We’ve seen that the far field electric and magnetic fields associated with a magnetic vector potential were

\BE = -j \omega \textrm{Proj}_\T \BA,
\BH = \inv{\eta} \kcap \cross \BE.

It’s worth a quick note that the duality transformation for this, referring to [1] tab. 3.2, is

\BH = -j \omega \textrm{Proj}_\T \BF
\BE = -\eta \kcap \cross \BH.

What does \( \BH \) look like in terms of \( \BA \), and \( \BE \) look like in terms of \( \BH \)?

The first is

= -\frac{j \omega}{\eta} \kcap \cross \lr{ \BA – \lr{\BA \cdot \kcap} \kcap },

in which the \( \kcap \) crossed terms are killed, leaving

= -\frac{j \omega}{\eta} \kcap \cross \BA.

The electric field follows again using a duality transformation, so in terms of the electric vector potential, is

\BE = j \omega \eta \kcap \cross \BF.

These show explicitly that neither the electric or magnetic far field have any radial component, matching with intuition for transverse propagation of the fields.


[1] Constantine A Balanis. Antenna theory: analysis and design. John Wiley & Sons, 3rd edition, 2005.