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Formula: **Magnetic Dipole in a Magnetic 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.