By Professor Dr. Sc. Sergey Ivanovich Molokovsky, Professor Dr. Sc. Aleksandr Danilovich Sushkov (auth.)

**Intense Ion and Electron Beams** treats extreme charged-particle beams utilized in vacuum tubes, particle beam expertise and experimental installations comparable to loose electron lasers and accelerators. It addresses, between different issues, the physics and uncomplicated concept of excessive charged-particle beams; computation and layout of charged-particle weapons and focusing platforms; multiple-beam charged-particle structures; and experimental equipment for investigating severe particle beams. The insurance is thoroughly balanced among the physics of excessive charged-particle beams and the layout of optical platforms for his or her formation and focusing. it may be urged to all scientists learning or making use of vacuum electronics and charged-particle beam expertise, together with scholars, engineers and researchers.

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**Example text**

2 ∂z ε0 The problem is set up in the following way. 3 Calculation of Electrostatic Field: Cauchy Problem 35 as well as the space-charge density distributions ρ (r, z) are supposed to be known. A solution of this problem can be found by power series expansion of the potential U (r, z) and space-charge density ρ (r, z): U (r, z) = f0 (z) + f2 (z) r2 + f4 (z) r4 + . . + f2n (z) r2n ρ (r, z) = ρ0 (z) + ρ2 (z) r2 + ρ4 (z) r4 + . . + ρ2n (z) r2n . 27) where f0 (z) – function of the axial potential distribution, f0 (z) and f0IV (z) – the second and the fourth derivatives of f0 (z) on the z-coordinate.

59) Bz0 r r r3 (−1)n Br = − Bz0 + B − . . n! 60) r The magnetic ﬂux ψ = 2π 0 Bz rdr can be also presented by a series. 48). This is reduced to computation of volume integrals. Let us consider the calculation of axially symmetric ﬁelds produced by solenoids and ring-shaped permanent magnets. 1 Field of a Solenoid The geometry of a solenoid is shown in Fig. 8. A square mesh divides it into a set of elementary ring volumes, which results in an elementary current ∆Ik = δk ∆Sk , where ∆Sk – cross section of the elementary volume, δk – average current density.

60) r The magnetic ﬂux ψ = 2π 0 Bz rdr can be also presented by a series. 48). This is reduced to computation of volume integrals. Let us consider the calculation of axially symmetric ﬁelds produced by solenoids and ring-shaped permanent magnets. 1 Field of a Solenoid The geometry of a solenoid is shown in Fig. 8. A square mesh divides it into a set of elementary ring volumes, which results in an elementary current ∆Ik = δk ∆Sk , where ∆Sk – cross section of the elementary volume, δk – average current density.