- #1
Mr. Rho
- 15
- 1
Hi people, I have a problem with some integral here.
I have a loop of radius a, with a current I = Ioe-iωt' and trying to calculate the radiating fields in the far zone, my procedement is:
Current density: J(r',t') = Ioδ(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)
I have a loop of radius a, with a current I = Ioe-iωt' and trying to calculate the radiating fields in the far zone, my procedement is:
Current density: J(r',t') = Ioδ(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)