3.2. Radiated Fields

A small loop in which loop current is everywhere in phase can be expressed a magnetic dipole with dipole moment \(M\). All the electromagnetic field components of the magnetic dipole are proportional to the \(M\) when the loop current varies sinusoidally. These fields vary with distance \(R\) as \(k^2/R\) for the radiation or far field components, as \(k/R^2\) for the transition field components, and as \(l/R^3\) for the induction or reactive field components. If the wavelength \(\lambda\), and any one of the three field components are given, the remaining components may be calculated. Alternately, if the \(M\) is given, all three field components may be calculated [57].

3.2.1. Thin Wire, Constant Current, Small Circumference

The wire is assumed to be very thin and the current spatial distribution is given by \(I_\phi = I_0\) where \(I_0\) is a constant. Although this type of current distribution is accurate only for a loop antenna with a very small circumference, a more complex distribution makes the mathematical formulation quite cumbersome [8].

(3.1)\[\vec A = j \frac{k \mu a^2 I_0 \sin \theta}{4 r} \left[ 1 + \frac{1}{jkr} \right] e^{-jkr} \vec a_\phi\]

(3.2)\[\begin{split}\begin{aligned} E_r &= 0 \\ E_\theta &= 0 \\ E_\phi &= \eta \frac{ (ka)^2 I_0 \sin \theta}{4 r} \left[ 1 + \frac{1}{jkr} \right] e^{-jkr} \end{aligned}\end{split}\]

(3.3)\[\begin{split}\begin{aligned} H_r &= j \frac{k a^2 I_0 \cos \theta}{2 r^2} \left[ 1 + \frac{1}{jkr} \right] e^{-jkr} \\ H_\theta &= - \frac{ (ka)^2 I_0 \sin \theta}{4 r} \left[ 1 + \frac{1}{jkr} - \frac{1}{(kr)^2} \right] e^{-jkr} \\ H_\phi &= 0 \end{aligned}\end{split}\]