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.552 Electrodynamics-Fieldsand Wavesso thatR/d 1Rd 1-(21)R/d + nwhich when solved for Rid requiresR 1-a--(22)d n-IPROBLEMSSection 7-11. For the following electric fields in a linear media ofpermittivity e and permeability Cj find the charge density,magnetic field, and current density.(a) E = Eo(xi. +yi,) sin wt(b) E = Eo(yi, -xi,) cos wt(c) E= Re[Eo e" \-•--&)i,]. How must k,, k,, and o berelated so that J = 0?2. An Ohmic conductor of arbitrary shape has an initialcharge distribution po(r) at t = 0.(a) What is the charge distribution for all time?(b) The initial charge distribution is uniform and isconfined between parallel plate electrodes of spacing d. Whatare the electric and magnetic fields when the electrodes areopened or short circuited?(c) Repeat (b) for coaxial cylindrical electrodes of innerradius a and outer radius b.(d) When does a time varying electric field not generate amagnetic field?3. (a) For linear media of permittivity e and permeability /,use the magnetic vector potential A to rewrite Faraday's lawas the curl of a function.(b) Can a scalar potential function V be defined? What isthe electric field in terms of V and A? The choice of V is notunique so pick V so that under static conditions E = -V V.(c) Use the results of (a) and (b) in Ampere's law withMaxwell's displacement current correction to obtain a singleequation in A and V. (Hint: Vx (Vx A) = V(V -A) -V2A.)(d) Since we are free to specify V *A, what value should wepick to make (c) an equation just in A? This is called settingthe gauge.(e) Use the results of (a)-(d) in Gauss's law for D to obtain asingle equation in V.Problem 553(f) Consider a sinusoidally varying point charge at r = 0,Se".'t. Solve (e) for r > 0.Hint:1 8 (ra 82ar -=-(rV)Define a new variable (rV). By symmetry, V only depends on rand waves can only propagate away from the charge and nottowards it. As r-0, the potential approaches the quasi-staticCoulomb potential.Section 7-24. Poynting's theorem must be modified if we have ahysteretic material with a nonlinear and double-valued rela-tionship between the polarization P and electric field E andthe magnetization M and magnetic field H.(a) For these nonlinear constitutive laws put Poynting'stheorem in the form8wV S+-= -Pd -Pp-PMatwhere Pp and PM are the power densities necessary topolarize and magnetize the material.(b) Sinusoidal electric and magnetic fields E = E, cos at andH = H, cos at are applied. How much energy density is dis-sipated per cycle?5. An electromagnetic field is present within a superconduc-tor with constituent relationaJf= wE8t(a) Show that Poynting's theorem can be written in theform8wV.s+-=08tWhat is w?IAMk r ,554 Electrodynamics-Fieldsand Waves(b) What is the velocity of the charge carriers each withcharge q in terms of the current density Jr? The numberdensity of charge carriers is n.(c) What kind of energy does the superconductor add?(d) Rewrite Maxwell's equations with this constitutive lawfor fields that vary sinusoidally with time.(e) Derive the complex Poynting theorem in the formV.-[½(r)XH*(r) + 2j < w > = 0What is <w>?6. A paradoxical case of Poynting's theorem occurs when astatic electric field is applied perpendicularly to a staticmagnetic field, as in the case of a pair of electrodes placedwithin a magnetic circuit.yy(a) What are E, H, and S?(b) What is the energy density stored in the system?(c) Verify Poynting's theorem.7. The complex electric field amplitude has real andimaginary partsE(r) = E, +jEiUnder what conditions are the following scalar and vectorproducts zero:(a) E E 10(b) E • -0(c) E xE* 0(d) E x E* 1 0Section 7.38. Consider a lossy medium of permittivity e, permeability ;.,and Ohmic conductivity or.(a) Write down the field equations for an x-directed elec-tric field.Problems 555(b) Obtain a single equation in E,.(c) If the fields vary sinusoidally with time,Ex= Re [PE(z) e"i t]what are the spatial dependences of the fields?(d) Specialize (c) to the (i) low loss limit (o/sE<< 1) and (ii)large loss limit (o•/e >1).(e) Repeat (a)-(c) if the medium is a plasma with constitu-tive lawaJ= eEat(f) A current sheet Ko cos wti, is placed at z = 0. Find theelectric and magnetic fields if the sheet is placed within anOhmic conductor or within a plasma.9. A uniformly distributed volume current of thickness 2d,Jo cos wti., is a source of plane waves.e0, 1O< 2d(a) From Maxwell's equations obtain a single differentialequation relating E, to J..(b) Find the electric and magnetic fields within and outsidethe current distribution.(c) How much time-average power per unit area isdelivered by the current?(d) How does this generated power compare to the elec-tromagnetic time-average power per unit area leaving thevolume current at z = ±d?10. A TEM wave (E., H,) propagates in a medium whosepermittivity and permeability are functions of z, e(z), and1A(Z).(a) Write down Maxwell's equations and obtain singlepartial differential equations in E. and H,.(b) Consider the idealized case where e(z)=ee" Ze andL(z)=L e-alzI.A current sheet Koe"'i. is at z =0. What areo6, 00556 Eklctrody~amics-Fieldsand Wavesthe resulting electric and magnetic fields on each side of thesheet?(c) For what values of a are the solutions spatiallyevanescent or oscillatory?11. We wish to compare various measurements between twoobservers, the second moving at a constant velocity vi, withrespect to the first.(a) The first observer measures simultaneous events at twopositions z and z2 so that tL =lt. What is the time intervalbetween the two events t' -t as measured by the secondobserver?(b) The first observer measures a time interval At = tl -t2between two events at the same position z. What is the timeinterval as measured by the second observer?(c) The first observer measures the length of a stick asL = z -z. What is the length of the stick as measured by thesecond observer?12. A stationary observer measures the velocity of a particleas u = ni. + u,i, + ui,.(a) What
View Full Document
Unlocking...