' for y direction (both integrals from 0 to 2π).

Maybe I'm doing something wrong, but I don't know, any help?

(r' are the source coordinates and r the observer coordinates)

 

[TSny]

Usually it is assumed that you are working with low enough frequencies and a small enough loop that you can take $a$ to be small in relation to the wavelength of the radiation. Then you can approximate $e^{-ika \sin \theta \cos (\phi - \phi ')}$.