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In Jackson's 9.14 we calculate the expressions for $E$ and $H$ of a loop antenna with current $I=I_0\cos(\omega t)$ in the $x-y$ plane in the radiation zone, where we approximate the distances to be large so:

$$\frac{1}{|\vec{r}-\vec{r}^{'}|}\approx\frac{1}{\vec{r}}$$

$$|\vec{r}-\vec{r}^{'}|\approx \vec{r}-\vec{r}.\vec{r}^{'}$$

But what if we want to derive an expression that is valid for lower modes of the quadrupole radiation in all distances without approximation? How we should approach the integral with $\frac{1}{|\vec{r}-\vec{r}^{'}|}$ in the denominator?

$$\vec{A}(\vec{x})=\frac{\mu_0}{4\pi}\int J(\vec{r}')\frac{\exp(ik|\vec{r}-\vec{r}'|)}{|\vec{r}-\vec{r}'|}d\vec{r}'$$

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  • $\begingroup$ What you need to look for is the near field expression for the radiation. This is quite complicated because of back-action effects etc. There are expressions in EEng textbooks on antenna theory but I recall reading the modelling is not so great. This type of calculation cannot be done exactly and must be done term by term. $\endgroup$ Jul 12, 2020 at 14:49
  • $\begingroup$ incidentally you have a number of typos in your question. One cannot divide by a vector so $1/\vec r$ does not make sense and should be $1/\vert \vec r \vert$. Likewise $\vert \vec r-\vec r’\vert$ is a scalar so you should have $\vert \vec r\vert$ on the right hand side. Finally, your $\vec A$ is really a phasor and should be indicated as such. $\endgroup$ Jul 12, 2020 at 14:52

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You can use the Laplace expansion of the inverse distance:

$$\frac{1}{|\vec{r}-\vec{r}'|}= \sum_{\ell=0}^\infty \frac{4\pi}{2\ell +1}\frac{r_<^\ell}{r_>^{\ell+1}} \sum_{m=-\ell}^{+\ell}Y_{\ell m}^*(\theta,\phi)Y_{\ell m}(\theta',\phi')$$ where

  • $r_<$ is the smaller one of $r$ and $r'$,
  • $r_>$ is the bigger one of $r$ and $r'$,
  • and $Y_{\ell m}$ are the spherical harmonics.

If $r_< \ll r_>$, then it is sufficient to take only the first few terms (with lowest $\ell$).


The expansion for $\frac{\exp(ik|\vec{r}-\vec{r}'|)}{|\vec{r}-\vec{r}'|}$ (which, apart from a factor $4\pi$, is actually the Green's function of the Helmholtz wave equation) is more difficult in the radial factors:

$$\frac{\exp(ik|\vec{r}-\vec{r}'|)}{|\vec{r}-\vec{r}'|}= 4\pi ik \sum_{\ell=0}^\infty j_\ell(kr_<)h_\ell^{(1)}{(kr_>)} \sum_{m=-\ell}^{+\ell}Y_{\ell m}^*(\theta,\phi)Y_{\ell m}(\theta',\phi') $$ where

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  • $\begingroup$ Thanks, and what about the |r-r'| term in the exponential? $\endgroup$
    – Ummagumma
    Jul 12, 2020 at 11:55
  • $\begingroup$ @Ummagumma Can you please add the relevant formulas into your question? I don't have Jackson's book available. $\endgroup$ Jul 12, 2020 at 11:59
  • $\begingroup$ Done, I added the formula for vector potential from which E and B can be derived $\endgroup$
    – Ummagumma
    Jul 12, 2020 at 12:24

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