bc thinks K&J should post the limits to the region of approximation.>
1) The circular loop
As I mentioned in my last post on this topic, Jackson, "Classical
Electrodynamics", 1962, pp 141-145, works out the exact magnetic field (in
cgs units) for a circular loop of radius a and constant current I. On axis,
at a distance r from the center of the loop, the solution is simple with a
spatially varying part given by:
f(r/a) = 1/(a(1 + r^2/a^2)^1.5)
It's easy to see that:
For r/a << 1, f(r/a) ~ 1/a or constant.
For r/a >> 1, f(r/a) ~ a^2/r^3.
It's also clear that for intermediate values of r/a, f(r/a) must fall off
from a constant behavior to the far field 1/r^3 dependence. However, it
wasn't clear to me just where the dependence would be 1/r and then 1/r^2 and
how wide these regions would be. I did a simple and crude graphical
analysis to approximately find these regions of r/a. I found that:
For r/a ~ 0.4 to 1.3, f(r/a) ~ 1/r
For r/a > 1.3 to 2.5, f(r/a) ~ 1/r^2
2) My earlier comments that the field should fall off like 1/r^2 in the
"near" field near a disc or bar magnet.
In the narrow sense above (r/a ~ 0.4 to 1.3) this statement seems to me to
be correct for a disc. However, a bar magnet is a 3-dimensional object.
Looking back at my old copy of Scott, "The Physics of Electricity and
Magnetism", 2nd Edition, 1972, he describes how at distances from a bar
magnet (or cylindrical electret) large compared to the radius a, but not
large with respect to the length L of the magnet, one can approximate the
outside field by assuming a point magnetic pole (or point electric charge)
at each end. If one then increases the distance a "little more" (so the
point charge approximation is still valid, but the dipole approximation is
not yet a good one, I would think that the field would fall off like 1/r^2
in some axial region where the field of the nearer pole predominated. I
haven't tried to calculate the length of such a region. However, this is
what I was thinking (but not well remembering) in my original 1/r^2 remarks.
Don
Dr. Donald Polvani
Adjunct Faculty, Physics
Anne Arundel Community College