Half-Wave Antenna
المؤلف:
Sidney B. Cahn Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 1 , p 73
9-8-2016
1535
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:
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