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Magnetic Dipoles & eq's

  1. Aug 19, 2004 #1

    pervect

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    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 [tex] \mu = i A [/tex]. The dipole moment is sometimes expressed as a vector [tex] \vec{\mu}[/tex] 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 [tex] \vec{\mu} \times \vec{B}[/tex]

    Energy in an external field [tex] -\vec{\mu} \cdot \vec{B}[/tex]

    Field from dipole at distant points along axis |B| = [tex] \frac {\mu_0}{2 \pi} \frac {\mu}{r^3}[/tex]

    Field from dipole at distant points along bisector |B| = [tex]\frac {\mu_0}{4 \pi} \frac {\mu}{r^3} [/tex]

    Field from dipole, vector form [tex] \vec{B} = \frac {\mu_0 \mu}{4 \pi r^3} (2 cos(\theta) \vec{r} + sin(\theta) \vec{\theta})[/tex]

    Net force on dipole from a constant magnetic field zero

    Net force on a dipole from a varying magnetic field [tex] \nabla (\vec{\mu} \cdot \vec{B}) [/tex]

    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.
     
    Last edited: Aug 19, 2004
  2. jcsd
  3. Aug 19, 2004 #2

    krab

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    "Constant" or "varying" usually means w.r.t. time. I think here you mean w.r.t. space, so common usage is "uniform" or "non-uniform".
     
  4. Aug 20, 2004 #3

    pervect

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    Yes, thats what I mean. To develop a net force, one needs the field to be different at the two ends of the dipole, which means that the field must be varying in space.
     
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