MIT OpenCourseWare http ocw mit edu Electromagnetic Field Theory A Problem Solving Approach For any use or distribution of this textbook please cite as follows Markus Zahn Electromagnetic Field Theory A Problem Solving Approach Massachusetts Institute of Technology MIT OpenCourseWare http ocw mit edu accessed MM DD YYYY License Creative Commons Attribution NonCommercial Share Alike For more information about citing these materials or our Terms of Use visit http ocw mit edu terms Problems 375 the force density can be written as IBo F 1 12 3 t 0 2 2b 2 r sin ki cos 0ix 27 The total force on the permeable wire is 2r b F1r dr do f 28 We see that the trigonometric terms in 27 integrate to zero so that only the first term contributes f IB 0ol 2 2 b r dr do 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 Boi At t 0 its velocity and displacement are v t 0 vxix o0i vUoiz r t 0 xoix yoiy zoi 0 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 2mlvl 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 Boi 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 I The Magnetic Field 1 L IICOI I Vo Y X GBoi E S a c A magnetron is built with coaxial electrodes where electrons are injected from r a 4 0 with zero initial velocity Using the relations from Table 1 2 ir cos i sin 4i sin 4i 4i cos Oi show that di d4 dt di di dt r vs r do dt v6 l r What is the acceleration of a charge with velocity V rir v i d Find the velocity of the electrons as a function of radial position Hint dv dv dr dt dr dt dv dv dr dt dr di Vr dv d dr dr 2 dvr 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 gi has an initial velocity voi A magnetic field Boi is applied What Problems 377 q Bvoi x mg 4 value of Bo will keep the particle moving at constant speed in the x direction 4 The charge to mass ratio of an electron e m was first measured by Sir J J Thomson in 1897 by the cathode ray tube device shown Electrons emitted by the cathode pass through a slit in the anode into a region with crossed electric and magnetic fields both being perpendicular to the electrons velocity The end of the tube is coated with a fluorescent material that produces a bright spot where the electron beam impacts Screen a What is the velocity of the electrons when passing through the slit if their initial cathode velocity is vo b The electric field E and magnetic field B are adjusted so that the vertical deflection of the beam is zero What is the initial electron velocity Neglect gravity c The voltage V2 is now set to zero What is the radius R of the electrons motion about the magnetic field d What is e m in terms of E B and R 5 A charge q of mass m at t 0 crosses the origin with velocity vo v oi v oi For each of the following applied magnetic fields where and when does the charge again cross the y 0 plane a B Boi b B Boi c B Boi vo vo ix cose i sin6 j iy sinO Boix B Boi a x cosO b Boi B c vo vo i 378 The Magnetic Field 6 In 1896 Zeeman observed that an atom in a magnetic field had a fine splitting of its spectral lines A classical theory of the Zeeman effect developed by Lorentz modeled the electron with mass m as being bound to the nucleus by a springlike force with spring constant k so that in the absence of a magnetic field its natural frequency was wo r a A magnetic field Boi is applied Write Newton s law for the x y and z displacements of the electron including the spring and Lorentz forces b Because these equations are linear guess exponential solutions of the form e s What are the natural frequencies c Because oa is typically in the optical range wh 10 5radian sec show that the frequency splitting is small compared to wk even for a strong field of B 0 1 tesla In this limit find approximate expressions for the natural frequencies of b 7 A charge q moves through a region where there is an electric field E and magnetic field B The medium is very viscous so that inertial effects are negligible pv q E vxB where 6 is the viscous drag coefficient What is the velocity of and the charge Hint vxB xB v B B B v B B q f E B v 8 Charges of mass m charge q and number density n move through a conducting material and collide with the host medium with a collision frequency v in the presence of an electric field E and magnetic field B a Write Newton s first law for the charge carriers along the same lines as developed in Section 3 2 2 with the addition of the Lorentz force b Neglecting particle inertia and diffusion solve for the particle velocity v c What is the constitutive law relating the current density J qnv to E and B This is the generalized Ohm s law in the presence of a magnetic field d What is the Ohmic conductivity r A current i is passed through this material in the presence of a perpendicular magnetic field A resistor RL is connected across the terminals What is the Hall voltage See top of page 379 e What value of RL maximizes the power dissipated in the load Problems 379 Section 5 2 9 A point charge q is traveling within the magnetic field of an infinitely long line current I At r ro its velocity is v t 0 Vrir Voi vzoiz Its subsequent velocity is only a function of r a What is the velocity of the charge as a function of position Hint See Problem 2c and 2d …
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