School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007
|
|
- Christian Pope
- 6 years ago
- Views:
Transcription
1 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 , 5.9, 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
2 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
3 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
4 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
5 ) 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
6 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
7 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
School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007
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
More informationElectric Potential. and Equipotentials
Electic Potentil nd Euipotentils U Electicl Potentil Review: W wok done y foce in going fom to long pth. l d E dl F W dl F θ Δ l d E W U U U Δ Δ l d E W U U U U potentil enegy electic potentil Potentil
More informationThis immediately suggests an inverse-square law for a "piece" of current along the line.
Electomgnetic Theoy (EMT) Pof Rui, UNC Asheville, doctophys on YouTube Chpte T Notes The iot-svt Lw T nvese-sque Lw fo Mgnetism Compe the mgnitude of the electic field t distnce wy fom n infinite line
More informationWinter 2004 OSU Sources of Magnetic Fields 1 Chapter 32
Winte 4 OSU 1 Souces Of Mgnetic Fields We lened two wys to clculte Electic Field Coulomb's Foce de 4 E da 1 dq Q enc ˆ ute Foce Clcultion High symmety Wht e the nlogous equtions fo the Mgnetic Field? Winte
More informationGeneral Physics II. number of field lines/area. for whole surface: for continuous surface is a whole surface
Genel Physics II Chpte 3: Guss w We now wnt to quickly discuss one of the moe useful tools fo clculting the electic field, nmely Guss lw. In ode to undestnd Guss s lw, it seems we need to know the concept
More informationPhysics 604 Problem Set 1 Due Sept 16, 2010
Physics 64 Polem et 1 Due ept 16 1 1) ) Inside good conducto the electic field is eo (electons in the conducto ecuse they e fee to move move in wy to cncel ny electic field impessed on the conducto inside
More informationLecture 11: Potential Gradient and Capacitor Review:
Lectue 11: Potentil Gdient nd Cpcito Review: Two wys to find t ny point in spce: Sum o Integte ove chges: q 1 1 q 2 2 3 P i 1 q i i dq q 3 P 1 dq xmple of integting ove distiution: line of chge ing of
More informationCHAPTER 18: ELECTRIC CHARGE AND ELECTRIC FIELD
ollege Physics Student s Mnul hpte 8 HAPTR 8: LTRI HARG AD LTRI ILD 8. STATI LTRIITY AD HARG: OSRVATIO O HARG. ommon sttic electicity involves chges nging fom nnocoulombs to micocoulombs. () How mny electons
More informationU>, and is negative. Electric Potential Energy
Electic Potentil Enegy Think of gvittionl potentil enegy. When the lock is moved veticlly up ginst gvity, the gvittionl foce does negtive wok (you do positive wok), nd the potentil enegy (U) inceses. When
More informationElectric Field F E. q Q R Q. ˆ 4 r r - - Electric field intensity depends on the medium! origin
1 1 Electic Field + + q F Q R oigin E 0 0 F E ˆ E 4 4 R q Q R Q - - Electic field intensity depends on the medium! Electic Flux Density We intoduce new vecto field D independent of medium. D E So, electic
More informationChapter 28 Sources of Magnetic Field
Chpte 8 Souces of Mgnetic Field - Mgnetic Field of Moving Chge - Mgnetic Field of Cuent Element - Mgnetic Field of Stight Cuent-Cying Conducto - Foce Between Pllel Conductos - Mgnetic Field of Cicul Cuent
More informationElectricity & Magnetism Lecture 6: Electric Potential
Electicity & Mgnetism Lectue 6: Electic Potentil Tody s Concept: Electic Potenl (Defined in tems of Pth Integl of Electic Field) Electicity & Mgnesm Lectue 6, Slide Stuff you sked bout:! Explin moe why
More informationPhysics 11b Lecture #11
Physics 11b Lectue #11 Mgnetic Fields Souces of the Mgnetic Field S&J Chpte 9, 3 Wht We Did Lst Time Mgnetic fields e simil to electic fields Only diffeence: no single mgnetic pole Loentz foce Moving chge
More informationSolutions to Midterm Physics 201
Solutions to Midtem Physics. We cn conside this sitution s supeposition of unifomly chged sphee of chge density ρ nd dius R, nd second unifomly chged sphee of chge density ρ nd dius R t the position of
More informationProf. Anchordoqui Problems set # 12 Physics 169 May 12, 2015
Pof. Anchodoqui Poblems set # 12 Physics 169 My 12, 2015 1. Two concentic conducting sphees of inne nd oute dii nd b, espectively, cy chges ±Q. The empty spce between the sphees is hlf-filled by hemispheicl
More informationChapter 21: Electric Charge and Electric Field
Chpte 1: Electic Chge nd Electic Field Electic Chge Ancient Gees ~ 600 BC Sttic electicit: electic chge vi fiction (see lso fig 1.1) (Attempted) pith bll demonsttion: inds of popeties objects with sme
More informationELECTROSTATICS. 4πε0. E dr. The electric field is along the direction where the potential decreases at the maximum rate. 5. Electric Potential Energy:
LCTROSTATICS. Quntiztion of Chge: Any chged body, big o smll, hs totl chge which is n integl multile of e, i.e. = ± ne, whee n is n intege hving vlues,, etc, e is the chge of electon which is eul to.6
More informationPhysics 1502: Lecture 2 Today s Agenda
1 Lectue 1 Phsics 1502: Lectue 2 Tod s Agend Announcements: Lectues posted on: www.phs.uconn.edu/~cote/ HW ssignments, solutions etc. Homewok #1: On Mstephsics this Fid Homewoks posted on Msteingphsics
More informationELECTRO - MAGNETIC INDUCTION
NTRODUCTON LCTRO - MAGNTC NDUCTON Whenee mgnetic flu linked with cicuit chnges, n e.m.f. is induced in the cicuit. f the cicuit is closed, cuent is lso induced in it. The e.m.f. nd cuent poduced lsts s
More information(A) 6.32 (B) 9.49 (C) (D) (E) 18.97
Univesity of Bhin Physics 10 Finl Exm Key Fll 004 Deptment of Physics 13/1/005 8:30 10:30 e =1.610 19 C, m e =9.1110 31 Kg, m p =1.6710 7 Kg k=910 9 Nm /C, ε 0 =8.8410 1 C /Nm, µ 0 =4π10 7 T.m/A Pt : 10
More informationPX3008 Problem Sheet 1
PX38 Poblem Sheet 1 1) A sphee of dius (m) contins chge of unifom density ρ (Cm -3 ). Using Guss' theoem, obtin expessions fo the mgnitude of the electic field (t distnce fom the cente of the sphee) in
More informationAnswers to test yourself questions
Answes to test youself questions opic Descibing fields Gm Gm Gm Gm he net field t is: g ( d / ) ( 4d / ) d d Gm Gm Gm Gm Gm Gm b he net potentil t is: V d / 4d / d 4d d d V e 4 7 9 49 J kg 7 7 Gm d b E
More informationCourse Updates. Reminders: 1) Assignment #8 available. 2) Chapter 28 this week.
Couse Updtes http://www.phys.hwii.edu/~vne/phys7-sp1/physics7.html Remindes: 1) Assignment #8 vilble ) Chpte 8 this week Lectue 3 iot-svt s Lw (Continued) θ d θ P R R θ R d θ d Mgnetic Fields fom long
More information3.1 Magnetic Fields. Oersted and Ampere
3.1 Mgnetic Fields Oested nd Ampee The definition of mgnetic induction, B Fields of smll loop (dipole) Mgnetic fields in mtte: ) feomgnetism ) mgnetiztion, (M ) c) mgnetic susceptiility, m d) mgnetic field,
More informationCollection of Formulas
Collection of Fomuls Electomgnetic Fields EITF8 Deptment of Electicl nd Infomtion Technology Lund Univesity, Sweden August 8 / ELECTOSTATICS field point '' ' Oigin ' Souce point Coulomb s Lw The foce F
More informationSPA7010U/SPA7010P: THE GALAXY. Solutions for Coursework 1. Questions distributed on: 25 January 2018.
SPA7U/SPA7P: THE GALAXY Solutions fo Cousewok Questions distibuted on: 25 Jnuy 28. Solution. Assessed question] We e told tht this is fint glxy, so essentilly we hve to ty to clssify it bsed on its spectl
More informationAlgebra Based Physics. Gravitational Force. PSI Honors universal gravitation presentation Update Fall 2016.notebookNovember 10, 2016
Newton's Lw of Univesl Gvittion Gvittionl Foce lick on the topic to go to tht section Gvittionl Field lgeb sed Physics Newton's Lw of Univesl Gvittion Sufce Gvity Gvittionl Field in Spce Keple's Thid Lw
More informationClass Summary. be functions and f( D) , we define the composition of f with g, denoted g f by
Clss Summy.5 Eponentil Functions.6 Invese Functions nd Logithms A function f is ule tht ssigns to ech element D ectly one element, clled f( ), in. Fo emple : function not function Given functions f, g:
More informationOptimization. x = 22 corresponds to local maximum by second derivative test
Optimiztion Lectue 17 discussed the exteme vlues of functions. This lectue will pply the lesson fom Lectue 17 to wod poblems. In this section, it is impotnt to emembe we e in Clculus I nd e deling one-vible
More information( ) ( ) ( ) ( ) ( ) # B x ( ˆ i ) ( ) # B y ( ˆ j ) ( ) # B y ("ˆ ( ) ( ) ( (( ) # ("ˆ ( ) ( ) ( ) # B ˆ z ( k )
Emple 1: A positie chge with elocit is moing though unifom mgnetic field s shown in the figues below. Use the ight-hnd ule to detemine the diection of the mgnetic foce on the chge. Emple 1 ˆ i = ˆ ˆ i
More informationFI 2201 Electromagnetism
FI 1 Electomgnetism Alexnde A. Isknd, Ph.D. Physics of Mgnetism nd Photonics Resech Goup Electosttics ELECTRIC PTENTIALS 1 Recll tht we e inteested to clculte the electic field of some chge distiution.
More informationHomework 3 MAE 118C Problems 2, 5, 7, 10, 14, 15, 18, 23, 30, 31 from Chapter 5, Lamarsh & Baratta. The flux for a point source is:
. Homewok 3 MAE 8C Poblems, 5, 7, 0, 4, 5, 8, 3, 30, 3 fom Chpte 5, msh & Btt Point souces emit nuetons/sec t points,,, n 3 fin the flux cuent hlf wy between one sie of the tingle (blck ot). The flux fo
More informationChapter 23 Electrical Potential
hpte Electicl Potentil onceptul Polems [SSM] A poton is moved to the left in unifom electic field tht points to the ight. Is the poton moving in the diection of incesing o decesing electic potentil? Is
More informationDEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING FLUID MECHANICS III Solutions to Problem Sheet 3
DEPATMENT OF CIVIL AND ENVIONMENTAL ENGINEEING FLID MECHANICS III Solutions to Poblem Sheet 3 1. An tmospheic vote is moelle s combintion of viscous coe otting s soli boy with ngul velocity Ω n n iottionl
More informationSchool of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007
School of Electical and Compute Engineeing, Conell Univesity ECE 303: Electomagnetic Fields and Waves Fall 007 Homewok 8 Due on Oct. 19, 007 by 5:00 PM Reading Assignments: i) Review the lectue notes.
More informationChapter 2: Electric Field
P 6 Genel Phsics II Lectue Outline. The Definition of lectic ield. lectic ield Lines 3. The lectic ield Due to Point Chges 4. The lectic ield Due to Continuous Chge Distibutions 5. The oce on Chges in
More informationRadial geodesics in Schwarzschild spacetime
Rdil geodesics in Schwzschild spcetime Spheiclly symmetic solutions to the Einstein eqution tke the fom ds dt d dθ sin θdϕ whee is constnt. We lso hve the connection components, which now tke the fom using
More informationPhysics 111. Uniform circular motion. Ch 6. v = constant. v constant. Wednesday, 8-9 pm in NSC 128/119 Sunday, 6:30-8 pm in CCLIR 468
ics Announcements dy, embe 28, 2004 Ch 6: Cicul Motion - centipetl cceletion Fiction Tension - the mssless sting Help this week: Wednesdy, 8-9 pm in NSC 128/119 Sundy, 6:30-8 pm in CCLIR 468 Announcements
More informationEinstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,
PE ELECTOSTATICS C Popeties of chges : (i) (ii) (iii) (iv) (v) (vi) Two kinds of chges eist in ntue, positive nd negtive with the popety tht unlike chges ttct ech othe nd like chges epel ech othe. Ecess
More informationProblems for HW X. C. Gwinn. November 30, 2009
Problems for HW X C. Gwinn November 30, 2009 These problems will not be grded. 1 HWX Problem 1 Suppose thn n object is composed of liner dielectric mteril, with constnt reltive permittivity ɛ r. The object
More informationContinuous Charge Distributions
Continuous Chge Distibutions Review Wht if we hve distibution of chge? ˆ Q chge of distibution. Q dq element of chge. d contibution to due to dq. Cn wite dq = ρ dv; ρ is the chge density. = 1 4πε 0 qi
More information(1) It increases the break down potential of the surrounding medium so that more potential can be applied and hence more charge can be stored.
Cpcito Cpcito: Cpcito ( o conense ) is evice fo stoing chge. It essentilly consists of two conucting sufces such s two pltes o two spheicl shell o two cylines etc. kept exctly pllel to ech othe septe y
More information2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum
2. Electostatics D. Rakhesh Singh Kshetimayum 1 2.1 Intoduction In this chapte, we will study how to find the electostatic fields fo vaious cases? fo symmetic known chage distibution fo un-symmetic known
More information6. Gravitation. 6.1 Newton's law of Gravitation
Gvittion / 1 6.1 Newton's lw of Gvittion 6. Gvittion Newton's lw of gvittion sttes tht evey body in this univese ttcts evey othe body with foce, which is diectly popotionl to the poduct of thei msses nd
More informationOn the Eötvös effect
On the Eötvös effect Mugu B. Răuţ The im of this ppe is to popose new theoy bout the Eötvös effect. We develop mthemticl model which loud us bette undestnding of this effect. Fom the eqution of motion
More informationSection 35 SHM and Circular Motion
Section 35 SHM nd Cicul Motion Phsics 204A Clss Notes Wht do objects do? nd Wh do the do it? Objects sometimes oscillte in simple hmonic motion. In the lst section we looed t mss ibting t the end of sping.
More informationChapter 22 The Electric Field II: Continuous Charge Distributions
Chpte The lectic Field II: Continuous Chge Distibutions Conceptul Poblems [SSM] Figue -7 shows n L-shped object tht hs sides which e equl in length. Positive chge is distibuted unifomly long the length
More informationFluids & Bernoulli s Equation. Group Problems 9
Goup Poblems 9 Fluids & Benoulli s Eqution Nme This is moe tutoil-like thn poblem nd leds you though conceptul development of Benoulli s eqution using the ides of Newton s 2 nd lw nd enegy. You e going
More information10 m, so the distance from the Sun to the Moon during a solar eclipse is. The mass of the Sun, Earth, and Moon are = =
Chpte 1 nivesl Gvittion 11 *P1. () The un-th distnce is 1.4 nd the th-moon 8 distnce is.84, so the distnce fom the un to the Moon duing sol eclipse is 11 8 11 1.4.84 = 1.4 The mss of the un, th, nd Moon
More informationLecture 10. Solution of Nonlinear Equations - II
Fied point Poblems Lectue Solution o Nonline Equtions - II Given unction g : R R, vlue such tht gis clled ied point o the unction g, since is unchnged when g is pplied to it. Whees with nonline eqution
More informationπ,π is the angle FROM a! TO b
Mth 151: 1.2 The Dot Poduct We hve scled vectos (o, multiplied vectos y el nume clled scl) nd dded vectos (in ectngul component fom). Cn we multiply vectos togethe? The nswe is YES! In fct, thee e two
More informationCh 26 - Capacitance! What s Next! Review! Lab this week!
Ch 26 - Cpcitnce! Wht s Next! Cpcitnce" One week unit tht hs oth theoeticl n pcticl pplictions! Cuent & Resistnce" Moving chges, finlly!! Diect Cuent Cicuits! Pcticl pplictions of ll the stuff tht we ve
More informationTwo dimensional polar coordinate system in airy stress functions
I J C T A, 9(9), 6, pp. 433-44 Intentionl Science Pess Two dimensionl pol coodinte system in iy stess functions S. Senthil nd P. Sek ABSTRACT Stisfy the given equtions, boundy conditions nd bihmonic eqution.in
More informationExample 2: ( ) 2. $ s ' 9.11" 10 *31 kg ( )( 1" 10 *10 m) ( e)
Emple 1: Two point chge e locted on the i, q 1 = e t = 0 nd q 2 = e t =.. Find the wok tht mut be done b n etenl foce to bing thid point chge q 3 = e fom infinit to = 2. b. Find the totl potentil eneg
More informationSTUDY OF THE UNIFORM MAGNETIC FIELD DOMAINS (3D) IN THE CASE OF THE HELMHOLTZ COILS
STUDY OF THE UNIFORM MAGNETIC FIED DOMAINS (3D) IN THE CASE OF THE HEMHOTZ COIS FORIN ENACHE, GHEORGHE GAVRIĂ, EMI CAZACU, Key wods: Unifom mgnetic field, Helmholt coils. Helmholt coils e used to estblish
More informationChapter 24. Gauss s Law
Chpte 24 Guss s Lw CHAPTR OUTLIN 24.1 lectic Flux 24.2 Guss s Lw 24.3 Appliction of Guss s Lw to Vious Chge Distibutions 24.4 Conductos in lectosttic uilibium 24.5 Foml Deivtion of Guss s Lw In tble-top
More informationGeneral Physics (PHY 2140)
Genel Physics (PHY 40) Lightning Review Lectue 3 Electosttics Lst lectue:. Flux. Guss s s lw. simplifies computtion of electic fields Q Φ net Ecosθ ε o Electicl enegy potentil diffeence nd electic potentil
More information2.2 This is the Nearest One Head (Optional) Experimental Verification of Gauss s Law and Coulomb s Law
2.2 This is the Neest One Hed 743 P U Z Z L R Some ilwy compnies e plnning to cot the windows of thei commute tins with vey thin lye of metl. (The coting is so thin you cn see though it.) They e doing
More informationSTD: XI MATHEMATICS Total Marks: 90. I Choose the correct answer: ( 20 x 1 = 20 ) a) x = 1 b) x =2 c) x = 3 d) x = 0
STD: XI MATHEMATICS Totl Mks: 90 Time: ½ Hs I Choose the coect nswe: ( 0 = 0 ). The solution of is ) = b) = c) = d) = 0. Given tht the vlue of thid ode deteminnt is then the vlue of the deteminnt fomed
More informationPreviously. Extensions to backstepping controller designs. Tracking using backstepping Suppose we consider the general system
436-459 Advnced contol nd utomtion Extensions to bckstepping contolle designs Tcking Obseves (nonline dmping) Peviously Lst lectue we looked t designing nonline contolles using the bckstepping technique
More informationPhysics 505 Fall 2005 Midterm Solutions. This midterm is a two hour open book, open notes exam. Do all three problems.
Physics 55 Fll 5 Midtem Solutions This midtem is two hou open ook, open notes exm. Do ll thee polems. [35 pts] 1. A ectngul ox hs sides of lengths, nd c z x c [1] ) Fo the Diichlet polem in the inteio
More informationMAGNETIC EFFECT OF CURRENT & MAGNETISM
TODUCTO MAGETC EFFECT OF CUET & MAGETM The molecul theo of mgnetism ws given b Webe nd modified lte b Ewing. Oested, in 18 obseved tht mgnetic field is ssocited with n electic cuent. ince, cuent is due
More informationFSK 116 Semester 1 Mathematics and Other Essentials. Priorities
FSK 6 Semeste Mthemtics nd Othe Essentils Pioities Know how YOUR clculto woks nd lwys hve YOUR clculto with you. Alwys hve pencil (nd n ese) t hnd when doing Physics. Geek Alphbet Alph Et Nu Tu Bet Thet
More informationEECE 260 Electrical Circuits Prof. Mark Fowler
EECE 60 Electicl Cicuits Pof. Mk Fowle Complex Numbe Review /6 Complex Numbes Complex numbes ise s oots of polynomils. Definition of imginy # nd some esulting popeties: ( ( )( ) )( ) Recll tht the solution
More informationWeek 8. Topic 2 Properties of Logarithms
Week 8 Topic 2 Popeties of Logithms 1 Week 8 Topic 2 Popeties of Logithms Intoduction Since the esult of ithm is n eponent, we hve mny popeties of ithms tht e elted to the popeties of eponents. They e
More informationWork, Potential Energy, Conservation of Energy. the electric forces are conservative: ur r
Wok, Potentil Enegy, Consevtion of Enegy the electic foces e consevtive: u Fd = Wok, Potentil Enegy, Consevtion of Enegy b b W = u b b Fdl = F()[ d + $ $ dl ] = F() d u Fdl = the electic foces e consevtive
More information1. The sphere P travels in a straight line with speed
1. The sphee P tels in stight line with speed = 10 m/s. Fo the instnt depicted, detemine the coesponding lues of,,,,, s mesued eltie to the fixed Oxy coodinte system. (/134) + 38.66 1.34 51.34 10sin 3.639
More informationProblem Set 3 SOLUTIONS
Univesity of Albm Deptment of Physics nd Astonomy PH 10- / LeCli Sping 008 Poblem Set 3 SOLUTIONS 1. 10 points. Remembe #7 on lst week s homewok? Clculte the potentil enegy of tht system of thee chges,
More informationChapter 4 Two-Dimensional Motion
D Kinemtic Quntities Position nd Velocit Acceletion Applictions Pojectile Motion Motion in Cicle Unifom Cicul Motion Chpte 4 Two-Dimensionl Motion D Motion Pemble In this chpte, we ll tnsplnt the conceptul
More information1 Using Integration to Find Arc Lengths and Surface Areas
Novembe 9, 8 MAT86 Week Justin Ko Using Integtion to Find Ac Lengths nd Sufce Aes. Ac Length Fomul: If f () is continuous on [, b], then the c length of the cuve = f() on the intevl [, b] is given b s
More informationModule 05: Gauss s s Law a
Module 05: Gauss s s Law a 1 Gauss s Law The fist Maxwell Equation! And a vey useful computational technique to find the electic field E when the souce has enough symmety. 2 Gauss s Law The Idea The total
More informationPhysics 2020, Spring 2005 Lab 5 page 1 of 8. Lab 5. Magnetism
Physics 2020, Sping 2005 Lab 5 page 1 of 8 Lab 5. Magnetism PART I: INTRODUCTION TO MAGNETS This week we will begin wok with magnets and the foces that they poduce. By now you ae an expet on setting up
More informationUnit 1. Electrostatics of point charges
Unit 1 Electosttics of point chges 1.1 Intoduction 1. Electic chge 1.3 Electosttic foces. Coulomb s lw 1.4 Electic field. Field lines 1.5 Flux of the electic field. Guss s lw 1.6 Wok of the foces of electic
More informationChapter 4 Kinematics in Two Dimensions
D Kinemtic Quntities Position nd Velocit Acceletion Applictions Pojectile Motion Motion in Cicle Unifom Cicul Motion Chpte 4 Kinemtics in Two Dimensions D Motion Pemble In this chpte, we ll tnsplnt the
More informationJEE(Advanced) 2018 TEST PAPER WITH SOLUTION PHYSICS. (HELD ON SUNDAY 20 th MAY, 2018) PART-1 : PHYSICS. (C) L = mkr ALLEN
JEE(Advnced) 08 TEST PAPE WITH SOUTION (HED ON SUNDAY 0 th MAY, 08) PAT- : JEE(Advnced) 08/Ppe-. The potentil enegy of pticle of mss m t distnce fom fixed point O is given by V () k /, whee k is positive
More information200 points 5 Problems on 4 Pages and 20 Multiple Choice/Short Answer Questions on 5 pages 1 hour, 48 minutes
PHYSICS 132 Smple Finl 200 points 5 Problems on 4 Pges nd 20 Multiple Choice/Short Answer Questions on 5 pges 1 hour, 48 minutes Student Nme: Recittion Instructor (circle one): nme1 nme2 nme3 nme4 Write
More informationELECTROSTATICS. Syllabus : Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road PE 1
PE ELECTOSTATICS Syllbus : Electic chges : Consevtion of chge, Coulumb s lw-foces between two point chges, foces between multiple chges; supeposition pinciple nd continuous chge distibution. Electic field
More informationSURFACE TENSION. e-edge Education Classes 1 of 7 website: , ,
SURFACE TENSION Definition Sufce tension is popety of liquid by which the fee sufce of liquid behves like stetched elstic membne, hving contctive tendency. The sufce tension is mesued by the foce cting
More informationChapter 25 Electric Potential
Chpte 5 lectic Potentil consevtive foces -> potentil enegy - Wht is consevtive foce? lectic potentil = U / : the potentil enegy U pe unit chge is function of the position in spce Gol:. estblish the eltionship
More informationLecture 13 - Linking E, ϕ, and ρ
Lecture 13 - Linking E, ϕ, nd ρ A Puzzle... Inner-Surfce Chrge Density A positive point chrge q is locted off-center inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on
More informationρ θ φ δ δ θ δ φ δ φ π δ φ π δ φ π
Physics 6 Fin Ex Dec. 6, ( pts Fou point chges with chge ± q e nged s in Figue. (5 pts. Wht is the chge density function ρ (, θφ,? (,, q ( ( cos ( / + ( ( / / ρ θ φ δ δ θ δ φ δ φ π δ φ π δ φ π b (5 pts.
More informationTopics for Review for Final Exam in Calculus 16A
Topics fo Review fo Finl Em in Clculus 16A Instucto: Zvezdelin Stnkov Contents 1. Definitions 1. Theoems nd Poblem Solving Techniques 1 3. Eecises to Review 5 4. Chet Sheet 5 1. Definitions Undestnd the
More informationFriedmannien equations
..6 Fiedmnnien equtions FLRW metic is : ds c The metic intevl is: dt ( t) d ( ) hee f ( ) is function which detemines globl geometic l popety of D spce. f d sin d One cn put it in the Einstein equtions
More informationLecture 4. Electric Potential
Lectue 4 Electic Ptentil In this lectue yu will len: Electic Scl Ptentil Lplce s n Pissn s Eutin Ptentil f Sme Simple Chge Distibutins ECE 0 Fll 006 Fhn Rn Cnell Univesity Cnsevtive Ittinl Fiels Ittinl
More informationToday s Plan. Electric Dipoles. More on Gauss Law. Comment on PDF copies of Lectures. Final iclicker roll-call
Today s Plan lectic Dipoles Moe on Gauss Law Comment on PDF copies of Lectues Final iclicke oll-call lectic Dipoles A positive (q) and negative chage (-q) sepaated by a small distance d. lectic dipole
More informationAPEX CARE INSTITUTE FOR TRB, SLET AND NET IN PHYSICS ELECTROMAGNETIC THEORY- SLET / NET QUESTION BANK
www.tbtnpsc.com ELECTROMAGNETC THEORY- SLET / NET QUESTON BANK. The electosttic potentil V(, ) in fee spce in egion whee the chge densit ρ is eo is given b V, e f. Given tht the -component of the electic
More informationRELATIVE KINEMATICS. q 2 R 12. u 1 O 2 S 2 S 1. r 1 O 1. Figure 1
RELAIVE KINEMAICS he equtions of motion fo point P will be nlyzed in two diffeent efeence systems. One efeence system is inetil, fixed to the gound, the second system is moving in the physicl spce nd the
More informationr a + r b a + ( r b + r c)
AP Phsics C Unit 2 2.1 Nme Vectos Vectos e used to epesent quntities tht e chcteized b mgnitude ( numeicl vlue with ppopite units) nd diection. The usul emple is the displcement vecto. A quntit with onl
More informationof Technology: MIT OpenCourseWare). (accessed MM DD, YYYY). License: Creative Commons Attribution- Noncommercial-Share Alike.
MIT OpenouseWe http://ocw.mit.edu 6.1/ESD.1J Electomgnetics nd pplictions, Fll 25 Plese use the following cittion fomt: Mkus Zhn, Eich Ippen, nd Dvid Stelin, 6.1/ESD.1J Electomgnetics nd pplictions, Fll
More informationGet Solution of These Packages & Learn by Video Tutorials on EXERCISE-1
FEE Downlod Study Pckge fom website: www.tekoclsses.com & www.mthsbysuhg.com Get Solution of These Pckges & Len by Video Tutoils on www.mthsbysuhg.com EXECISE- * MAK IS MOE THAN ONE COECT QUESTIONS. SECTION
More informationUniversity of Illinois at Chicago Department of Physics. Electricity & Magnetism Qualifying Examination
E&M poblems Univesity of Illinois at Chicago Depatment of Physics Electicity & Magnetism Qualifying Examination Januay 3, 6 9. am : pm Full cedit can be achieved fom completely coect answes to 4 questions.
More informationCHAPTER 2 ELECTROSTATIC POTENTIAL
1 CHAPTER ELECTROSTATIC POTENTIAL 1 Intoduction Imgine tht some egion of spce, such s the oom you e sitting in, is pemeted by n electic field (Pehps thee e ll sots of electiclly chged bodies outside the
More informationSample Exam 5 - Skip Problems 1-3
Smple Exm 5 - Skip Problems 1-3 Physics 121 Common Exm 2: Fll 2010 Nme (Print): 4 igit I: Section: Honors Code Pledge: As n NJIT student I, pledge to comply with the provisions of the NJIT Acdemic Honor
More information(a) Counter-Clockwise (b) Clockwise ()N (c) No rotation (d) Not enough information
m m m00 kg dult, m0 kg bby. he seesw stts fom est. Which diection will it ottes? ( Counte-Clockwise (b Clockwise ( (c o ottion ti (d ot enough infomtion Effect of Constnt et oque.3 A constnt non-zeo toque
More informationChapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s:
Chpte 7 Kleene s Theoem 7.1 Kleene s Theoem The following theoem is the most impotnt nd fundmentl esult in the theoy of FA s: Theoem 6 Any lnguge tht cn e defined y eithe egul expession, o finite utomt,
More informationFlux. Area Vector. Flux of Electric Field. Gauss s Law
Gauss s Law Flux Flux in Physics is used to two distinct ways. The fist meaning is the ate of flow, such as the amount of wate flowing in a ive, i.e. volume pe unit aea pe unit time. O, fo light, it is
More informationRadiowave Propagation Modelling using the Uniform Theory of Diffraction
Deptment of lecticl nd lectonic ngineeing Pt IV Poject Repot Ye 2003 inl Repot Rdiowve Popgtion Modelling using the Unifom Theoy of Diffction chool of ngineeing The Univesity of Aucklnd Cho-Wei Chng 2365708
More informationSupplementary material for " Coherent and Tunable Terahertz Radiation from Graphene Surface Plasmon Polarirons Excited by Cyclotron Electron Beam "
Suppleenty teil fo " Coheent nd Tunble Tehet Rdition fo Gphene Sufce Plson Polions Excited by Cycloton Electon Be " To Zho,, Sen Gong,, Min Hu,, Renbin Zhong,,Diwei Liu,,Xioxing Chen,, Ping hng,, Xinn
More informationSchool of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007
School of Electical and Compute Engineeing, Conell Univesity ECE 33: Electomagnetic Fields and Waves Fall 7 Homewok 6 Due on Oct. 5, 7 by 5: PM Reading Assignments: i) Review the lectue notes. ii) Review
More informationThe Wave Equation I. MA 436 Kurt Bryan
1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string
More information