- #1

wrobel

Science Advisor

- 743

- 455

This is not a very famous effect but it is really amazing. Consider a pendulum which consists of a massless rod of length ##r## and a point of mass ##m##; the system is in the standard gravitational field ##\boldsymbol g##. So the point ##m## moves on the sphere of radius ##r##.

It is clear, the equilibrium when the point rests in the North Pole of the sphere is unstable.

Introduce a Cartesian inertial frame ##OXYZ## with origin in the point of suspension and the axis ##OZ## is vertical such that ##\boldsymbol g=-g\boldsymbol e_z##.

Now let us switch on a Lorentz force ##\boldsymbol F=\boldsymbol B\times\boldsymbol v## which acts on ##m##. The vector ##\boldsymbol B=B\boldsymbol e_z## is constant.

Theorem. Assume that ##B## is sufficiently big:

##\frac{B^2}{8m}>\frac{mg}{2r}##

then the North Pole equilibrium is stable.