Week $14-$ Problems to solve B$18$&B$19($Radiation and Antenna$)$

Problem $1$

In this week lecture, we have demonstrated how to obtain the radiation from wire antenna, especially for a thin dipole. The magnetic vector potential $A_{?}$ that relates to a phasor current distribution $J$ inside the volume $v^{\text{'}}$ is given by

$A_{?}=\frac{{}_0}{4}{\displaystyle \underset{v^{\text{'}}}{\overset{\phantom{}}{}}}\frac{J_e^{-jR}}{R}d{v}^{\text{'}},$

where the $r$ is the distance from the origin to the observation point $P$ and $R$ is the distance between the source point $(r^{\text{'}})$ to the observation point $(r).$

As we know, for a short dipole $(l$ is small$)$ with small radius, we may assume that the current only flows in the $z$ direction as a constant, $I_0.$ The current density is simply $J_{?}=$hat$(z)(\frac{I_0}{s}),$ where $s$ is the cross section area of the dipole, $d{v}^{\text{'}}=sdz$ and the limits of integration goes from $z=-\frac{l}{2}$ to $z=+\frac{l}{2}.$

Rigorously, $R$ does not equal to the distance from the origin, which means $Rr.$ Can you:

Determine the magnetic vector potential $A_{?}$ under the condition $Rr,$ and based on your results, for a short dipole $(l$ is sufficiently small$),$ will our treatment for this problem in the last week still be valid?

Determine the electric field $E$ as the far$-$field contribution to the radiation?

$[$Hint: you may use the fact that the magnitude difference between $\frac{1}{R}$ and $\frac{1}{r}$ is insignificant in far$-$field approximation$]$

$[$Hint: You may ignore the $E-$field radial contribution as we have discussed in the lecture$]$

$[$Hint: You may use $(1){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{e}^{ax}dx=\frac{1}{a}{e}^{ax}$ and $(2)sin(x)x$ for sufficiently small $x]$