By Pierre Grivet, P.W. Hawkes
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Additional info for Electron Optics
Grad(log/0. We obtain an analogy between the two equations, and hence between 0(x, y) in the beam and V(x, y) in the tank, if we write £ • grad (log/*) = - * - 7 = r . \0 In a tank the bottom of which is shaped in such a way that h varies and this relation is satisfied (see Fig. 10 a), the potential distribution across the surface of the electrolyte (h = 0) will provide a chart of the potential 0(x,y). 52 ELECTRON OPTICS FIG. 10a. Cross-section of the tank with a shaped bottom, as used by MussonGenon.
1) The electrode system; (2) the same system after transformation. ) (£-&) K / TvT - 1 ) (ttn/w-1) (£~£n) , <*£ + £. The points & = ± oo do not enter into this formula. Consider a simple example: the outline of the parallel plate condenser shown in Fig. 8b. From symmetry considerations, we need only calculate the function u(x, y) in a simpler system consisting of two electrodes: one a plane, held at potential 4> = K(), and the other a half-plane, distant y = h from the first for x > 0, and held at
After interpolation between the values found at the meshpoints, the equipotentials can be inserted. } as boundary conditions. These methods were originally employed without the assistance of elec tronic computers, and a great deal of mathematical work went into determining rules for increasing the speed of "convergence" towards the final solution. Rotationally symmetric systems. Here, only the r and z coordinates are involved. 26), when y and x are replaced by r and z respectively. We first consider a point M such that r0 = ph (Fig.