Classical Electrodynamics · Problem 9.18

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · short-antenna-reactance

Problem

A qualitative understanding of the result for the reactance of a short antenna whose radiation fields are described by the electric dipole fields of Section 9.2 can be achieved by considering the idealized dipole fields (9.18). (a) Show that the integral over all angles at fixed distance

r

\epsilon_0|\mathbf{E}|^2 - \mu_0|\mathbf{H}|^2

\int [\epsilon_0 |\mathbf{E}|^2 - \mu_0 |\mathbf{H}|^2]\, d\Omega = \frac{1}{2\pi\epsilon_0}\frac{|\mathbf{p}|^2}{r^6}

(b) Using (6.140) for the reactance, show that the contribution

X_a

to the reactance from fields at distances

r > a

X_a = -\frac{\omega |\mathbf{p}|^2}{6\pi\epsilon_0 |I_i|^2 a^3}

where

I_i

is the input current. (c) For the short center-fed antenna of Section 9.2 show that

X_a \approx -d^2/24\pi\epsilon_0 \omega a^3

, corresponding to an effective capacitance

24\pi\epsilon_0 a^3/d^2

. With

a = d/2

X_a

gives only a small fraction of the total negative reactance of a short antenna. The fields close to the antenna, obviously not dipole in character, contribute heavily. For calculations of reactances of short antennas, see the book by Schelkunoff and Friis.

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