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Jackson, "Classical Electrodynamics", 1962 works out (unfortunately, in cgs
units) the exact magnetic field from a circular loop carrying a constant
current I and of radius a. Although the general solution involves the
complete elliptic integrals K and E, the exact B field, on axis (B_r), is
given simply by:
B_r = (2I*pi*a^2/cr^3)*1/(1 + a^2/r^2)^3/2
Where c = speed of light, pi = the usual math constant 3.14...
From this, we see that on axis:
1) For r >> a, B_r varies as 1/r^3
2) For r << a, B_r is constant
3) For r ~ a, B_r varies, approximately, as 1/r
Don