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 .
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 .
(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}