Electron Beams, Lenses, and Optics by A. B. El-Kareh

By A. B. El-Kareh

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Example text

It is clear that in a relatively weak lens where the intersection occurs outside the lens, the projective and objective cardinal points are identical. 4 Sign Convention Generally speaking, any sign convention will be satisfactory so long as it is consistent. The following is the convention we shall use in this text: (a) A plane of reference is chosen near or at the geometrical center of the lens. All distances measured on the left of this reference plane are negative, and those on the right are positive.

3). A similar derivation can lead to Laplace's equation in cylindrical coordinates. 4 Conformal Transformation 25 Let us now divide throughout by F(x)G(y). We get F(x) ox G(y) 1 oy 1 Since the two variables are independent, each term of the equation must be equal to a constant. 12) 2 _ . 16) *=o We have been able to eliminate C without loss of generality. 19) and ir = χ + y>. 22) in which the equipotential consists of two planes cutting at an angle π/η. Now suppose η = 1. Then φ = y, and we have a uniform field of force.

If we let V = Av in Eq. 131) 2 This is valid for k = 2. Hence, Eq. 132) 2 A solution is U = u. Since Eq. 138) U (\ + Μ ) <*U 2 Thus m = / ϊ ^(1Τ +Ϊ M) ) 5 2 2 " = - C l a r c t a n u + c * ( 3 1 3 9 ) where C is again a constant of integration. 141) 2 or x 2 Here, /T and K are new constants. 143) where B = aK , A = aK , and C are all arbitrary constants. Their values depend on the boundary conditions. Now for ν = 0, we must have φ = V , thus C = K . From Eq. 145) Q We still have to express // as a function of r and z.

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