ρ θ φ δ δ θ δ φ δ φ π δ φ π δ φ π
|
|
- Cecilia Dickerson
- 5 years ago
- Views:
Transcription
1 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. Wht e the utipoes q (do s ny s you cn; thee is usefu infotion in the HW soutions? In pticu, wht is the fist non vnishing utipoe? ( θ, φ ρ(, θ, φ q Y dd Ω * ( ( + ( ( + ( ( + ( ( + +! q P e + e e π! iπ/ iπ iπ/ +! iπ / q P e + ( π! +! q P ( i + ( π! +! q P π! / fo even P vnishes uness+ is even so ust be even so. The fist non vnishing utipoe needs to hve nd is 5 5 q q q π! π c ( pts. Do utipoes fo odd wys vnish? Cn you expin why o why not using the ottion syety in the souce? Yes. Notice tht ottion bout the z xis by π eves the chge density identic to the ρ, θφ, + π ρ, θφ,. But unde ottion by π initi density, i.e. iπ ( θ, φ + π ( θ, φ ( ( θ, φ Y Y e Y * * * Theefoe, fo ottiony syetic distibutions
2 * ( θ, φ ρ(, θ, φ ( θ, φ π ρ(, θ, φ π ( ( θ, φ ρ(, θ, φ ( q q Y dd Ω * Y + + dd Ω * Y dd Ω Fo odd, the utipoe oents ust vnish. d (5 pts. Wht is the potenti t octions with > nd <? By the ddition theoe q π Φ + + πε + < * * * * (, θ, φ Y (, Y (, π / Y (, π Y (, π / Y ( θ, φ + > ( q +! ε < P + + > π +! / ( Y ( θφ, whee < nd > e the se nd ge of nd. e (5 pts. Suppose the chges e encosed in gounded conducting sphee of dius b centeed on the oigin. Wht e the potentis fo > nd <? The esiest ethod is to sipy dd to the soution in d the negtive of the soution of Lpce s eqution tht hs the coect boundy potenti vue ( q +! Φ,,, < tot ( θφ P Y + ε + > π +! ( P + + b π +! / ( θφ q +! + ε / ( Y ( θ, φ q! < + / P ( Y( θφ, + + ε + > b π ( +! whee < nd > e the se nd ge of nd. This soution woud so be obtined by dding the potentis of the fou ige chges with stength q qb/ t octions b /.
3 f ( pts. Fo the soution in e, wht is the chge density on the inside sufce of the sphee? σ ε E ( ( + / ( Y ( θ, φ (! E P Y, Φ q b + ( b ε + b b π +! q +! σ P π! b b / ( θ φ g ( pts. Wht is the fied outside the sphee? Becuse the chge is inside the conducting sphee, by Guss s Lw the fied outside the sphee is shieded wy to zeo. (5 pts. Sove the sque D potenti pobe given by Figue. The foowing steps y be hepfu. (5 pts. Wht e the fundent soutions in Ctesin coodintes? An expe of n expnsion set is ( xy, Φ ( π + ( π + ( π + ( π ( π + ( π + ( π + ( π A sin x/ e B cos x/ e C sin x/ e D cos x/ e πy / πy / πy / πy / E sin y/ e F cos y/ e G sin y/ e H cos y/ e πx / πx / πx / πx / b (5 pts Sove the boundy vue pobe with the potenti Φ t x, < y< nd x, < y < equ to V nd with the othe sides hving Φ. Hee becuse of the boundy condition t y nd y πx (, sin ( π / sin ( π / Φ xy E y e + G y e Becuse of the boundy conditions t x nd x Othogonity gives / πx / ( π ( π V E sin y/ + G sin y/ π ( π ( π V E sin y/ e + G sin y/ e π
4 V ( cos π + ( E + G π V π π ( cos π + ( Ee + Ge π V π π V π E ( cos π + ( e / ( e ( cos π + ( e / sinh π π π V V G cos π + e / e cos π + e / sinh π π π π π π V π ( π π / Φ xy, sin y/ sinh x/ sinh x π sinh π odd Note tht ust be odd fo non zeo expnsion coefficient. c (5 pts. Sove n ppopite boundy vue pobe to incude the fces t Φ V. The soution is the se switching x y nd evesing the sign of the fce vues: V Φ ( x, y sin ( πx/ sinh πy/ sinh π ( y / π sinh π odd d (5 pts. Wht is the tot potenti? By supeposition tot odd ( xy, ( xy, ( xy, Φ Φ +Φ ( ( ( π π π ( sin y/ sinh x/ sinh x / V π sinh π sin ( πx/ sinh πy/ sinh π ( y / e (5 pts. Put pied coodinte syste with its oigin t x ( /, / the θ fied?. Ne wht is dependence of the potenti? Ne wht is the θ dependence of the eectic (5 pts. A unifoy chged infinitey thin spheic she of chge q nd dius is spun ound the z xis with constnt ngu fequencyω. J, θ, φ is (5 pts. Show tht the cuent density function qω J(, θ, φ δ ( sin θφˆ. π The esy wy to get this is to eize the chge density is
5 q ρ π ( δ, nd the ottion veocity is v ωsinθφˆ. Mutipying these two esuts gives the expession. Anothe wy is to conside the cuent pssing though the e eeent ddθ t po ngeθ. The tot chge in the ing of chge t this oction is ( q nd / sinθdθ. This chge psses the oction on the sphee t θ with fequency ω /π, so the oc cuent is qω /π sinθdθ. The no to the e eeent is ˆ φ. Now qω qω Jdd φ θ sinθdθ Jφ δ ( sinθ π π b (5 pts. Wht is the vecto potenti in spce fo this cuent souce ssuing the vecto potenti vnishes s? J( x qω δ ( sinθ ˆ A d x φ d x π xx 6π xx ˆ qω δ ( sinθ Aφ A φ [ cosφcosφ + sinφsinφ ] dd Ω 6π x x ( φ( Y ( θ φ Y ( θ φ ( ( cos φ Y ( θ, φ Y ( θ, φ + ( + ( 8π qω δ π dd Ω x x sin φ Y θ, φ Y θ, φ / i 8π qω cos,, < δ ( d 8π > + sin φ Y θ, φ + Y θ, φ / i qωsinθ δ < ( d π > qωsinθ < π Aφ qω sinθ > π c (5 pts. Wht is the gnetic induction inside the she?
6 ˆ ˆ θ B A ( sinθ Aφ ( Aφ sinθ θ cos ( sin cos ˆ sin sin ˆ cos ˆ qω θ θ φx + θ φy+ θz π sinθ( cosθcosφxˆ+ cosθsinφyˆsinθzˆ qω σω zˆ ˆ z π wheeσ is the (unifo! sufce chge density. d (5 pts. Wht is the gnetic induction outside the she nd the gnetic oent? qω σω cosθ + sinθθ cosθ + sin θθˆ, π B ˆ ˆ ˆ cssic dipoe fied distibution. Coping to Eqn. 5., the oent is coud so be obtined by diect integtion too. qω /. This esut d diπ sin ω ω q π π π di dq d cos d ωq ωq ωq sin θdcosθ [ / ] θ θ φ e (5 pts. Wht is the gnetic enegy inside nd outside the she? The gnetic enegy inside the she is esy B B π σ ω T V 7 5 σω π 8 The gnetic enegy outside the she is ony sighty oe txing.
7 T B B d x cos + sin π σω θ θ dd cosθ π σω cos θ + dd cosθ σω one hf the enegy inside! d 8π σω π 8, ( pts. In ou fin hoewok set we det with cceeto dipoe gnets. In this pobe, sove sii pobe fo D cceeto qudupoe gnets with ode cuent density NI J z (, θ cos θδ (. ( pts. Wht is the vecto potenti geneted by this souce ssuing the vecto potenti vnishes s? The vecto potenti is A z (, θ ( sin θ cos θ ( sin θ cos θ Continuity nd othogonity t A z (, θ A + B < C + D > ipy ( sin θ cos θ ( sin θ cos θ A + B < A + B > The jup condition t nd othogonity ipy A, B fo, nd
8 Az Az NIcos θ + ε ε ( ( B B NI B NI Theefoe A z (, θ NI NI cos θ cos θ < > b ( pts. Wht is the gnetic induction inside nd outside? Tking the cu of the vecto potenti B B θ (, θ (, θ NI Az θ NI NI cos θ Az NI cos θ sin θ < sin θ > < > c ( pts. Wite the induction fo < in tes of the Ctesin coodintes x nd y nd the Ctesin unit vectos ˆx nd ŷ. Wht e the powes of x nd/o y tht ppe? B B ˆ ˆ + Bθθ NI sinθcosθ ( cosθxˆ sinθyˆ ( cos θ sin θ( sinθxˆ cosθyˆ NI NI [ sinθxˆ+ cosθyˆ] [ yxˆ+ xyˆ] The fied is ine in the Ctesin vibes. d ( pts. Wht is the gnetic enegy pe unit ength inside nd outside? The gnetic enegy inside
9 T The gnetic enegy outside B B ddθ NI π d NI N I π π 6 NI T NI π π ( the se s inside (this is esut ike Pobe 5. in Jckson! 5 sin θ + cos θ ddθ N I π 6, e ( pts. Wht is the sef inductnce pe unit ength of the gnet ssuing I is the tot cuent enteing (nd eving! the gnet? LI N I N I T π +π 6 6 N L π 8 N I π 8 Ext cedit (5 pts.: Given wht you edy know, wht is function fo of the potenti s function of ndθ, nd so of x nd y fo sextupoe gnet? (you ve done di(twopoes nd qud(fouupoes edy now!
10 Figue Figue
Classical Electrodynamics
Fist Look t Quntu hysics Cssic Eectoynics Chpte gnetosttics Fy s Lw Qusi-Sttic Fies Cssic Eectoynics of. Y. F. Chen Contents Fist Look t Quntu hysics. The etionship between eectic fie n gnetic fie. iot
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 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 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 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 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 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 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 informationSchool 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 3 Due on Sep. 14, 007 by 5:00 PM Reding Assignments: i) Review the lectue notes. ii) Relevnt
More informationSchool 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 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 informationSolutions. V in = ρ 0. r 2 + a r 2 + b, where a and b are constants. The potential at the center of the atom has to be finite, so a = 0. r 2 + b.
Solutions. Plum Pudding Model (a) Find the coesponding electostatic potential inside and outside the atom. Fo R The solution can be found by integating twice, 2 V in = ρ 0 ε 0. V in = ρ 0 6ε 0 2 + a 2
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 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 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 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 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 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 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 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 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 informationELECTROMAGNETISM. at a point whose position vector with respect to a current element i d l is r. According to this law :
ELECTROMAGNETISM ot-svt Lw: Ths w s used to fnd the gnetc fed d t pont whose poston vecto wth espect to cuent eeent d s. Accodng to ths w : µ d ˆ d = 4π d d The tot fed = d θ P whee ˆ s unt vecto n the
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 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 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 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 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 informationCHAPTER (6) Biot-Savart law Ampere s Circuital Law Magnetic Field Density Magnetic Flux
CAPTE 6 Biot-Svt w Ampee s Ciuit w Mgneti Fied Densit Mgneti Fu Soues of mgneti fied: - Pemnent mgnet - Fow of uent in ondutos -Time ving of eeti fied induing mgneti fied Cuent onfigutions: - Fiment uent
More informationPhysics 505 Electricity and Magnetism Fall 2003 Prof. G. Raithel. Problem Set 7 Maximal score: 25 Points. 1. Jackson, Problem Points.
Physics 505 Eecticity and Magnetism Fa 00 Pof. G. Raithe Pobem et 7 Maxima scoe: 5 Points. Jackson, Pobem 5. 6 Points Conside the i-th catesian component of the B-Fied, µ 0 I B(x) ˆx i ˆx i d (x x ) x
More informationCh.9. Electromagnetic Induction
PART Ch.9. Eectomgnetic nuction F. Mutu nuctnce between the Two Cicuits G. Exmpes of nuctnce Ccution H. Enegy Stoe in the Coi. Wok by Eectomgnetic Foce J. Ey Cuent n Skin Effect Yong-Jin Shin, Pofesso
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 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 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 information[Griffiths Ch.1-3] 2008/11/18, 10:10am 12:00am, 1. (6%, 7%, 7%) Suppose the potential at the surface of a hollow hemisphere is specified, as shown
[Giffiths Ch.-] 8//8, :am :am, Useful fomulas V ˆ ˆ V V V = + θ+ φ ˆ and v = ( v ) + (sin θvθ ) + v θ sinθ φ sinθ θ sinθ φ φ. (6%, 7%, 7%) Suppose the potential at the suface of a hollow hemisphee is specified,
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 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 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 information= ρ. Since this equation is applied to an arbitrary point in space, we can use it to determine the charge density once we know the field.
Gauss s Law In diffeentia fom D = ρ. ince this equation is appied to an abita point in space, we can use it to detemine the chage densit once we know the fied. (We can use this equation to ve fo the fied
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 informationMAGIC058 & MATH64062: Partial Differential Equations 1
MAGIC58 & MATH646: Prti Differenti Equtions 1 Section 4 Fourier series 4.1 Preiminry definitions Definition: Periodic function A function f( is sid to be periodic, with period p if, for, f( + p = f( where
More informationPhysics 2D Lecture Slides Lecture 28: Mar 9th
This is the fist couse t UCSD fo which Lectue on Demnd hve been mde vibe. Minutes of Steming video seved to hundeds of demnds without inteuption (4/7) P. tke minutes to fi out the Steming Video Suvey sent
More informationPLEASE DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO DO SO THEN ENSURE THAT YOU HAVE THE CORRECT EXAM PAPER
OLLSCOIL NA ÉIREANN, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA OLLSCOILE, CORCAIGH UNIVERSITY COLLEGE, CORK 4/5 Autumn Suppement 5 MS Integ Ccuus nd Diffeenti Equtions Pof. P.J. Rippon
More informationDynamically Equivalent Systems. Dynamically Equivalent Systems. Dynamically Equivalent Systems. ME 201 Mechanics of Machines
ME 0 Mechnics of Mchines 8//006 Dynmicy Equivent Systems Ex: Connecting od G Dynmicy Equivent Systems. If the mss of the connecting od m G m m B m m m. Moment out cente of gvity shoud e zeo m G m B Theefoe;
More informationBEAM DIAGRAMS AND FORMULAS. Nomenclature
BEA DIAGAS AND FOULAS Nomencture E = moduus of esticity of stee t 9,000 ksi I = moment of inerti of em (in. 4 ) L = tot ength of em etween rection points (ft) m = mimum moment (kip-in.) = mimum moment
More information16.1 Permanent magnets
Unit 16 Magnetism 161 Pemanent magnets 16 The magnetic foce on moving chage 163 The motion of chaged paticles in a magnetic field 164 The magnetic foce exeted on a cuent-caying wie 165 Cuent loops and
More informationPhysics 235 Final Examination December 4, 2006 Solutions
Physics 35 Fi Emitio Decembe, 6 Soutios.. Fist coside the two u quks. They e idetic spi ½ ptices, so the tot spi c be eithe o. The Pui Picipe equies tht the ove wvefuctio be echge tisymmetic. Sice the
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 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 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 information1 Spherical multipole moments
Jackson notes 9 Spheical multipole moments Suppose we have a chage distibution ρ (x) wheeallofthechageiscontained within a spheical egion of adius R, as shown in the diagam. Then thee is no chage in the
More informationu(r, θ) = 1 + 3a r n=1
Mth 45 / AMCS 55. etuck Assignment 8 ue Tuesdy, Apil, 6 Topics fo this week Convegence of Fouie seies; Lplce s eqution nd hmonic functions: bsic popeties, computions on ectngles nd cubes Fouie!, Poisson
More informationChapter 22: Electric Fields. 22-1: What is physics? General physics II (22102) Dr. Iyad SAADEDDIN. 22-2: The Electric Field (E)
Geneal physics II (10) D. Iyad D. Iyad Chapte : lectic Fields In this chapte we will cove The lectic Field lectic Field Lines -: The lectic Field () lectic field exists in a egion of space suounding a
More informationCharged particle motion in magnetic field
Chaged paticle otion in agnetic field Paticle otion in cued agnetic fieldlines We diide the equation of otion into a elocity coponent along the agnetic field and pependicula to the agnetic field. Suppose
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 informationCoordinate Geometry. Coordinate Geometry. Curriculum Ready ACMNA: 178, 214, 294.
Coordinte Geometr Coordinte Geometr Curricuum Red ACMNA: 78, 4, 94 www.mthetics.com Coordinte COORDINATE Geometr GEOMETRY Shpes ou ve seen in geometr re put onto es nd nsed using gebr. Epect bit of both
More informationLecture 6 Notes, Electromagnetic Theory I Dr. Christopher S. Baird University of Massachusetts Lowell
Lecture 6 Notes, Eectrognetic Theory I Dr. Christopher S. Bird University of Msschusetts Lowe. Associted Legendre Poynois - We now return to soving the Lpce eqution in spheric coordintes when there is
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 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 informationSOLUTIONS TO CONCEPTS CHAPTER 11
SLUTINS T NEPTS HPTE. Gvittionl fce of ttction, F.7 0 0 0.7 0 7 N (0.). To clculte the gvittionl fce on t unline due to othe ouse. F D G 4 ( / ) 8G E F I F G ( / ) G ( / ) G 4G 4 D F F G ( / ) G esultnt
More informationPHYS 301 HOMEWORK #10 (Optional HW)
PHYS 301 HOMEWORK #10 (Optional HW) 1. Conside the Legende diffeential equation : 1 - x 2 y'' - 2xy' + m m + 1 y = 0 Make the substitution x = cos q and show the Legende equation tansfoms into d 2 y 2
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 informationFI 2201 Electromagnetism
F Eectomagnetism exane. skana, Ph.D. Physics of Magnetism an Photonics Reseach Goup Magnetostatics MGNET VETOR POTENTL, MULTPOLE EXPNSON Vecto Potentia Just as E pemitte us to intouce a scaa potentia V
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 informationB da = 0. Q E da = ε. E da = E dv
lectomagnetic Theo Pof Ruiz, UNC Asheville, doctophs on YouTube Chapte Notes The Maxwell quations in Diffeential Fom 1 The Maxwell quations in Diffeential Fom We will now tansfom the integal fom of the
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 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 informationElectromagnetic Theory 1
/ lectomagnetic Theoy uestion : lectostatic Potential negy A sphee of adius caies a positive chage density ρ constant Obviously the spheical coodinates system is appopiate hee Take - C m - and cm τ a)
More informationFARADAY'S LAW. dates : No. of lectures allocated. Actual No. of lectures 3 9/5/09-14 /5/09
FARADAY'S LAW No. of lectues allocated Actual No. of lectues dates : 3 9/5/09-14 /5/09 31.1 Faaday's Law of Induction In the pevious chapte we leaned that electic cuent poduces agnetic field. Afte this
More informationPhys 201A. Homework 6 Solutions. F A and F r. B. According to Newton s second law, ( ) ( )2. j = ( 6.0 m / s 2 )ˆ i ( 10.4m / s 2 )ˆ j.
7. We denote the two foces F A + F B = ma,sof B = ma F A. (a) In unit vecto notation F A = ( 20.0 N)ˆ i and Theefoe, Phys 201A Homewok 6 Solutions F A and F B. Accoding to Newton s second law, a = [ (
More information3.8.1 Electric Potential Due to a System of Two Charges. Figure Electric dipole
3.8 Solved Poblems 3.8.1 Electic Potential Due to a System o Two Chages Conside a system o two chages shown in Figue 3.8.1. Figue 3.8.1 Electic dipole Find the electic potential at an abitay point on the
More information1 Fundamental Solutions to the Wave Equation
1 Fundamental Solutions to the Wave Equation Physical insight in the sound geneation mechanism can be gained by consideing simple analytical solutions to the wave equation. One example is to conside acoustic
More informationObjectives. We will also get to know about the wavefunction and its use in developing the concept of the structure of atoms.
Modue "Atomic physics and atomic stuctue" Lectue 7 Quantum Mechanica teatment of One-eecton atoms Page 1 Objectives In this ectue, we wi appy the Schodinge Equation to the simpe system Hydogen and compae
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 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 informationMark Scheme (Results) January 2008
Mk Scheme (Results) Jnuy 00 GCE GCE Mthemtics (6679/0) Edecel Limited. Registeed in Englnd nd Wles No. 4496750 Registeed Office: One90 High Holbon, London WCV 7BH Jnuy 00 6679 Mechnics M Mk Scheme Question
More informationPhysics 313 Practice Test Page 1. University Physics III Practice Test II
Physics 313 Pactice Test Page 1 Univesity Physics III Pactice Test II This pactice test should give you a ough idea of the fomat and oveall level of the Physics 313 exams. The actual exams will have diffeent
More information7.5-Determinants in Two Variables
7.-eteminnts in Two Vibles efinition of eteminnt The deteminnt of sque mti is el numbe ssocited with the mti. Eve sque mti hs deteminnt. The deteminnt of mti is the single ent of the mti. The deteminnt
More informationJackson 4.7 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jackson 4.7 Homewok obem Soution D. Chistophe S. Baid Univesity of Massachusetts Lowe ROBLEM: A ocaized distibution of chage has a chage density ρ()= 6 e sin θ (a) Make a mutipoe expansion of the potentia
More informationChapter 31 Faraday s Law
Chapte 31 Faaday s Law Change oving --> cuent --> agnetic field (static cuent --> static agnetic field) The souce of agnetic fields is cuent. The souce of electic fields is chage (electic onopole). Altenating
More informationJackson 3.3 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jackson 3.3 Homewok Pobem Soution D. Chistophe S. Baid Univesity of Massachusetts Lowe POBLEM: A thin, fat, conducting, cicua disc of adius is ocated in the x-y pane with its cente at the oigin, and is
More informationPHYS 2421 Fields and Waves
PHYS 242 Felds nd Wves Instucto: Joge A. López Offce: PSCI 29 A, Phone: 747-7528 Textook: Unvesty Physcs e, Young nd Feedmn 23. Electc potentl enegy 23.2 Electc potentl 23.3 Clcultng electc potentl 23.4
More informationVector Spherical Harmonics and Spherical Waves
DEPARTMENT OF PHYSICS INDIAN INSTITUTE OF TECHNOLOGY, MADRAS PH5020 Eectomagnetic Theoy Mach 2017 by Suesh Govinaajan, Depatment of Physics, IIT Maas Vecto Spheica Hamonics an Spheica Waves Let us sove
More information9.4 The response of equilibrium to temperature (continued)
9.4 The esponse of equilibium to tempetue (continued) In the lst lectue, we studied how the chemicl equilibium esponds to the vition of pessue nd tempetue. At the end, we deived the vn t off eqution: d
More informationChapter 6 Differential Analysis of Fluid Flow
1 Chapte 6 Diffeential Analysis of Fluid Flow Inviscid flow: Eule s equations of otion Flow fields in which the sheaing stesses ae zeo ae said to be inviscid, nonviscous, o fictionless. fo fluids in which
More information( ) ( ) Physics 111. Lecture 13 (Walker: Ch ) Connected Objects Circular Motion Centripetal Acceleration Centripetal Force Sept.
Physics Lectue 3 (Wlke: Ch. 6.4-5) Connected Objects Cicul Motion Centipetl Acceletion Centipetl Foce Sept. 30, 009 Exmple: Connected Blocks Block of mss m slides on fictionless tbletop. It is connected
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 informationQualifying Examination Electricity and Magnetism Solutions January 12, 2006
1 Qualifying Examination Electicity and Magnetism Solutions Januay 12, 2006 PROBLEM EA. a. Fist, we conside a unit length of cylinde to find the elationship between the total chage pe unit length λ and
More informationPhys 1215, First Test. September 20, minutes Name:
Phys 115, Fist Test. Septembe 0, 011 50 minutes Name: Show all wok fo maximum cedit. Each poblem is woth 10 points. k =.0 x 10 N m / C ε 0 = 8.85 x 10-1 C / N m e = 1.60 x 10-1 C ρ = 1.68 x 10-8 Ω m fo
More informationSuggested Solution to Assignment 5
MATH 4 (5-6) prti diferenti equtions Suggested Soution to Assignment 5 Exercise 5.. () (b) A m = A m = = ( )m+ mπ x sin mπx dx = x mπ cos mπx + + 4( )m 4 m π. 4x cos mπx dx mπ x cos mπxdx = x mπ sin mπx
More informationFARADAY'S LAW dt
FAADAY'S LAW 31.1 Faaday's Law of Induction In the peious chapte we leaned that electic cuent poduces agnetic field. Afte this ipotant discoey, scientists wondeed: if electic cuent poduces agnetic field,
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 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 informationMutipy by r sin RT P to get sin R r r R + T sin (sin T )+ P P = (7) ffi So we hve P P ffi = m (8) choose m re so tht P is sinusoi. If we put this in b
Topic 4: Lpce Eqution in Spheric Co-orintes n Mutipoe Expnsion Reing Assignment: Jckson Chpter 3.-3.5. Lpce Eqution in Spheric Coorintes Review of spheric por coorintes: x = r sin cos ffi y = r sin sin
More informationChapter 8. Ch.8, Potential flow
Ch.8, Voticit (epetition) Velocit potentil Stem function Supeposition Cicultion -dimensionl bodies Kutt-Joukovskis lift theoem Comple potentil Aismmetic potentil flow Rotting fluid element Chpte 4 Angul
More informationThe Formulas of Vector Calculus John Cullinan
The Fomuls of Vecto lculus John ullinn Anlytic Geomety A vecto v is n n-tuple of el numbes: v = (v 1,..., v n ). Given two vectos v, w n, ddition nd multipliction with scl t e defined by Hee is bief list
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 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 informationReview: Electrostatics and Magnetostatics
Review: Electostatics and Magnetostatics In the static egime, electomagnetic quantities do not vay as a function of time. We have two main cases: ELECTROSTATICS The electic chages do not change postion
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 informationPhysics 235 Chapter 5. Chapter 5 Gravitation
Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus
More information