Question

A circular aperture of radius a is mounted on an infinite electric ground plane. Assuming the opening is on the x-y plane and its field distribution is given by 12.44. 4 a find the far-zone electric and magnetic field components radiated by the antenna.

Answer 1

E_a = a_y^- E_0 [1 - p^'/a)^2] rightarrow m_s = -2 n^- times E_a = ax^- 2 E_0 [1 - p^'/a)^2] L_theta integral integral (M_X Cos theta Cos phi + M_Y Cos theta sin phi - M_Z sin phi) e^j k r Cos psi ds = 2 E_0 Cos theta Cos phi integral_0^a p^' [1 - (p^'/a)^2] integral_0^2 pi e^j k p^' sin theta Cos phi d theta d p^' After separating in to two integrals to L_ theta = 4 pu E_0 Cos theta Cos phi [integral_0^a p^' J_0 (k p^' sin theta) d p^' = 1/a^2 integral_0^a p^' 3 J_0 (k p^' sin theta) d p^'] making a change of variable of the form x = k p^' sin theta rightarrow dx = k sin theta d p^' we can write it as L theta = 4 pi E_0 Cos theta Cos phi [integral_0^a p^' J_0 (k p^' sin theta) d p^' = 1/a^2 (1/(k sin theta)^4 integral_0^k a sin theta x^3 J_0 (x) dx]