The time$-$harmonic complex electric field radiated in free space by a linear radiating element is given by:

$E=$hat$(r)E_r+$hat$()E_{}$

where

$E_r=E_e\frac{cos}{r^2}[1+\frac{1}{j{}_{e}r}]e^{-j{}_{e}r}$

$E_{}=j{E}_o\frac{{}_{e}sin}{2r}[1+\frac{1}{j{}_{o}r}-\frac{1}{({}_{o}r{)}^2}]e^{-j{}_{e}r}$

where hat$(r)$ and hat$()$ are unit vectors in the spherical directions $r$ and $,E_0$ is a constant, and ${}_o=\sqrt[\phantom{2}]{{}_{o}lo{n}_o}$

$($a$)$ Determine the time$-$harmonic spherical magnetic field components.

$($b$)$ Show that the fields are TEM mode in the far$-$field. Hint: for large distances from the radiating element, all field components that "fall off" faster than $\frac{1}{R}$ will be negligible.