Half-Wave Antenna

Consider the half-wave antenna shown in Figure 1.1. The current distribution shown as a broken line is I = I₀ cos(2πz/λ) cos(ωt)

Figure 1.1

a) Find the vector potential in the radiation zone due to the complex current I = I₀ cos(2πz/λ) exp(iωt).

b) Find the electric field E and the magnetic induction B in the radiation zone.

c) Show that the time-averaged power radiated per unit solid angle is

Hint:

SOLUTION

a) The vector potential may be found from the integral :

(1)

The current density may be written with a complex time dependence (taking the real part at the end of the calculation):

(2)

Substituting (2) into (1) and integrating over x' and y', we obtain

(3)

where we have used the assumption that we are in the radiation zone, so that

Expanding the square root in (3) to order z'/r we find

(4)

Letting z = r cos θ and performing the integral in (4) (write cos as sum of exponentials), we get

(5)

b) The electric field E in the radiation zone may be found directly from

using (5). The magnetic induction B in the radiation zone is given by

So

c) The power radiated is calculated using the hint in the problem: