Question in Antenna Theory by Balanis

See attached scanned pages of page 118(left side) and page 125(right side) of Antenna Theory by Balanis.

On the left side, it is just some basic EM where

\[\nabla\times \vec {E}=-\frac{\partial {\vec {B}}}{\partial {t}}\;\Rightarrow\;\vec {E}=-\nabla {V}-\frac{\partial {\vec {A}}}{\partial {t}}\]

Then to Lorentz thingy \[\nabla\cdot \vec {A}=-\mu\epsilon \frac{\partial {V}}{\partial {t}}\].

And so on......

The only thing different is it is in time harmonic form where \[\frac{\partial{}}{\partial {t}}\rightarrow\;j\omega\].

Now going to the right side. I underlined the sentence that I have no idea where that comes from!! What is \[\frac {1}{r^n}=0\] where n=,2,3,4.....

Please explain what is that. This supposed to be very basic thing that

\[\vec {B}=\nabla\times\vec {A}\]
Then for far field where it is assumed plane wave,\[\vec {E}=\eta\hat {R} \times \vec {H}\]

Please help.

Thanks

I since find out what Balanis is talking about. He used Multipole Expansion for charges from great distance!!! But why? What's wrong with the following?


If you look at (3-58a) \[\tilde {E}=-j\omega \vec{A}\]

This is nothing more than
\[\nabla\times \vec{E}=-\frac{\partial{\vec{B}}}{\partial{t}}=-\nabla \times \frac{\partial{\vec{A}}}{\partial{t}}\;\Rightarrow\;\nabla\times \vec{E}+\nabla \times \frac{\partial{\vec{A}}}{\partial{t}}=0\;\Rightarrow\;\nabla\times\left[\vec{E}+\frac{\partial{\vec{A}}}{\partial{t}}\right]=0\]

\[\Rightarrow\;\vec{E}=-\frac{\partial{\vec{A}}}{\partial{t}}=-j\omega\vec{A}\]

Why all the fuzz?