Axial Field of a Finite Solenoid
|The magnetic field in the central region of a solenoid can be approximated as a simple polynomial involving Legendre polynomials. Montgomery and Terrell state that "out to within a few percent of the inner radius of the coil, the field can be found quite accurately with only a few ... terms."|
Solenoid in cross section view.
polynomial can be written as:
BO is the central field strength of the solenoid,
and the first four even numbered error coefficients, e n, are:
, , , ,
r1, r2, and l are inner, and outer coil radii, and coil length, respectively.
d is the distance from the coil center to the field measurement point,
q is the angle between the measurement point position vector and the coil axis,
and the corresponding Legendre polynomials, rn, are:
|But What About Off-Axis Radial Field Components?||According
is a corresponding series for obtaining the radial
component of the magnetic field in the central region of
a solenoid, which looks like this:
Where r ln are derivatives of the even Legendre polynomials, such as:
It looks wonderful, except for the fact that the computed radial fields are wrong. I don't know why they're wrong, since the math is a little over my head. If you know the answer, please drop me a line.
formulas on this page are adapted from:
SOME USEFUL INFORMATION FOR THE DESIGN OF AIR-CORE SOLENOIDS by D.Bruce Montgomery and J. Terrell., published November, 1961, under Air Force Contract AF19(604)-7344.
Montgomery and Terrell, in turn, credit:
M.W. Garrett, J. Appl. Phys. 22, 9, Sept. 1951.