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Electromagnetic Field Theory: A Problem Solving Approach Part 41

Electromagnetic Field Theory: A Problem Solving Approach Part 41. Electromagnetic field theory is often the least popular course in the electrical engineering curriculum. Heavy reliance on vector and integral calculus can obscure physical phenomena so that the student becomes bogged down in the mathematics and loses sight of the applications. This book instills problem solving confidence by teaching through the use of a large number of worked problems. To keep the subject exciting, many of these problems are based on physical processes, devices, and models. This text is an introductory treatment on the junior level for a two-semester electrical engineering. | Problems 375 the force density can be written as ZB0 Z 1 2 z z0 _ F -2 ---- 2 2 r sin d iy cos d ix 27 770 2 770 The total force on the permeable wire is f i i F rdrtty 28 J A o o We see that the trigonometric terms in 27 integrate to zero so that only the first term contributes ZB0 f2 f6 7T r dr d p ITO J o r O IBol 29 The total force on the wire is independent of its magnetic permeability. PROBLEMS Section 5-1 1. A charge q of mass m moves through a uniform magnetic field Boh. At t 0 its velocity and displacement are v t 0 vlOix v oi Vzoiz r t 0 xoix yoi z0i2 a What is the subsequent velocity and displacement b Show that its motion projected onto the xy plane is a circle. What is the radius of this circle and where is its center c What is the time dependence of the kinetic energy of the charge 2 t v 2 2. A magnetron is essentially a parallel plate capacitor stressed by constant voltage Vo where electrons of charge e are emitted at x 0 y 0 with zero initial velocity. A transverse magnetic field Boiz is applied. Neglect the electric and magnetic fields due to the electrons in comparison to the applied field. a What is the velocity and displacement of an electron injected with zero initial velocity at t 0 b What value of magnetic field will just prevent the electrons from reaching the other electrode This is the cut-off magnetic field. 376 The Magnetic Field M c c A magnetron is built with coaxial electrodes where electrons are injected from r a 0 with zero initial velocity. Using the relations from Table 1-2 show that ir cos sin sin j ix cos i dir_. d f dt dt r di fr . dd v t . ---- lr ---- -------li- dt dt r What is the acceleration of a charge with velocity v vrir v i d Find the velocity of the electrons as a function of radial position. Hint dvT_dvTdr_ dvr d . 2 dt dr dt Vt dr drzVr dv dv dr dv --- ------- fr---- dt dr dt dr e What is the cutoff magnetic field Check your answer with b in the limit b a s where s a. 3. A charge q of mass m within a gravity field .