Question in Antenna Theory by Balanis

[Alan0354]

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

 

[Alan0354]

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?