Geometrical Electron Optics 7 



moving electron is at right angles both to the direction of the 

 field and to the direction of motion. The efifect is a twisting or 



Equipotontial surfaces 



Fig. 1. Example of electrostatic lens 



rotation of the electron trajectories around the axis, as shown 

 in figure 2, an effect which has no counterpart in ordinary 

 optics. 



Forces of this kind have not been considered by Hamilton. 

 However, starting from pioneer work by K. Schwarzschild, 

 W. Glaser -^ could show that the motion of electrons in a mag- 

 netic field can be also derived from Fermat's principle by means 

 of a suitable refractive index. This is of a very unfamiliar type, 

 as it depends not only on the value of the velocity, but also on 

 its direction, and little is gained by using it. Fortunately in 

 the case of axial symmetry, it can be replaced by a much simpler 

 law of a more familiar type. 



This simplification is obtained by abstracting from the rota- 

 tion around the axis and considering, instead of the trajectories, 

 their projections by coaxial circles on a meridian plane, i.e., on a 

 plane passing through the axis. The rotation obeys a simple 

 law. Let us call M the magnetic flux through a coaxial circle 

 of radius r, and let z't be the tangential velocity of an electron. 

 If the electron proceeds from a position 1 to a position 2, its 

 angular momentum iiirz'i around the axis will sufter a change 



(>nrvt)i — (inrvt)2 =2Z? (^^^ — ^^^^ 



(2) 



