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 375the force density can be written asIBo 12(3t + -0)F 1-, (2b2)2 r (sin ki, +cos 0ix) (27)The total force on the permeable wire is2r bf= F1r dr do (28)We see that the trigonometric terms in (27) integrate to zeroso that only the first term contributes:IB0ol 2, bf = 2 r dr do= IBol (29)The total force on the wire is independent of its magneticpermeability.PROBLEMSSection 5-11. A charge q of mass m moves through a uniform magneticfield Boi,. At t = 0 its velocity and displacement arev(t = 0) = vxix + o0i•++ vUoizr(t = 0) = xoix + yoiy + zoi0(a) What is the subsequent velocity and displacement?(b) Show that its motion projected onto the xy plane is acircle. What is the radius of this circle and where is its center?(c) What is the time dependence of the kinetic energy ofthe charge 2mlvl 2?2. A magnetron is essentially a parallel plate capacitorstressed by constant voltage Vo where electrons of charge -eare emitted at x = 0, y = 0 with zero initial velocity. A trans-verse magnetic field Boi, is applied. Neglect the electric andmagnetic fields due to the electrons in comparison to theapplied 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 elec-trons from reaching the other electrode? This is the cut-offmagnetic field.-I(a)(c) A magnetron is built with coaxial electrodes whereelectrons are injected from r = a, 4 = 0 with zero initial veloc-ity. Using the relations from Table 1-2,ir = cos 4i. + sin 4i,i, = -sin 4i, +cos Oi,show thatdi, .d4 vs.dt di rdi .do v6.--= '-r- ---- ldt dt rWhat is the acceleration of a charge with velocityV = rir, + v$i,?(d) Find the velocity of the electrons as a function of radialposition.Hint:dv, dv, dr dv, d 2dt dr dt Vr dr drdv, dv dr dvrdt dr di ' dr(e) What is the cutoff magnetic field? Check your answerwith (b) in the limit b = a + s where s << a.3. A charge q of mass m within a gravity field -gi, has aninitial velocity voi.. A magnetic field Boi, is applied. What376 The Magnetic FieldI _ '-1+ ILVoY./---,. XGBoiý-E S--_____~IICO.Problems 377qBvo --i•x4 value of Bo will keep the particle moving at constant speed inmg the x direction?4. The charge to mass ratio of an electron e/m was firstmeasured by Sir J. J. Thomson in 1897 by the cathode-raytube device shown. Electrons emitted by the cathode passthrough a slit in the anode into a region with crossed electricand magnetic fields, both being perpendicular to the elec-trons velocity. The end of the tube is coated with a fluorescentmaterial that produces a bright spot where the electron beamimpacts.Screen(a) What is the velocity of the electrons when passingthrough the slit if their initial cathode velocity is vo?(b) The electric field E and magnetic field B are adjusted sothat the vertical deflection of the beam is zero. What is theinitial electron velocity? (Neglect gravity.)(c) The voltage V2 is now set to zero. What is the radius Rof 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 withvelocity vo = v.oi +v,oi,. For each of the following appliedmagnetic fields, where and when does the charge again crossthe y = 0 plane?(a) B=Boi.(b) B = Boi,(c) B = Boi.vo= vo[ix cose + i , sin6](a) B = Boix(b) B = Boi,(c)B = Boi•j vo= vo[ixcosO + iy sinO]378 The Magnetic Field6. In 1896 Zeeman observed that an atom in a magnetic fieldhad a fine splitting of its spectral lines. A classical theory ofthe Zeeman effect, developed by Lorentz, modeled the elec-tron with mass m as being bound to the nucleus by a spring-like force with spring constant k so that in the absence of amagnetic field its natural frequency was wo = r,-.(a) A magnetic field Boi, is applied. Write Newton's law forthe x, y, and z displacements of the electron including thespring and Lorentz forces.(b) Because these equations are linear, guess exponentialsolutions 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 smallcompared to wk even for a strong field of B0 = 1 tesla. In thislimit, find approximate expressions for the natural frequen-cies of (b).7. A charge q moves through a region where there is anelectric field E and magnetic field B. The medium is veryviscous so that inertial effects are negligible,pv=q(E+vxB)where 6 is the viscous drag coefficient. What is the velocity ofthe charge? (Hint: (vxB)xB= -v(B-B)+B(v*B) andv .B = (q/f)E -B.)8. Charges of mass m, charge q, and number density n movethrough a conducting material and collide with the hostmedium with a collision frequency v in the presence of anelectric field E and magnetic field B.(a) Write Newton's first law for the charge carriers, alongthe same lines as developed in Section 3-2-2, with the additionof the Lorentz force.(b) Neglecting particle inertia and diffusion, solve for theparticle velocity v.(c) What is the constitutive law relating the current densityJ = qnv to E and B. This is the generalized Ohm's law in thepresence of a magnetic field.(d) What is the Ohmic conductivity r? A current i is passedthrough this material in the presence of a perpendicularmagnetic field. A resistor RL is connected across theterminals. What is the Hall voltage? (See top of page 379).(e) What value of RL maximizes the power dissipated in theload?Problems 379+Section 5.29. A point charge q is traveling within the magnetic field ofan infinitely long line current I. At r = ro its velocity isv(t = 0) = Vrir + Voi, + vzoizIts subsequent velocity is only a function of r.(a) What is the velocity of the charge as a function ofposition? Hint: See Problem 2c and 2d,-ldx = (Inx)2(b) What is the kinetic energy of the charge?(c) What is the closest distance that the charge canapproach the line current if v,0 = 0?10. Find the magnetic field at the point P shown for thefollowing
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