Formula: Magnetic Dipole in a Magnetic Field Force Magnetic flux density (B-field)
$$\boldsymbol{F} ~=~ \nabla \left( \class{red}{\boldsymbol{\mu}} \cdot \class{violet}{\boldsymbol{B}} \right)$$
$$\boldsymbol{F} ~=~ \nabla \left( \class{red}{\boldsymbol{\mu}} \cdot \class{violet}{\boldsymbol{B}} \right)$$
Force
$$ \boldsymbol{F} $$ Unit $$ \mathrm{N} = \frac{\mathrm{kg} \, \mathrm{m}}{\mathrm{s}^2} $$
Force on a magnetic dipole in an external magnetic field \(\class{violet}{\boldsymbol{B}}\). The force can also be written as follows:
\[ \boldsymbol{F} ~=~ \boldsymbol{\mu} \, \nabla \class{violet}{\boldsymbol{B}} \]
So the force is determined by the gradient of the magnetic field: \(\nabla \boldsymbol{B}\). Here \(\nabla\) is the nabla operator.
Magnetic dipole moment
$$ \class{red}{\boldsymbol{\mu}} $$ Unit $$ \mathrm{A} \cdot \mathrm{m}^2 $$
Magnetic dipole moment is a measure for the "strength" of a magnetic dipole. Here the scalar product between \(\boldsymbol{\mu}\) and \(\class{violet}{\boldsymbol{B}}\) is formed.
The \(\boldsymbol{\mu}\) vector points in the same direction as the normal vector \(\boldsymbol{A}\). Dipole moment is therefore orthogonal to the surface enclosed by the current loop.
Magnetic flux density
$$ \class{violet}{B} $$ Unit $$ \mathrm{T} = \frac{\mathrm{kg}}{\mathrm{A} \, \mathrm{s}^2} $$
The external magnetic field in which the magnetic dipole is located. If the magnetic field is homogeneous, the magnetic dipole does not experience a force (but only a torque). In an inhomogeneous field, however, the dipole moves in the direction of the larger magnetic field.