Paraxial calculations
BiotSavart implements paraxial field calculations for loops and solenoids using the method of R.H. Jackson [1]. This allows fast and accurate computuation of magnetic fields when the probe point is sufficiently close to the axis of the coil. In the case of a solenoid the expansion can be quite accurate right up to the inner radius of the solenoid.
The paraxial calculation is used when the distance of the probe point to the axis is the coil is less than paraxialR. This is found on the Calc tab of the edit dialog for axially symmetric conductors.
Implementation details
Jackson derives a series expansion for the field about the axis. Each term in the series involves a polynomial obtained from lower-order polynomials by a recurrence relation. As the order increases the integer coefficients of the polynomial become seriously large. Here are the polynomials to twentieth order in the case of a current loop (click on the image for a closer look):
In the above table the polynomials Pn(x) are written as an pn(x) with the greatest common factor an factored out; still the coefficients in pn(x) are large. However, with careful implementation the expansion is stable and keeping many terms in the expansion allows accurate calculation of magnetic field and gradient even quite far from the axis.
References
[1] R.H. Jackson, Off-axis expansion solution of Laplace's equation: Application to accurate and rapid calculation of coil magnetic fields. IEEE Transactions on Electron Devices, 46(5):1050-1062, May 1999.
