Grad B drift

A particle moving in a gradually changing magnetic field will develop a drift velocity in a direction perpendicular to both the magnetic field and the direction in which it grows. Let us consider a charged particle going round in a non uniform magnetic field. When it feels a stronger magnetic field its curvature radius would be smaller, and vice versa. Thus, each in cycle the center of each circle will be separated by the difference of radii

$\Delta r = \frac{m c v}{q B^2} \Delta B$

Multiplying by the frequency we can obtain the average velocity of the drift

$v_d = \omega \Delta r = v \frac{\Delta |B|}{|B|}$

If the gradient in the magnitude of the magnetic field is constant, then

$v_d = v \frac{\Delta B}{B} \approx \frac{\nabla |B|}{|B|} r \approx \frac{m c v^2 \nabla |B|}{q |B|^2}$