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Re: [Phys-L] spherical waves, near and far --> term by term

Hi,
Now I looked in my own Spam folder and found there 3 messages from the Phys-L members, starting from Apr 15.So I am not alone!
Moses FayngoldNJIT 


On Tuesday, May 3, 2016 1:05 PM, Donald Polvani <dgpolvani@verizon.net> wrote:


On : Saturday, April 30, 2016 7:39 AM, John Denker wrote:

Start with the equation:
    E    =      (1/4πє0) k^3    (n×p)×n        e^ikr (1/kr)      (1a)
          −i (1/4πє0) k^3    [3n(n·p) − p]    e^ikr (1/kr)^2      (1b)
          +  (1/4πє0) k^3    [3n(n·p) − p]    e^ikr (1/kr)^3      (1c)

where:
    p =    electric dipole moment
        including the e^iωt time dependence
    r =    radial coordinate
    n =    unit vector in the dr direction

Reference: Jackson equation 9.18.

1) I was interested to see this, since I was taught graduate level electrodynamics, long ago, from Jackson's original edition (1962).  The same equation, with the same equation number, is given in the original edition but in Gaussian units.  Unfortunately, I don't recall why the 1/(kr)^2 (intermediate zone) term was also called the "induction" zone term.
2) I found it useful to make an "order of magnitude" estimate of the relative importance of the three terms by plotting 1/kr, 1/(kr)^2, and 1/(kr)^3 versus kr as a log-log plot.  Of course,  I obtained three straight lines, all intersecting at kr = 1 with the 1/(kr)^3 term dominating for kr <1 and the 1/kr term dominating for kr > 1.  The 1/(kr)^2 term nicely stayed "intermediate" between the other two lines for all kr (except, of course, kr = 1).  To provide a further simple estimate of relative importance, I then added a fourth plot which was a log-log plot of the simple sum of the three terms versus kr.  The sum plot made clear that, for kr <0.1, the sum was not noticeably different from just the 1/(kr)^3 term, and, for kr > 10, the sum plot was not noticeably different from the 1/kr term alone.  However, for 0.1 < kr < 10, the sum plot was significantly different from either the 1/(kr)^3 or 1/kr term lines yielding an estimate of where the "intermediate" term is important.

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