Maxwell's analytical formula for the mutual inductance of two coaxial circular loops

mutual_inductance.py

from numpy import sqrt
from scipy.special import ellipe, ellipk
import pint

ureg = pint.UnitRegistry()

def mutual_inductance(r1: float, r2: float, h: float, length_units: str = "m") -> pint.Quantity:
    """Calculates the mutual inductance between two coaxial circular loops.
    
    All lengths, ``r1``, ``r2``, and ``h`` are assumed to be given in ``length_units``.
    
    Reference: https://nvlpubs.nist.gov/nistpubs/bulletin/08/nbsbulletinv8n1p1_A2b.pdf, page 6.
    
    Args:
        r1: The radius of the first circle
        r2: The radius of the second circle
        h: The distance between the two circles along their shared axis
        length_units: The units in which the first three arguments are given
        
    Returns:
        The mutual inductance as a pint Quantity
    """
    # k is the modulus of the elliptic integrals
    k = 2 * sqrt(r1 * r2) / sqrt((r1 + r2)**2 + h**2)
    # m = k**2 is the argument accepted by the scipy functions
    m = k**2
    K = ellipk(m)
    E = ellipe(m)
    return ureg("mu_0") * ureg(length_units) * sqrt(r1 * r2) * ((2 / k - k) * K - 2 / k * E)