The Magnetic Dipole Moment — Question 7

Question 7

A field gradient and a guide can provide stable axial dipole equilibrium against gravity

A dipole of mass m and fixed upward moment mu moves on a frictionless vertical guide that also fixes its orientation. Along the guide, $B_z(z)=B_0+Gz-Hz^2/2$, with H positive. Gravity is downward with acceleration g. The expression specifies only the on-axis field, not its off-axis components.

1. Find the equilibrium height and small-oscillation angular frequency along the guide.

2. Evaluate them for m=.0200 kg, mu=.500 ampere square metre, G=2.00 T/m, H=10.0 T/m squared and g=9.81 m/s squared.

Question 7 - Solution

A field gradient and a guide can provide stable axial dipole equilibrium against gravity

(a) Method and calculation

$$U(z)=mgz-\mu B_z(z),\quad F_z=\mu(G-Hz)-mg.$$$$z_{\mathrm{eq}}=\frac{G-mg/\mu}{H},\quad F_z=-\mu H(z-z_{\mathrm{eq}}),\quad\Omega=\sqrt{\mu H/m}.$$

(b) Interpretation and further calculation

$$z_{\mathrm{eq}}=\frac{2.00-(.0200)(9.81)/.500}{10.0}=.16076\,\mathrm m.$$$$\Omega=\sqrt{(.500)(10.0)/.0200}=15.8114\,\mathrm{rad/s}.$$

Physical check

This establishes stability only along the imposed guide; it is not a claim of unconstrained three-dimensional magnetic levitation.