School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007

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1 School of Electicl nd Compute Engineeing, Conell Univesity ECE 303: Electomgnetic Fields nd Wves Fll 007 Homewok 4 Due on Sep. 1, 007 by 5:00 PM Reding Assignments: i) Review the lectue notes. ii) Relevnt sections of the online Hus nd Melche book fo this week e , 8.6, Note tht the book contins moe mteil thn you e esponsible fo in this couse. Detemine elevnce by wht is coveed in the lectues nd the ecittions. The book is ment fo those of you who e looking fo moe depth nd detils. Tble of Solutions of Lplce s Eqution Spheicl Coodinte System Cylindicl Coodinte System φ ( ) = A Constnt potentil φ ( ) = A Constnt potentil A φ φ ( ) ( ) = Aln( ) Cylindiclly symmetic potentil = Spheiclly symmetic potentil φ( ) = A cos( θ ) Potentil fo unifom z-diected φ( ) = A cos( φ ) Potentil fo unifom -diected E-Field E-Field φ( ) = A sin( φ) Potentil fo unifom y-diected E-Field cos ( ) ( θ ) φ cos = A Potentil fo point-chge-dipolelike solution oiented long the z-is like solution oiented long the -is ( ) ( φ) φ = A Potentil fo line-chge-dipole- sin ( ) ( φ ) φ = A Potentil fo line-chge-dipolelike solution oiented long the y-is Poblem 4.1: (Metl wie ove pefect metl plne) Conside thin metl wie of dius ove n infinite pefect metl gound plne, s shown in the figue below. The distnce of the wie fom the gound plne is much lge thn the dius of the wie (i.e. d >> ). The stuctue is infinite in the z-diection. In this poblem you will need to find the inductnce pe unit length (i.e. the length in the z-diection) between the metl wie nd the gound plne. The metl wie is ssumed to cy time-vying cuent I ( time-vying in the mgnetoqusisttic sense) in the +z-diection. ) Sketch the imge metl wie using the method of imges. Indicte the oienttion nd the cuent tht the imge wie cies on you sketch. 1

2 b) Assuming tht the entie cuent I in the metl wie is cied by line cuent t the cente of the metl wie, find the vecto potentil A( ) eveywhee in the egion outside the pefect metl (use esult in you lectue hndouts). c) Using you esult in pt (b) bove, nd the vecto potentil A( ), wite n epession tht eltes the cuent I to the mgnetic flu pe unit length tht psses between the metl wie nd the gound plne. dius y d pefect metl d) Find the inductnce pe unit length L (units: Heny/m) between the metl wie nd the gound plne by tking the tio of the mgnetic flu pe unit length (clculted in pt (c) bove) to the cuent I. The lst two pts e elted to the concept of imge cuents. e) If the top metl wie is cying time-vying cuent ( t ) sense) in the +z-diection then thee must be position dependent sufce cuent density K (, t ) I ( time-vying in the mgnetoqusisttic on the sufce of the pefect metl. Find this sufce cuent density (mgnitude nd diection). flowing f) Integte the epession fo the sufce cuent density found in pt (e) bove to find the totl cuent tht flows on the sufce of the pefect metl. Poblem 4.: (A cylinde with sufce cuent density) Conside sufce cuent density on the sufce of cylinde of dius, s shown in the figue, nd given by the epession: K = Ko cos( φ )zˆ The stuctue shown in the figue is infinite in the z-diection. You need to find the mgnetic field eveywhee. One cn pehps use the Biot-Svt lw diectly but I cn imgine the integtions involved will not be plesnt. Insted, you will solve the vecto Poisson s Eqution diectly. Since the cuent hs only z-component nd the stuctue is infinite in the z-diection, the vecto potentil will only hve z- component A z (,φ) tht is function of the dil coodinte nd the ngle φ in cylindicl coodinte system.

3 y K = Ko cos φ ˆ ( ) z ) Given the vecto potentil A = Az (,φ )zˆ, wite down simplified epessions fo the H-field components using the tbles t the bck of you tetbook. The vecto potentil stisfies the eqution: A = µ oj Since thee is no volume cuent density in the poblem only sufce cuent density - nd since the vecto potentil hs only z-component the bove eqution becomes: A z = 0 So the z-component of the vecto potentil stisfies the Lplce s Eqution. b) Wite down til solutions Az in (,φ) nd A out (,φ ) z fo < nd > tht go to zeo t infinity nd do not blow up t the oigin. You solutions should hve undetemined constnts. c) The inside nd outside solutions must be stitched t = by using the boundy conditions on the components of the H-field noml nd tngentil to the sufce of the cylinde. Wht e these boundy conditions in tems of the vecto potentils Az in (,φ ) nd Az out (,φ)? d) Use the boundy conditions in pt (c) to find ll the unknown constnts in you solution of pt (b). e) Wite down the finl epessions fo ll the components of the H-field. f) Sketch the mgnetic field lines inside nd outside the cylinde. Poblem 4.3: (Electomgnetic induction) ) Conside pefect metl wie in the fom of loop. A mgnetic field pointing in the +z-diection psses though the cente of the loop s shown below. Thee is smll i gp of length in the metl wie. The mgnetic field is time vying nd the totl mgnetic flu though the loop is function of time nd is given by the epession: λ ( t ) = gt The dius of the metl wie loop is c nd the dimete of the wie is b, nd << b << c. Find the mgnitude s well s diection of the E-field E within the i gp in the metl wie. 3

4 y b H z c b) Now conside conductive wie (not pefect-metl wie) in the fom of loop. A mgnetic field pointing in the +z-diection psses though the cente of the loop s shown below. The mgnetic field is time vying nd the mgnetic flu though the loop is function of time nd is given by the epession: λ ( t ) = gt The conductivity of the wie mteil is σ, the dius of the wie loop is c nd the dimete of the wie is b, nd << b << c. Find the E-field E (mgnitude nd diection) nd the totl cuent I (mgnitude nd diection) in the wie. y σ b H z c Poblem 4.4: (Inductos nd electomgnetic induction) Conside co-il cylindicl inducto of length W cying time vying cuent I () t, s shown in the figue below (nd lso discussed in the lectue notes). The inne cylinde hs dius nd the oute cylindicl shell hs dius b. The gp between the inne nd oute cylindes is closed off by pefect metllic wll t z=w to povide etun pth fo the cuent. You should wok in cylindicl co-odintes in this poblem. The mgnetic field hs only φ-component. ) Find the mgnetic field H φ (, t ) s function of the cuent ( t ) b) Find the totl mgnetic flu () t I. λ enclosed within the inducto s function of the cuent I () t. c) Find the inductnce L of the inducto using you nswe in pt (b). 4

5 I () t H φ + () t V _ b Now comes the inteesting pt: 0 W z d) Cicuit theoy tells us tht if the cuent is chnging with time in n inducto then thee should be mesuble voltge V () t coss the two ends of the inducto given by: d I () ( t ) V t = L dt If thee is mesuble voltge then thee hs to be dil E-field t z=0 whose line integl coss the input teminls gives the voltge V () t. By using Fdy s lw diectly nd choosing n ppopite contou, veify tht the dil component of the E-field t z=0 must stisfy: b d I ( ) () t E z t d L, = 0, = dt e) Find the time-dependent nd the position dependent E-field inside the inne nd oute cylindes s function of the co-odinte z nd the co-odinte. Hints: The E-field must stisfy ll pefect metl boundy conditions nd lso Mwell s following two equtions: µ o H ( ) ( ) (, t ). E, t = 0 E, t = t Poblem 4.5: (Powe dissiption in conductos nd Poynting s theoem) Conside conducto in the fom of od of dius nd length L, s shown. The conducto cies unifom time-peiodic cuent density given by J = Jo sin( ω t )zˆ. y σ L z ) Find the time-dependent electic field E (, t ) Fdy s lw hee). within the conducto (Hint: You e not supposed to use 5

6 b) Find the time-dependent mgnetic field H (, t ) within the conducto. Note tht you e woking in the mgnetoqusisttic limit whee the mgnetic fields e poduced by only electic cuents nd not by time-vying electic fields. c) Using you esult in (), find the time-vege powe dissiption (units: Joules/sec) within the entie volume of the conducto. (Hint: you hve to use J (, t ). E(, t )dv whee the ngled bckets indicte time-veging ove one peiod T given by, T = π ω. You esult fo time-vege powe dissiption will be independent of time). The following fomuls might pove helpful: 1 T 1 1 T 1 sin ( ω t ) = sin ( ω t ) dt = cos ( ω t ) = cos ( ω t ) dt = T T 0 d) Find the totl esistnce R of the od nd the totl time-dependent cuent I ( t ) flowing in the od, nd then show tht the time-vege powe dissiption given by I ( t )R bove. 0 is the sme s tht found in pt (c) e) Now clculte the time-vege electomgnetic enegy flow (units: Joules/sec) into the conducto using the Poynting vecto (Hint: you need to clculte the sufce integl ( E(, t ) H(, t )). d ight t the sufce of the conducto nd then time-vege the esult ove one peiod). f) The Poynting s theoem sttes tht: ( E (, t ) H(, t )). d = W (, t ) dv + J(, t ). E(, t )dv t In the pesent cse, when one is deling with time-peiodic fields, the time-vege vesion of Poynting s theoem is: E (, t ) H(, t ). d = J, t. E, t ( ) ( ) ( )dv t whee the ngled-bckets indicte time-veging ove one peiod. Notice tht, W (, t ) dv = 0 becuse the fields e peiodic in time nd the totl electomgnetic enegy in closed volume t the beginning nd t the end of peiod must be the sme. Using you esults in pt (c) nd pt (e) veify the time-vege vesion of the Poynting s theoem fo time-peiodic fields. Poblem 4.6: (Dielectic imge chge foces) Conside point chge + Q sitting in fee spce t distnce d bove dielectic medium of pemittivity ε, s shown below. You hve ledy solved this poblem in homewok 3. 6

7 z d + Q ε o ε In this poblem you will eploe the foces eeted by dielectic intefces on chges. As you hve ledy seen in homewok 3, chge density due to pied chges t cn eist t dielectic intefces nd this chge density cn lso eet foces. ) Using you esults fom homewok 3, clculte the foce (mgnitude nd diection) on the point chge +Q. Specify whethe the foce clculted is epulsive o ttctive (i.e. is the point chge +Q being pulled towds the dielectic intefce o being pushed wy fom it). Now we conside slightly diffeent spin on the sme poblem. Conside point chge + Q sitting inside dielectic medium of pemittivity ε t distnce d below the intefce, s shown below. z ε o d + Q ε b) Bsed upon wht you lened in homewok 3, find epessions fo the potentil out ( ) dielectic nd fo the potentil φ ( ) inside the dielectic. in φ outside the c) Using you esult in pt (b), find the net foce (mgnitude nd diection) being eeted on the chge + Q. Specify whethe the foce clculted is epulsive o ttctive (i.e. is the point chge +Q being pulled towds the dielectic intefce o being pushed wy fom it). 7

8 Bonus Chllenge Poblem (30 dditionl points TAs nd Instuctos cnnot be consulted bout this poblem): A pefect metl cylinde of dius is plced inside time-vying unifom mgnetic field pointing in the -diection, s shown below. The cylinde is vey long (infinite) in the z-diection. H = Ho () t ˆ y Of couse, the ctul H-field lines cnnot be s shown bove since thee cnnot be time vying H-field within pefect metl. Find the ctul mgnetic field eveywhee nd sketch it. 8