Theory
            A magnetic dipole in the presence of an external magnetic field has a potential energy:
U = -msB.  The dipole will experience a force in the direction of decreasing potential energy:
F = -sU.  If the field is constant in time or slowly varying, the dipole will align itself with the field, and the force is:  F = mB
 
            The dipole will feel a force in the direction of increasing field magnitude.  Therefore the dipole would be stably confined at any local maximum in B because there would be a restoring force in every direction.  Unfortunately such a local maximum would require more magnetic field lines entering a region than leaving that region which would require a magnetic monopole, and Maxwell's equations forbid such monopoles.

            Since no maximum in field strength can occur in free space, it is not possible to confine a dipole using constant (in time) magnetic fields.  It is possible however to use an oscillating quadrupole potential to magnetically confine the dipole.

In our experiment we used an axially symmetric magnetic field with the equation:
 
        Bz(r,z,t) = Boz + 1/2 (kdc + kac cos Wt)(z2 - r2/2)
    Br(r,z,t) = - (kdc + kac cos Wt)zr
 
            If the constant Boz is much larger than the second term, the magnitude of the field can be approximated by Bz and will be constant in time.  The magnetic dipole will align itself with the large Bz and will experience a force mBz.  The resulting equations of motion of the dipole are in the form of Mathieu equations:

 
where (m is the mass of the particle):    az = 4mkdc/mW2 ,  q = 2mkac/mW2 ,  ar = -az/2 ,  qr = -qz/2,  and  T = Wt/2
            Solutions to the Mathieu equations can be found in literature on Mathematical Equations, but the significance of these equations is that their solutions are bounded for certain values of a and q.  There are certain values of az and qz , for which the solutions are bounded in both the axial and radial coordinates, given by the following graph:
        We chose values of current and separation for our coils which gave curvatures of the B field that have az and qz within the region above.

<<Intro -- Apparatus>>  

Main Menu