The magnetic field from a magnetic dipole along a given line, and in any given direction can be described by the following basis functions:
which are known as Anderson functions.[1]
Definitions:
- is the dipole's strength and direction
- is the projected direction (often the Earth's magnetic field in a region)
- is the position along the line
- points in the direction of the line
- is a vector from the dipole to the point of closest approach (CPA) of the line
- , a dimensionless quantity for simplification
The total magnetic field along the line is given by
where is the magnetic constant, and are the Anderson coefficients, which depend on the geometry of the system. These are[2]
where and are unit vectors (given by and , respectively).
Note, the antisymmetric portion of the function is represented by the second function. Correspondingly, the sign of depends on how is defined (e.g. direction is 'forward').
The total field measurement resulting from a dipole field in the presence of a background field (such as earth magnetic field) is
The last line is an approximation that is accurate if the background field is much larger than contributions from the dipole. In such a case the total field reduces to the sum of the background field, and the projection of the dipole field onto the background field. This means that the total field can be accurately described as an Anderson function with an offset.