Magnetic dipole
In electromagnetism, a magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the size of the source is reduced to zero while keeping the magnetic moment constant.
It is a magnetic analogue of the electric dipole, but the analogy is not perfect. In particular, a true magnetic monopole, the magnetic analogue of an electric charge, has never been observed in nature.
Because magnetic monopoles do not exist, the magnetic field at a large distance from any static magnetic source looks like the field of a dipole with the same dipole moment. For higher-order sources with no dipole moment, their field decays towards zero with distance faster than a dipole field does.
External magnetic field produced by a magnetic dipole moment
In classical physics, the magnetic field of a dipole is calculated as the limit of either a current loop or a pair of charges as the source shrinks to a point while keeping the magnetic moment constant. For the current loop, this limit is most easily derived from the vector potential:where μ0 is the vacuum permeability constant and is the surface of a sphere of radius.
The magnetic flux density is then
Alternatively one can obtain the scalar potential first from the magnetic pole limit,
and hence the magnetic field strength is
The magnetic field strength is symmetric under rotations about the axis of the magnetic moment.
In spherical coordinates, with, and with the magnetic moment aligned with the z-axis, then the field strength can more simply be expressed as
Internal magnetic field of a dipole
The two models for a dipole, give the same predictions for the magnetic field far from the source. However, inside the source region they give different predictions. The magnetic field between poles is in the opposite direction to the magnetic moment, while inside a current loop it is in the same direction. Clearly, the limits of these fields must also be different as the sources shrink to zero size. This distinction only matters if the dipole limit is used to calculate fields inside a magnetic material.If a magnetic dipole is formed by making a current loop smaller and smaller, but keeping the product of current and area constant, the limiting field is
where is the Dirac delta function in three dimensions. Unlike the expressions in the previous section, this limit is correct for the internal field of the dipole.
If a magnetic dipole is formed by taking a "north pole" and a "south pole", bringing them closer and closer together but keeping the product of magnetic pole-charge and distance constant, the limiting field is
These fields are related by, where
is the magnetization.
Forces between two magnetic dipoles
The force exerted by one dipole moment on another separated in space by a vector can be calculated using:or
where is the distance between dipoles. The force acting on is in the opposite direction.
The torque can be obtained from the formula