Question in Antenna Theory by Balanis

时间：04-05 整理：3721RD 点击：

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

$3S\nabla\times\vec{E}=-\frac{\partial{\vec{B}}}{\partial{t}}\;\Rightarrow\;\vec{E}=-\nabla{V}-\frac{\partial{\vec{A}}}{\partial{t}}$

Then to Lorentz thingy

$3S\nabla\cdot\vec{A}=-\mu\epsilon\frac{\partial{V}}{\partial{t}}$

.

And so on......

The only thing different is it is in time harmonic form where

$3S\frac{\partial{}}{\partial{t}}\rightarrow\;j\omega$

.

Now going to the right side. I underlined the sentence that I have no idea where that comes from! What is

$3S\frac{1}{r^n}=0$

where n=,2,3,4.....

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

$3S\vec{B}=\nabla\times\vec{A}$

Then for far field where it is assumed plane wave,

$3S\vec{E}=\eta\hat{R}\times\vec{H}$

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)

$3S\tilde{E}=-j\omega\vec{A}$

This is nothing more than

$3S\nabla\times\vec{E}=-\frac{\partial{\vec{B}}}{\partial{t}}=-\nabla\times\frac{\partial{\vec{A}}}{\partial{t}}\;\Rightarrow\;\nabla\times\vec{E}+\nabla\times\frac{\partial{\vec{A}}}{\partial{t}}=0\;\Rightarrow\;\nabla\times\left[\vec{E}+\frac{\partial{\vec{A}}}{\partial{t}}\right]=0$

$3S\Rightarrow\;\vec{E}=-\frac{\partial{\vec{A}}}{\partial{t}}=-j\omega\vec{A}$

Why all the fuzz?

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