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Second reply to Bob Sciamanda who wrote:
The derivation of the equation you are using assumes r<<lambda and so
ignores any "resonance effects of standing waves". It should serve your
purpose for comparisons at "low frequencies" (r<<lambda).
I just came home and looked at the derivation of the formula for R_rad.
I see a condition r>>lambda but in this case r is the distance from the
center of the loop, not the radius of the antenna. This condition is used
to calculate H for regions where "E and H are again perpendicular to each
other and have the proper relationship for an outward-going electromagnetic
wave." All emitted waves eventually enter the large r region. The value
of R_rad is calculated from the expression for the "time-average Poynting
vector". I note that the derived formula,
R_rad=2.46e6*(a/lambda)^4, where a is the radius of the loop,
has a much larger numerical coefficient than that posted by Bob. I do
not have another book at home to resolve this conflict. Can somebody look
for the coefficient in another reference for us? The textbook I am using
is "Electromagnetic Theory" by D. R. Frankl (Prentice-Hall, 1986, page 323).
It is quite possible that the condition a<<lambda is also implied but it
is not mentioned. Please help to resolve this issue.
1) What is the coefficient in the (a/lambda)^4 formula?
2) What are the restrictions for the validity of this formula?
Ludwik