School of Electicl nd Compute Engineeing, Conell Univesity ECE 303: Electomgnetic Fields nd Wves Fll 007 Homewok 3 Due on Sep. 14, 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 5.0-5.3, 5.9, 6.0-6.. 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. ii) This homewok is long stt ely. 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 -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 -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 1
Poblem 3.1: (A nno-stuctued dielectic medium) These dys nno-technology is being used to design mteils (s opposed to elying on ntue) tht hve some desied chcteistics. In this poblem you will eploe one such mteil mde of nnodots. Conside mteil mde up of nno-sied dielectic sphees (o nno-dots) of dielectic pemittivity ε 1 embedded in nothe mteil tht hs the sme pemittivity s tht of fee-spce ( ε o ), s shown below. ε o ε 1 We will conside this poblem in diffeent steps. Conside fist single dielectic sphee of dius nd of dielectic pemittivity ε 1 embedded in medium of pemittivity ε o, s shown below. A constnt nd unifom E-field hs been pplied in the +-diection fom side. E = Eo ˆ ε o θ ε 1, fo the potentils inside nd side the dielectic sphee. φ must hve tem tht hs the sme fom s dipole potentil). ) Find til solutions, φ in ( ) nd φ ( ) (Hint: ( ) b) Wite down ll the boundy conditions (t lest s mny s the numbe of unknown constnts in you φ. nswe to pt () bove) elevnt to solving fo the potentils, φ ( ) nd ( ) c) Find ll the unknown constnts in you solutions in pt () bove by using ll the boundy conditions in pt (b) bove. in
d) Compe the dipole-like tem in you solution ( ) φ to tht of point-chge dipole potentil (see you homewok poblem.1 solutions) nd fom tht compison figue the dipole-moment p (smll p ) of the polied dielectic sphee. Mke sue you get the coect units. (Hint: the dipole moment must be popotionl to the pplied E-field mgnitude). This dipole moment hs been induced in the dielectic sphee due to the etenl E-field. Now come bck to the medium mde up of nno-sied dielectic sphees of dielectic pemittivity ε 1 embedded in nothe medium of pemittivity ε o, s shown in n elie pictue. Suppose the dots e spced esonbly f pt nd so the field fom one dot does not intect with the field of the othe dots. Suppose tht thee e ppoimtely N dots pe unit volume. e) Wht is the polition vecto P (cpitl P ) of the nno-dot medium given tht you know the dipole-moment of ech dot? (Hint: the polition vecto must be popotionl to the pplied E-field mgnitude). f) Fom you nswe to pt (e), find the electicl susceptibility χ e of the nno-dot medium. g) Fom you nswe in pt (f), find the dielectic pemittivity ε of the nno-dot medium. Poblem 3.: (Dielectic imge chges) Conside point chge + Q sitting in fee spce t distnce d bove dielectic medium of pemittivity ε, s shown below. d + Q ε o ε The electic field fom the chge will get ptilly sceened by the sufce polition chge density (pied sufce chge density) tht will eist t the sufce of the dielectic medium. But unlike the pefect metl cse, the electic field will not get fully sceened of the dielectic mteil. In ode to solve this poblem, one needs to elie tht the ctul field solution, both inside nd side the dielectic medium, must be supeposition of the field due to the point chge nd the field due to the sufce polition chge density (i.e. the pied chge density t the sufce of the dielectic medium). A pioi, we don t know wht this sufce chge density looks like so we will ty to constuct guess solution. 3
We will ssume tht OUTSIDE the dielectic, the potentil looks like the supeposition of the potentil of the oiginl chge + Q nd the potentil due to n imge chge of stength Q sitting distnce d below the dielectic intefce nd tht the whole spce is filled with fee spce. The imge chge hs been ssumed to hve diffeent stength then the oiginl chge becuse dielectic sceening, unlike pefect metl sceening, is not epected to be pefect. Fo the potentil INSIDE the dielectic we will ssume tht it looks like tht of chge of stength + Qb sitting side the dielectic t distnce d wy fom the intefce nd tht the whole spce is filled with mteil of pemittivity ε. This is becuse the ctul field fom the chge + Q will get ptilly sceened by the polition (o pied) sufce chged density t the sufce of the dielectic. φ side the dielectic in tems of the distnces + φ inside the ) Wite n epession fo the guess potentil ( ) Q Q, espectively, nd fo the guess potentil ( ) nd fom the chges + nd dielectic in tems of the distnce + fom the chge + Qb. b) You hve two unknowns in you solution (the stength of the chges Q nd + Qb ) nd you need two boundy conditions. Wht e these two boundy conditions? c) Using the boundy conditions fom pt (b) find the stength of the chges Q nd + Qb in tems of the chge stength + Q nd the pemittivities ε nd ε o. d) Show fom you esult in pt (c) tht if ε = ε o then Q = 0 nd Q b = Q which is wht one would epect on physicl gounds. e) Show fom you esult in pt (c) tht when ε then the potentil OUTSIDE looks s if the dielectic mteil wee pefect metl. Poblem 3.3: (A pefect metl cylinde in unifom electic field) Conside n infinitely long (in the -diection) pefect metl od of dius plced in unifom nd constnt electic field pointing in the +-diection s shown below. The figue below shows only the pplied E-field lines s if the metl od wee not pesent. y in E o φ 4
) Find til solutions, φ in ( ) nd φ ( ) ( ), fo the potentils inside nd side the metl od. (Hint: φ must hve tem tht hs the sme fom s line-chge dipole potentil). b) Wite down ll the boundy conditions (t lest s mny s the numbe of unknown constnts in you φ. nswe to pt () bove) elevnt to solving fo the potentils, φ ( ) nd ( ) c) Find ll the unknown constnts in you solutions in pt () bove by using ll the boundy conditions in pt (b) bove. d) Find the sufce chge density on the metl od s function of the ngle φ. e) Sketch the totl E-field lines (note the figue bove shows only the pplied E-field lines s if the metl od wee not pesent). in Poblem 3.4: (A concentic spheicl dielectic cpcito) Conside pefect metl sphee suounded by pefect metl spheicl shell nd connected to voltge souce s shown below. Completely ignoe the physicl pesence of the voltge souce nd the connecting wies othe thn the fct tht they estblish fied potentil diffeence. The spce between the inne nd e sphees consists of two diffeent dielectic lyes s shown in the pictue below. c ε ε 1 b ) Wite til solutions fo the potentils, φ ( ) nd ( ) b c, espectively. + V - 1 φ, in the two dielectic egions b nd b) Wite down ll the boundy conditions (t lest s mny s the numbe of unknown constnts in you φ. nswe to pt () bove) elevnt to solving fo the potentils, φ ( ) nd ( ) c) Find ll the unknown constnts in you solutions in pt () bove by using ll the boundy conditions in pt (b) bove. 1 5
d) Find the sheet chge density (sign nd mgnitude) due to the pied chges t the intefce between the two dielectics. e) Find the sufce chge densities (sign nd mgnitude) on the inne sufce of the e spheicl metl shell nd on the e sufce of the inne metl sphee. f) Find the totl chge (sign nd mgnitude) on the inne sufce of the e metl shell nd lso on the e sufce of the inne metl sphee. g) Find the cpcitnce C (units: Fds) between the inne nd e sphees by tking the tio of the totl chge (found in pt (e) bove) nd the pplied voltge V. Poblem 3.5: (Feoelectics) Feoelectics (s opposed to dielectics) e mteils tht hve thei toms/molecules ll polied in the sme diection even when no etenl electic field is pesent. Tht is, feoelectic mteil hs builtin non-eo fied polition vecto P tht is independent of ny etenl field. Some impotnt semiconductos like Gllium Nitide (which is used these dys in lmost ll the high powe RF tnsmittes t bse sttions fo mobile/wieless systems) e feoelectic. In this poblem you will eploe the consequences of such built-in polition. Conside cicul disc of feoelectic mteil of thickness d tht is much smlle thn the dius R, s shown in the figue. The built-in polition vecto is given by P = Po ˆ. R P = P o ˆ ) Find the sufce chge density due to the pied chges t the uppe flt sufce of the disc. b) Find the sufce chge density due to the pied chges t the lowe flt sufce of the disc. c) Find the sufce chge density due to the pied chges t the cuved e sufce of the disc. d) Find the electic field (mgnitude nd diection) inside the feoelectic disc. Hint: Use you nswes fom pts () nd (b) nd (c). e) Find the D-field (mgnitude nd diection) inside the disc. d Poblem 3.6: (Mgnetic field of cicul cuent loop) Conside line-cuent in the fom of cicul loop of dius nd cying cuent I, s shown below. The loop is in the -y plne. You need to find the mgnetic field t the point P given by (0,0,). 6
y I P ) Wht is the -component of the mgnetic field t the loction P, s shown in the figue bove? b) Wht is the y-component of the mgnetic field t the loction P, s shown in the figue bove? c) Wht is the -component of the mgnetic field t the loction P, s shown in the figue bove? 7