Electrons in an inhomogeneous magnetic field

The following relationships can be observed in the experiment (for I ≠ 0 A): At low accelerating voltages (for example 1.0 kV and 1.5 A) and high coil current, however, it is also possible to make the shadow of the cross almost disappear, or to focus all electrons on the point in the center of the screen. If the coil current is increased further, a shadow of the cross is again visible on the screen. However, this shadow is reversed compared to before.

The coil as a magnetic lens for electrons

The behavior can be explained if one understands the coil as a magnetic lens for electrons. The hole in the anode of the electron gun corresponds to a pixel P from which divergent light rays emanate. After passing through the coil, which in the model corresponds to a converging lens, the electron beams are again focused on a point P'. In ray optics, all light rays are accordingly imaged from a point P to an image point P'. However, the paths of the electrons there are not straight lines, but complex curved paths.
Bahn eines Elektrons beim Flug durch eine magnetische Linse
If parallel electron beams fall through the magnetic lens, they are united or cross at the focal point F at a distance of the focal length $f$. As in optical imaging, when the screen is behind the focal point of the lens, the image of an object is laterally reversed. Additionally, magnetic lenses twist the image even further.

Special features of magnetic lenses

The focal length $f$ of a magnetic lens can be changed by the coil current. In this case, a lens does not have to be physically replaced as in optics. Further, the focal length of a lens at constant coil current is different for particles of different velocities or masses.
A thin magnetic lens has the refractive power for electrons$$\frac{1}{f}=\frac{e^2}{8\cdot m_e\cdot E_{\rm{kin}}}\int {B_z}^2 \rm{d}z$$

Use in electron microscopes

In technology, magnetic lenses are used in many ways. They are particularly important in electron microscopes (for developments and advances in this field, Ruska, Rohrer and Binning received the Nopel Prize for Physics in 1986). This uses coils in a specially shaped iron shell that generate very strong inhomogeneous magnetic fields.
Aufbau einer kurzen magnetischen Linse
A simulation of electron trajectories as they pass through a magnetic lens and images using magnetic lenses is provided by Matthias Borchardt.