Diamagnetism — Question 4

Question 4

The vertical field-gradient product sets a levitation balance

A small weakly diamagnetic sample of density rho has susceptibility chi less than zero. The applied field magnitude varies with upward coordinate z. Neglect air buoyancy and demagnetization. Use $F_z=\chi V B(dB/dz)/\mu_0$.

1. Derive the field-gradient product required to balance gravity.

2. Evaluate it for $\rho=1000\,\mathrm{kg/m^3}$, $\chi=-9.0\times10^{-6}$, $g=9.81\,\mathrm{m/s^2}$, and $\mu_0=4\pi\times10^{-7}\,\mathrm{H/m}$. Determine the sign of dB/dz if B is positive.

Question 4 - Solution

The vertical field-gradient product sets a levitation balance

(a) Method and calculation

$$F_z-\rho Vg=0\Rightarrow\boxed{B\frac{dB}{dz}=\frac{\mu_0\rho g}{\chi}}.$$$$V\text{ cancels: the required product is independent of sample volume.}$$

(b) Interpretation and further calculation

$$B\frac{dB}{dz}=\frac{(4\pi\times10^{-7})(1000)(9.81)}{-9.0\times10^{-6}}=\boxed{-1.370\times10^3\,\mathrm{T^2/m}}.$$$$B>0\Rightarrow\boxed{dB/dz<0}.$$

Physical check

Force balance alone does not establish stability; the spatial derivative of the net force must also be checked.