Lenses fundamental lens equation

In order to derive the full properties of a lens system, the fundamental lens equation has to be solved for individual rays subject to the influence of the given potential d>(z). However, without going into the details for such a procedure, some aspects, which are important for axial-symmetrical electrostatic lenses with paraxial rays, can be listed by using results from light optics ... 

The passage of electrons or other particles with charge q and mass m through an electrostatic lens system is governed by their motion under the action of the electric field. In the case considered here, cylindrical symmetry around the optical axis (z-axis) and paraxial rays will be assumed. Of the cylindrical coordinates only the transverse radial coordinate p and the distance coordinate z are of relevance, and the electrostatic potential of the lens is given by q>(p, z). As shown in Section 10.3.1, in the paraxial approximation the potential q>(p, z) is fully determined by the potential symmetry axis. Hence, the equations of motion and the fundamental differential equation of an electrostatic lens depend only on this potential. The fundamental lens equation is given by (see equ. (10.38))... 

Theories of interparticle forces play a fundamental part in many theoretical aspects of colloidal behaviour. It is therefore of great importance to have experimental evidence for the validity of these theories. One approach to this is to study the forces between macroscopic objects, to which the same theoretical equations should apply. Since these forces arc exceedingly small until the bodies come into very close proximity, work in this area has faced considerable experimental difficulties. Experiments on the force between two plates and between a plate and a lens have been of limited validity because of the difficulty in achieving adequate surface smoothness and in completely eliminating dust. The... 

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