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Magnetic Dipole and Electric Quadrupole Radiation Fields

The next term in the multipolar expansion is the $ \ell = 1$ term:

$\displaystyle {\bf A}(\vx) = i k \mu_0 h_1^+(kr) \sum_{m = -1}^1 Y_{1,m}(\hat{r}) \int_0^\infty \vJ({\bf x'}) j_1(kr') Y_{1,m}(\hat{r'})^\ast d^3x'$ (13.66)

When you (for homework, of course)
  1. $ m$ -sum the product of the $ Y_{\ell,m}$ 's
  2. use the small $ kr$ expansion for $ j_1(kr')$ in the integral and combine it with the explicit form for the resulting $ P_1(\theta)$ to form a dot product
  3. cancel the $ 2 \ell + 1$ 's
  4. explicitly write out the hankel function in exponential form
you will get equation (J9.30, for - recall - distributions with compact support):

$\displaystyle {\bf A}(\vx) = \frac{\mu_0}{4\pi} \frac{e^{ikr}}{r} \left ( \frac{1}{r} - ik \right ) \int_0^\infty \vJ({\bf x'}) ({\bf n} \cdot {\bf x'}) d^3x'.$ (13.67)

Of course, you can get it directly from J9.9 (to a lower approximation) as well, but that does not show you what to do if the small $ kr$ approximation is not valid (in step 2 above) and it neglects part of the outgoing wave!

There are two important and independent pieces in this expression. One of the two pieces is symmetric in $ \vJ$ and $ \vx'$ and the other is antisymmetric (get a minus sign when the coordinate system is inverted). Any vector quantity can be decomposed in this manner so this is a very general step:

$\displaystyle \vJ({\bf n} \cdot {\bf x'}) = \frac{1}{2} [({\bf n} \cdot {\bf x'...
...\bf n} \cdot \vJ) {\bf x'}] + \frac{1}{2} ({\bf x'} \times \vJ) \times {\bf n}.$ (13.68)



Subsections
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Next: Magnetic Dipole Radiation Up: Radiation Previous: Energy radiated by the   Contents
Robert G. Brown 2017-07-11