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 Waves so that R d Rd R d 11n 21 which when solved for Rid requires R 1 a d n I 22 PROBLEMS Section 7 1 1 For the following electric fields in a linear media of permittivity 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 related so that i How must k k and o be J 0 2 An Ohmic conductor of arbitrary shape has an initial charge distribution po r at t 0 a What is the charge distribution for all time b The initial charge distribution is uniform and is confined between parallel plate electrodes of spacing d What are the electric and magnetic fields when the electrodes are opened or short circuited c Repeat b for coaxial cylindrical electrodes of inner radius a and outer radius b d When does a time varying electric field not generate a magnetic field 3 a For linear media of permittivity e and permeability use the magnetic vector potential A to rewrite Faraday s law as the curl of a function b Can a scalar potential function V be defined What is the electric field in terms of V and A The choice of V is not unique so pick V so that under static conditions E V V c Use the results of a and b in Ampere s law with Maxwell s displacement current correction to obtain a single equation in A and V Hint Vx Vx A V V A V 2A d Since we are free to specify V A what value should we pick to make c an equation just in A This is called setting the gauge e Use the results of a d in Gauss s law for D to obtain a single 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 r 82a rV Define a new variable rV By symmetry V only depends on r and waves can only propagate away from the charge and not towards it As r 0 the potential approaches the quasi static Coulomb potential Section 7 2 4 Poynting s theorem must be modified if we have a hysteretic material with a nonlinear and double valued relationship between the polarization P and electric field E and the magnetization M and magnetic field H M I A k r a For these nonlinear constitutive laws put Poynting s theorem in the form 8w V S at Pd Pp PM where Pp and PM are the power densities necessary to polarize and magnetize the material b Sinusoidal electric and magnetic fields E E cos at and H H cos at are applied How much energy density is dissipated per cycle 5 An electromagnetic field is present within a superconductor with constituent relation aJf wE 8t a Show that Poynting s theorem can be written in the form 8w V s 0 8t What is w 554 Electrodynamics Fieldsand Waves b What is the velocity of the charge carriers each with charge q in terms of the current density Jr The number density of charge carriers is n c What kind of energy does the superconductor add d Rewrite Maxwell s equations with this constitutive law for fields that vary sinusoidally with time e Derive the complex Poynting theorem in the form V r XH r 2j w 0 What is w 6 A paradoxical case of Poynting s theorem occurs when a static electric field is applied perpendicularly to a static magnetic field as in the case of a pair of electrodes placed within a magnetic circuit y y 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 imaginary parts amplitude has real and E r E jEi Under what conditions are the following scalar and vector products zero a E E 10 b E 0 c E xE 0 d E x E 1 0 Section 7 3 8 Consider a lossy medium of permittivity e permeability and Ohmic conductivity or a Write down the field equations for an x directed electric field Problems 555 b Obtain a single equation in E c If the fields vary sinusoidally with time i t Ex Re PE z e 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 constitutive law aJ eE at f A current sheet Ko cos wti is placed at z 0 Find the electric and magnetic fields if the sheet is placed within an Ohmic 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 o6 00 2d a From Maxwell s equations obtain a single differential equation relating E to J b Find the electric and magnetic fields within and outside the current distribution c How much time average power per unit area is delivered by the current d How does this generated power compare to the electromagnetic time average power per unit area leaving the volume current at z d 10 A TEM wave E H propagates in a medium whose permittivity and permeability are functions of z e z and 1A Z a Write down Maxwell s equations and obtain single partial differential equations in E and H b Consider the idealized case where e z eee Z and L z L e a l zI A current sheet Koe i is at z 0 What are 556 Eklctrody amics Fieldsand Waves the resulting electric and magnetic fields on each side of the sheet c For what values of a are the solutions spatially evanescent or oscillatory 11 We wish to compare various measurements between two observers the second moving at a constant velocity vi with respect to the first a The first observer measures simultaneous events at two positions z and z2 so that tL lt What is the time interval between the two events t t as measured by the second observer b The first observer measures a time interval At tl t2 between two events at the same position z What is the time interval as measured by the second observer c The first observer measures the length of a stick as L z z What is the length of the stick as measured by the second …
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