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## Electron Beam Collimation by Axial Magnetic Field

This animation shows how the electron beam is formed and collimated by a small
axial magnetic field. Without the magnetic field, the beam would diverge due to the normal
radial velocities of the electrons emitted by the cathode.
If the radial velocities are designated `v_y` and `v_z`, then the magnetic field `B_x`
causes the velociites to change as given by the following equations:

`deltav_z=e*v_yB_x/m`

`deltav_y=-e*v_zB_x/m`

where `e` is the electron charge and `m` is its mass.
This causes radial trajectories of the electrons to be circles or spirals and the circles are smaller
if `B_x` is larger.
An expression for the radius, rc, of the transverse circles is:

`r_c=p_t/(eB_x)`

where `p_t` is the magnitude of the transverse component of the momentum or

`p_t=m(v_y^2+v_z^2)^(1/2)`

The rotation frequency, wc, of the electrons is independent of their transverse speed and is

`omega_c=eB_x/m`.

From the above equations, the transverse speed is just

`v_t=omega_cr_c`

as expected for any rotational motion.
A single magnetic field line from each axial permanent magnet is plotted in blue.
In the animation program, I just set `e` and `m` to 1 but, in a real case,
if we have thermal speeds for the emitted electrons, then, with `B_x=0.001 "Tesla"`,
the radii of the circles will be of order a few millimeters which is a very manageable beam diameter.
The trajectories of a select few of the electrons are plotted in green with a variable radial magnification factor.
It is clear that, inside the magnet, the ranges of these trajectories are greatly reduced.
It is obvious that these trajectories become elongated after the electrons pass through the anode and are accelerated.
The normalized root mean square (RMS) of the beam radius is also ploted Vs x.

The parameters available are the following:

1. DC voltage level applied to the anode

2. Length of collimating magnet

3. Length of electrode elements

4. Spacing between electrode element pairs

5. Intensity of collimating magnetic field

6. Trajectory Magnification