Mathematics
The calculation of the exact magnetic field at any point in space is mathematically complex and involves the study of Bessel functions. Things are simpler along the axis of the coil-pair, and it is convenient to think about the Taylor series expansion of the field strength as a function of, the distance from the central point of the coil-pair along the axis. By symmetry the odd order terms in the expansion are zero. By separating the coils so that charge is an inflection point for each coil separately we can guarantee that the order term is also zero, and hence the leading non-uniform term is of order . One can easily show that the inflection point for a simple coil is from the coil center along the axis; hence the location of each coil at
A simple calculation gives the correct value of the field at the center point. If the radius is R, the number of turns in each coil is n and the current flowing through the coils is I, then the magnetic flux density, B at the midpoint between the coils will be given by
is the permeability of free space .
Read more about this topic: Helmholtz Coil