Magnetic Dipole and its Radiation Field

Consider radiation emitted from two magnetic dipoles. Have them be two circular loop antennas each of area of $\pi a^2$ aligned parallel to the $(x,y)$-plane with center on the $z$-axis. Fix their location in Rindler sectors $I$ and $II$ by having them located at $\xi'=\xi'_I$ and $\xi'=\xi'_{II}$ so that they are accelerated into opposite directions. Suppose each antenna has proper current

$$i_{I,II}(\tau')=\frac{1}{\xi'_{I,II}} \frac{d~q_{I,II}(\tau'... ...d\quad \left[\frac{\textrm{charge}}{\textrm{length}}\right] ~.$$

Then its magnetic moment is

$$\quad\quad\quad\quad\quad\quad\quad\quad i_{I,II}(\tau')\pi ... ...ent)}\times \textrm{(length)}^2=\textrm{dipole moment}~\right]$$

its charge-flux four-vector obtained from Eq.(29) is

$$\left( \hat S_{\tau'} ,\hat S_{\xi'} ,\hat S_{r'} ,\hat S_{\theta'} \right) = \left(0,0,\frac{1}{r'} \frac{\partial S}{\partial \theta'}, \frac{\partial S}{\partial r'} \right)$$

$$= i_{I,II}(\tau')\delta(\xi'-\xi'_{I,II}) \delta(r' -a) \left(0,0,0... ...)~,\quad\quad\quad\quad \left[\frac{\textrm{charge}}{\textrm{length}^3} \right]$$ (59)

and the corresponding scalar source function $S$ for Eq.(27) is

$$S:\quad S_{I,II}(\tau',\xi',r',\theta')= \frac{d~q_{I,II}(... ...uad \left[\frac{\textrm{charge}}{\textrm{length}^2} \right]~.$$ (60)

Here $\Theta$ is the Heaviside unit step function. The proper magnetic dipole is the proper volume integral of this source,

$$\int_0^\infty \int_0^\infty \int _0^{2\pi} S_{I,II}(\tau',\x... ...left[~\textrm{(area)} \times \textrm{(proper current)}~\right]$$

Being symmetric around its axis, such a source produces only radiation which is independent of the polar angle $\theta$. Consequently, all partial derivatives w.r.t. $\theta$ vanish, and the full scalar field, Eq.(54), in $F$ becomes with the help of Eq.(47)

$$\psi_F(\xi,\tau,r,\theta)=2\pi a^2 \!\!\!\!\!\!\!\! \left[ \frac{1}{\sqrt{(\xi^2-\xi^{'2}_I -r^2)^2+(2\xi \xi'_I)^2}} \frac{dq_I(\tau+\sinh^{-1}u_I)}{d\tau} \right.$$

$$- \left. \frac{1}{\sqrt{(\xi^2-\xi^{'2}_{II} -r^2)^2+(2\xi \xi'_{II... ...ac{dq_{II}(\tau-\sinh^{-1}u_{II})}{d\tau} \right]~,\quad\quad [\textrm{charge}]$$ (61)

where

$$u_{I,II} =\frac{\xi^2-\xi^{'2}_{I,II} -r^2}{2\xi \xi'_{I,II}}$$

This is the T.E. scalar field due to a localized pair of axially symmetric loop antennas, each one with its own time dependent current. By setting one of them to zero one obtains the radiation field due to the other.