Solution

We will use the same invariant.

At the beginning, the angular momentum is zero.

Now we are talking about the minimum width condition, obviously, since this is the minimum, the radial component of the velocity is 0. But also the minimum condition is that the longitudinal velocity is 0, because we are considering the boundary case when the beam remains focused after the transition, which means:

$$\frac{eB_1R^2}{2} = \frac{eB_2r^2}{2}$$

$$r = R\sqrt{\frac{B_1}{B_2}}$$

Savchenko, apparently, has a typo.