Can we apply Conservation of angular momentum to a circulating charged particle in a magnetic field? In the question there is a charged particle undergoing pure circular motion at velocity v and radius r in a magnetic field of field strength B. The strength is instantaneously decreased to B/2, what would be the new velocity of the charged particle in terms of v.
Here do we consider conservation of energy thus taking the speed after the strength changed as equal to the initial speed, or do we apply the conservation of angular momentum of the charge?
Can we apply conservation of angular momentum to a charged particle undergoing circular motion in a magnetic field?
 A: I would like to say that since the magnetic force is radial, there would be no change in the tangential velocity; however, if the magnetic field is cylindrical, a sudden change in the field would cause a rapid change in flux, which would cause E field loops of very high emf.  The charged particle could experience a very large external impulse.
A: The force due to a magnetic field cannot do any work on a charged particle, because the force is perpendicular to its velocity. Thus conservation of energy can be used.
The torque is
$$\vec{r}\times \vec{F} = q\vec{r}\times (\vec{v}\times \vec{B}) = q(\vec{r}\cdot \vec{B})\vec{v} - q(\vec{r}\cdot \vec{v})\vec{B}  $$
Since the velocity, the B-field and the position vector are all perpendicular then there is no torque and angular momentum is conserved.
However, if you change the B-field then an electric field is induced which curls around the changing B-field. This E-field would be parallel to the velocity vector and thus would do work on the particle, decelerating it and decreasing its kinetic energy. This would change the speed, but the velocity vector is still perpendicular to the displacement so no torque is produced and the angular momentum is constant.
The radius of gyration is given by
$$r = \frac{mv_{\perp}}{qB}$$
and angular momentum
$$ L = \frac{(mv_\perp)^2}{qB}\ .$$
Thus, since $L$ remains constant we see that the kinetic energy must decrease by the same factor as the magnetic field. That means that the radius of gyration increases by the square root of this factor and the total magnetic flux within the "orbit" is conserved.
