pervect
Aug19-04, 12:54 AM
here seems to be some interest in magnetic dipoles, such as spinning electrons, and current loops. So I thought I would start a thread and present some of the relevant equations that describe the forces and fields generated by magnetic dipoles. These equations are very similar to those for electric dipoles, BTW.
A current loop with an area A and carrying a current i has a
magnetic dipole moment of \mu = i A . The dipole moment is sometimes expressed as a vector \vec{\mu} in which case the vector is perpendicular to the area A.
Some useful properties of the diople moment are given below
Torque generated by an external field \vec{\mu} \times \vec{B}
Energy in an external field -\vec{\mu} \cdot \vec{B}
Field from dipole at distant points along axis |B| = \frac {\mu_0}{2 \pi} \frac {\mu}{r^3}
Field from dipole at distant points along bisector |B| = \frac {\mu_0}{4 \pi} \frac {\mu}{r^3}
Field from dipole, vector form \vec{B} = \frac {\mu_0 \mu}{4 \pi r^3} (2 cos(\theta) \vec{r} + sin(\theta) \vec{\theta})
Net force on dipole from a constant magnetic field zero
Net force on a dipole from a varying magnetic field \nabla (\vec{\mu} \cdot \vec{B})
Note that the force between two dipoles will drop off with the 4th power of the distance - as the field generated by a dipole is proportional to 1/r^3, the gradient of the field is proportional to 1/r^4, and the force will be the dipole moment multiplied by the field gradient.
A current loop with an area A and carrying a current i has a
magnetic dipole moment of \mu = i A . The dipole moment is sometimes expressed as a vector \vec{\mu} in which case the vector is perpendicular to the area A.
Some useful properties of the diople moment are given below
Torque generated by an external field \vec{\mu} \times \vec{B}
Energy in an external field -\vec{\mu} \cdot \vec{B}
Field from dipole at distant points along axis |B| = \frac {\mu_0}{2 \pi} \frac {\mu}{r^3}
Field from dipole at distant points along bisector |B| = \frac {\mu_0}{4 \pi} \frac {\mu}{r^3}
Field from dipole, vector form \vec{B} = \frac {\mu_0 \mu}{4 \pi r^3} (2 cos(\theta) \vec{r} + sin(\theta) \vec{\theta})
Net force on dipole from a constant magnetic field zero
Net force on a dipole from a varying magnetic field \nabla (\vec{\mu} \cdot \vec{B})
Note that the force between two dipoles will drop off with the 4th power of the distance - as the field generated by a dipole is proportional to 1/r^3, the gradient of the field is proportional to 1/r^4, and the force will be the dipole moment multiplied by the field gradient.