Helmholtz spacing of thin rectangular magnetic field coils

In this Note, we discuss the Helmholtz spacing for a pair of thin rectangular coils of arbitrary aspect ratio and consider how best to use such coils to compensate for Earth's magnetic field along the coils' Cartesian symmetry axes. Such coils are frequently used in conjunction with charged-particle beam machines. The Helmholtz spacing varies non-monotonically between that for square coils and that for four optimally spaced infinite wires. We consider other coil spacings that extend the length over which the field varies by less than some tolerance along the Cartesian symmetry axes. The calculations also provide a convenient means to evaluate when the length of the coils is sufficiently long to be considered infinite at the center point within a fixed tolerance.