Motion of charged particles in complex fields > Motion in a homogeneous magnetic field > Problem 10.1.28

Statement

$10.1.28.$ Determine the minimum radius that an electron beam can have when moving from the field with induction $B_1$ to the field with induction $B_2$. The axes of symmetry of the transition field and the beam coincide. The beam radius in the B1 field is $R$, and the beam velocity in the $B_1$ field is parallel to the induction.

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.

Answer

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