Dipole force

The dipole force represents the interaction between a magnetic or electric dipole and an external electromagnetic field .

After the Coulomb force, it is the second-leading order of the force exerted by an electric field on any charge distribution and - in the absence of magnetic monopoles - the leading order of the force exerted by a magnetic field on any current distribution.

Derivation

In general, the force of an electric field on a charge distribution with the charge density is given by ${\displaystyle {\vec {F}}}$ ${\displaystyle {\vec {E}}}$ ${\displaystyle \rho ({\vec {x}})}$ ${\displaystyle \rho }$

${\displaystyle {\vec {F}}=\int {\vec {E}}_{\text{ext}}({\vec {x}})\,\rho ({\vec {x}})\,\mathrm {d} ^{3}x}$

If one develops the external electric field in a Taylor series :

${\displaystyle {\vec {E}}_{\text{ext}}({\vec {x}})\approx {\vec {E}}_{\text{ext}}({\vec {x}}_{0})+\left(({\vec {x}}-{\vec {x}}_{0})\cdot {\vec {\nabla }}\right){\vec {E}}_{\text{ext}}{\big |}_{{\vec {x}}_{0}}}$

with the Nabla operator as a measure of the inhomogeneity of the field between two points, ${\displaystyle {\vec {\nabla }}}$

so it results with

• the definition of the electric dipole as the first moment of the charge distribution and ${\displaystyle {\vec {D}}({\vec {x}}_{0})=\int ({\vec {x}}-{\vec {x}}_{0})\,\rho ({\vec {x}})\,\mathrm {d} ^{3}x}$
• of the total charge : ${\displaystyle Q=\int \rho ({\vec {x}})\,\mathrm {d} ^{3}x}$

${\displaystyle \Rightarrow {\vec {F}}\approx Q\,{\vec {E}}_{\text{ext}}({\vec {x}}_{0})+\left({\vec {D}}\cdot {\vec {\nabla }}\right){\vec {E}}_{\text{ext}}{\big |}_{{\vec {x}}_{0}}}$

Correspondingly, the following applies to the interaction of a magnetic field with a current distribution:

${\displaystyle {\vec {F}}({\vec {x}}_{0})\approx \left({\vec {\nabla }}\otimes {\vec {B}}\right){\big |}_{{\vec {x}}_{0}}{\vec {m}}({\vec {x}}_{0})}$

With

• the magnetic flux density ${\displaystyle {\vec {B}}}$
• the magnetic moment ${\displaystyle {\vec {m}}}$
• the dyadic product . ${\displaystyle \otimes }$

A dipole force therefore only acts in an inhomogeneous external field and is only a good approximation if the electric or magnetic field does not vary too much with the location.

application

In the case of a charge distribution without an intrinsic dipole moment, an external electric field can induce a dipole with the dipole moment . This is given by paraelectrics and dielectrics ${\displaystyle {\vec {p}}}$

${\displaystyle {\vec {p}}=\alpha {\vec {E}},}$

where the parameter is called polarizability . ${\displaystyle \alpha }$

Depending on the sign of its polarizability, a substance is moved by the dipole force in the direction of increasing or decreasing field strength . In the range of resonance frequencies , the polarizability of atoms changes sign. Therefore, by detuning the field that z. B. is irradiated by a laser , control whether the atom is moved by the dipole force in the direction of the intensity minimum or maximum of the field. In dielectrophoresis , the dipole force is used to manipulate particles or particles such as B. Viruses used.

Individual evidence

1. ↑ dipole force. In: Techniklexikon.net. Hamra Webservices, accessed January 19, 2013 . 
2. ↑ Laser cooling ( Memento of the original from June 13, 2007 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. , P. 18.