Lesson 07 Electrostatics in Materials


 Juniper Powers
 11 months ago
 Views:
Transcription
1 Electomagnetcs P71 Lesson 07 Electostatcs n Mateals Intoducton Mateals consst of atoms, whch ae made u of nucle (oste chages) and electons (negate chages). As a esult, the total electc feld must be modfed n the esence of mateals. We usually classfy mateals nto thee tyes: 1) Conductos: The electons n the outemost shells of the atoms ae ey loosely held and can feely mgate among atoms due to themal exctaton at oom temeatues. These electons ae shaed by all atoms. ) Delectcs: All electons ae confned wthn the nne shells of atoms, and can hadly mgate een by alyng a stong exctaton. 3) emconductos: Outemost electons ae modeately confned, whch may not mgate due to themal exctaton but become moable when alyng an extenal electc feld. Fg (a) Enegy leels and bands. (b) Band dagams of nsulato, semconducto, and metal. In tems of quantum theoy, the electons of an atom can only stay at dscete enegy leels (Fg. 71a, left). When a lage numbe of atoms aggegate n odeed manne (cystallne sold), each enegy leel slts nto densely saced leels (enegy band ). The allowed enegy bands can be oelaed o seaated by fobdden enegy ga (Fg. 71a, ght). Electc chaactestcs of mateals deend on the band stuctue and how they ae flled by the electons at the temeatue of 0 K (Fg. 71b):
2 Electomagnetcs P71) Conductos: Bands ae oelaed, o the conducton band s atally flled. ) Delectcs: Enegy ga s lage, and the conducton band s emty. 3) emconductos: Enegy ga s small, and the conducton band s emty. In ths lesson, we wll dscuss the feld behaos both nsde some mateal and on the nteface of dffeent mateals. 7.1 tatc Electc Feld n the Pesence of Conductos Chage and electc feld nsde a conducto Assume that some oste (o negate) chages ae ntoduced n the nteo of a conducto. Intally an electc feld wll be set u, whch wll ush the fee chages away fom one anothe by Coulomb s foce and modfy the electc feld dstbuton by tuns. Ths chagefeld nteacton ocess wll contnue untl: (1) all the chages each the conducto suface (no way to leae): ρ = 0, (7.1) and () the suface chages edstbute themseles such that no electc feld exsts nsde the conducto: E = 0 (7.) Eq. (7.) can be justfed by: 1) If E 0 somewhee nsde the conducto, the electc otental V ( ) s nonunfom. Ths wll esult n a contadcton that wok has to be done to moe fee chages. ) The fee chage dstbuton that leads to E = 0 actually coesonds to the lowest system enegy,.e., a state of equlbum (Lesson 9). Fo good conductos lke coe, the tme equed to each the equlbum s only n the ode of sec (Lesson 10).
3 Electomagnetcs P73 Aconducto nteface nce the chages on the conducto suface wll not be at est f thee s tangental electc feld comonent, the suface electc feld only has nomal comonent n the equlbum. goous bounday condtons (BCs) ae deed by alyng eq s (6.3), (6.4) wth esect to the dffeental contou abcda and the thn ll box (Fg. 7) acoss the aconducto nteface, esectely. Fg. 7. Dffeental contou and ll box used to dee BCs on the aconducto nteface (afte DKC). 1) Tangental BC: By eq s (6.4), (7.), E dl = E Δw + 0 Δw = 0, abcda Δh 0 t ) Nomal BC: BC: By eq s (6.3), (7.13), E = 0 (7.3) t E ds = E ρ Δ Δ =, s n Δh 0 ε 0 ρ s ε 0 E =, (7.4) n whee E n s n the nomal decton fom the conducto to the a. Examle 71: Consde a ont chage + located at the cente of a shecal conductng shell of fnte thckness (Fg. 73a). Fnd E, V nsde and outsde the shell.
4 Electomagnetcs P74 Fg (a) The geomety of a conductng shee. The coesondng (b) adal comonents of E (dashed), and (c) electc otental V. Ans: Because of the shecal symmety of the souce, we hae E a = E () eeywhee, and the Gaussan sufaces ae concentc shecal sufaces centeed at the ogn. < : E ds = E ( ) ( 4π ) =, E ( ) =. 3 ε 0 4 πε 0 () Fo < < o : By eq. (7.), E = 0, E ds = 0 (1) Fo fo any enclosed suface nsde the shell. Ths mles that fee chages of must be nduced on the nne suface of the shell. By the conseaton of chages and eq. (7.1), fee chages of + must be nduced on the oute suface of the shell. (3) Fo > o : The total chage enclosed by a Gaussan suface 1 s +, E ( ) =. 4 πε 0 The cue of E = E () s shown n Fg. 73b. One can dee the electc otental dstbuton V () by lne ntegal of E (Fg. 73c).
5 Electomagnetcs P tatc Electc Feld n the Pesence of Delectcs Concet of nduced doles A delectc molecule could be nonola (e.g., H, CH 4 ) o ola (e.g., H O, HCl, NH 3, O 3 ), deendng on whethe thee s nonzeo electc dole moment [eq. (6.17)]. Howee, a delectc bulk made u of a lage numbe of andomly oented molecules (ola o nonola) tycally has no macoscoc dole moment n the absence of extenal electc feld. When an electc feld s aled, the bound chages of the delectcs cannot feely mgate to the suface. Instead, (1) each nonola molecule s olazed fo the ostely chaged nucle and negately chaged electons ae slghtly dslaced n ooste dectons (dstoton of electon cloud). () Most of the nddual dole moments (ectos) ae algned n the same decton due to the electc toque. In ethe case, a macoscoc dole moment emeges. Fg A coss secton of a olazed delectc medum (afte DKC). Although an electc dole s neutal n ensemble, t odes nonzeo otental and electc feld [eq s (6.18), (6.16)], whch wll modfy the total feld nsde and outsde delectcs. Polazaton ecto and equalent chage denstes To analyze the effect of nduced doles, we defne a (mcoscoc) olazaton ecto P as
6 Electomagnetcs P76 the olume densty of electc dole moment: k P lm (7.5) Δ 0 Δ whee k denotes the kth dole moment nsde a dffeental olume Δ. If the olazaton ecto P s nhomogeneous (.e., aes wth oston) somewhee, thee must exst net (bound) olazaton chage at that oston. Ths henomenon can be llustated n two cases. Fg (a) The model to deduce the olazaton suface chage densty. (b) Net olazaton olume chage exsts whee the olazaton ecto P s nhomogeneous. 1) The olazaton ecto P s dscontnuous on the adelectc nteface, whee thee must exst net olazaton chage (Fg. 75a). To quanttately model ths henomenon, consde a dffeental aalleleed V bounded by a closed suface b, whee the dole moment of each molecule s = qd. The to suface of has a unt outwad b nomal ecto a n a n and an aea of Δ. The effecte heght of the aalleleed V s d. The net olazaton chage wthn V s: Δ = nq ( d a ) Δ = ( P a ) Δ n whee n (1/m 3 ) s the numbe densty of molecules. The coesondng olazaton n,
7 Electomagnetcs P77 suface chage densty Δ Δ s: s P a ρ = n (C/m ) (7.6) ) Consde two doles wth dffeent olazaton ectos P 1, P n the nteo of a dffeental olume V bounded by a closed suface (Fg. 75b). Although each nddual dole s electcally neutal, some net chage can exst on the nteface due to ncomlete cancellaton of the olazaton chage between adjacent doles. To mantan the electc neutalty, must be equal n magntude but ooste n sgn to the total suface olazaton chage s1 + s. Let ρ eesent the olazaton olume chage densty, ( P an ) ds = P ds = ( P) V ρ d. By eq s (7.6), (5.4), = d = V. The equement of ( + ) ( P) s1 s s1 + s = esults n: ρ = (C/m 3 ) (7.7) <Comment> 1) Eq. (7.6) can be egaded as a secal case of eq. (7.7) on the adelectc nteface, whee the degence of the olazaton ecto P s nfnte. ) Equalent olazaton chage denstes ρ s, ρ can be used n collaboaton wth eq s (6.10), (6.15) to ealuate the electc feld and otental contbuted by olazed delectcs. Electc flux densty In the esence of delectc mateals, the total electc feld would be ceated by both the fee and olazaton chages. Fundamental ostulate eq. (6.1) s thus modfed as:
8 Electomagnetcs P78 ρ + ρ E =. By eq. (7.7), we hae: D = ρ, (7.8) D ε 0 ε E + P (C/m ), (7.9) = 0 whee the electc flux densty D chaactezes the contbuton fom the fee chages. The ntegal fom of eq. (7.8) (Gauss s law) becomes: D ds = (7.10) Fo lnea, homogeneous, and sotoc delectcs, the olazaton ecto s ootonal to the electc feld (by the elastc sng model): P = ε 0 χee, (7.11) whee the electc suscetblty χ e s a dmensonless quantty ndeendent of magntude (lnea), oston (homogeneous), and decton (sotoc) of E. By eq s (7.9), (7.11), D = εe, (7.1) whee the emttty of the medum ε s defned as: ( 1+ χe ) ε 0 ε = (7.13) <Comment> The stategy s usng a sngle constant ε to elace the tedous nduced doles, olazaton ecto, and equalent olazaton chages n detemnng the total electc feld. Examle 7: Consde a aallellate caacto. (1) D descbes the suface densty of fee chages on the conductng lates. () P descbes the suface densty of olazaton chages.
9 Electomagnetcs P79 (3) ε E 0 descbes the suface densty of total chage o uncomensated fee chage. Fg Physcal meanngs of electc flux densty D, olazaton ecto P, and electc feld ntensty E llustated n the examle of a aallellate caacto (afte C. C. u). Examle 73: A ont chage + at the cente of a shecal delectc shell of emttty ε ε 0 (Fg. 77a), whee χe ε = 1 + s the elate emttty. Fnd D, E, V, P, and ρ s. Fg (a) The geomety of a delectc shee. The coesondng (b) adal comonents of D (sold), ε E (dashed), P (dashdot), and (c) electc otental 0 V.
10 Electomagnetcs P710 Ans: (1) By shecal symmety, D a = D (). By eq. (7.10) and the fact that delectc mateals contbute to no fee chage: D ds = D ( ) ( 4π ) = D ( ) =, fo all > 0. 4 π () By shecal symmety, E a = E (). By eq. (7.1),, E ( ) =, whee 4 πε ε ε 0, fo < < ε = ε 0, othewse 0. (3) V () s deed by ntegaton of E (). (4) By eq. (7.9), P = D ε E = a P ( ), whee P ( ) = D ( ) ε 0E ( ), 0 1 ( 1 ε ) P ( ) = 4π 0, othewse, fo < < 0. (5) By eq (7.6), the suface olazaton chage densty becomes: ρ s a = 1 1 ( 1 ε ) ( a ) = ( 1 ε ) ( 1 1 ε ) π 4 4π o ( > 0), fo = o 4π ( < 0), fo = ; The total olazaton suface chage s: s = ρ = s 1 1 ( 1 ε ) ( 4π ) = ( 1 ε ) 4π 1 ( 1 ε ) ( > 0), fo = o ( < 0), fo = ; They ceate a adally nwad E to educe the total electc feld n the delectcs. 7.3 Geneal Bounday Condtons fo Electc Felds Deaton As n aconducto nteface, we aly the ntegal foms of the two fundamental ostulates on the dffeental contou abcda and the thn ll box wth Δh 0 acoss the nteface of
11 Electomagnetcs P711 two delectc meda (Fg. 78) to dee the BCs fo the tangental and nomal comonents of the electc feld. Fg Dffeental contou and ll box used to dee geneal BCs (afte DKC). E dl = E Δw + E Δw, 1) Tangental BC: By eq. (6.4), ( ) ( ) 0 abcda E t Et ) Nomal BC: BC: By eq. (7.10), D ds = a 1 t t = 1 = (7.14), ( D a + D a )( Δ ) = ρ Δ 1 n ( D1 D ) = ρs n n1 s, (7.15) Eq. (7.15) can also be wtten as: D ρ 1 n D n = s, whee n D ( = 1, ) denotes comonent of medum 1). D n the decton of a n (unt nomal ecto dected fom medum to <Comment> 1) Eq s (7.3), (7.4) ae secal cases of eq s (7.14), (7.15), whee { E D } 0, = ae used. ) Only fee suface chage densty ρ s counts n eq. (7.15). If the two ntefacng meda ae both delectcs, ρ s = 0, D1 n = Dn. 3) Eq s (7.14), (7.15) eman ald een the felds ae tmeayng (Lesson 14, o Ch 7 of the textbook). BCs exlan why most otcal comonents (wth mateal ntefaces) ae olazaton deendent.
Chapter 23: Electric Potential
Chapte 23: Electc Potental Electc Potental Enegy It tuns out (won t show ths) that the tostatc foce, qq 1 2 F ˆ = k, s consevatve. 2 Recall, fo any consevatve foce, t s always possble to wte the wok done
More informationScalars and Vectors Scalar
Scalas and ectos Scala A phscal quantt that s completel chaacteed b a eal numbe (o b ts numecal value) s called a scala. In othe wods a scala possesses onl a magntude. Mass denst volume tempeatue tme eneg
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 unsymmetic known
More informationRigid Bodies: Equivalent Systems of Forces
Engneeng Statcs, ENGR 2301 Chapte 3 Rgd Bodes: Equvalent Sstems of oces Intoducton Teatment of a bod as a sngle patcle s not alwas possble. In geneal, the se of the bod and the specfc ponts of applcaton
More informationToday s Plan. Electric Dipoles. More on Gauss Law. Comment on PDF copies of Lectures. Final iclicker rollcall
Today s Plan lectic Dipoles Moe on Gauss Law Comment on PDF copies of Lectues Final iclicke ollcall lectic Dipoles A positive (q) and negative chage (q) sepaated by a small distance d. lectic dipole
More informationGauss Law. Physics 231 Lecture 21
Gauss Law Physics 31 Lectue 1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing
More informationUNIVERSITÀ DI PISA. Math thbackground
UNIVERSITÀ DI ISA Electomagnetc Radatons and Bologcal l Inteactons Lauea Magstale n Bomedcal Engneeng Fst semeste (6 cedts), academc ea 2011/12 of. aolo Nepa p.nepa@et.unp.t Math thbackgound Edted b D.
More informationObjectives: After finishing this unit you should be able to:
lectic Field 7 Objectives: Afte finishing this unit you should be able to: Define the electic field and explain what detemines its magnitude and diection. Wite and apply fomulas fo the electic field intensity
More information(Sample 3) Exam 1  Physics Patel SPRING 1998 FORM CODE  A (solution key at end of exam)
(Sample 3) Exam 1  Physics 202  Patel SPRING 1998 FORM CODE  A (solution key at end of exam) Be sue to fill in you student numbe and FORM lette (A, B, C) on you answe sheet. If you foget to include
More informationChapter 22 The Electric Field II: Continuous Charge Distributions
Chapte The lectic Field II: Continuous Chage Distibutions A ing of adius a has a chage distibution on it that vaies as l(q) l sin q, as shown in Figue 9. (a) What is the diection of the electic field
More informationPhysical & Interfacial Electrochemistry 2013
Physcal & Intefacal Electochemsty 013 Lectue 3. Ionon nteactons n electolyte solutons. Module JS CH3304 MoleculaThemodynamcs and Knetcs IonIon Inteactons The themodynamc popetes of electolyte solutons
More information$ i. !((( dv vol. Physics 8.02 Quiz One Equations Fall q 1 q 2 r 2 C = 2 C! V 2 = Q 2 2C F = 4!" or. r ˆ = points from source q to observer
Physics 8.0 Quiz One Equations Fall 006 F = 1 4" o q 1 q = q q ˆ 3 4" o = E 4" o ˆ = points fom souce q to obseve 1 dq E = # ˆ 4" 0 V "## E "d A = Q inside closed suface o d A points fom inside to V =
More informationReview of Vector Algebra and Vector Calculus Operations
Revew of Vecto Algeba and Vecto Calculus Opeatons Tpes of vaables n Flud Mechancs Repesentaton of vectos Dffeent coodnate sstems Base vecto elatons Scala and vecto poducts Stess Newton s law of vscost
More information17.1 Electric Potential Energy. Equipotential Lines. PE = energy associated with an arrangement of objects that exert forces on each other
Electic Potential Enegy, PE Units: Joules Electic Potential, Units: olts 17.1 Electic Potential Enegy Electic foce is a consevative foce and so we can assign an electic potential enegy (PE) to the system
More information8.022 (E&M)  Lecture 2
8.0 (E&M)  Lectue Topics: Enegy stoed in a system of chages Electic field: concept and poblems Gauss s law and its applications Feedback: Thanks fo the feedback! caed by Pset 0? Almost all of the math
More informationReview. Electrostatic. Dr. Ray Kwok SJSU
Review Electostatic D. Ray Kwok SJSU Paty Balloons Coulomb s Law F e q q k 1 Coulomb foce o electical foce. (vecto) Be caeful on detemining the sign & diection. k 9 10 9 (N m / C ) k 1 4πε o k is the Coulomb
More informationModule 18: Outline. Magnetic Dipoles Magnetic Torques
Module 18: Magnetic Dipoles 1 Module 18: Outline Magnetic Dipoles Magnetic Toques 2 IA nˆ I A Magnetic Dipole Moment μ 3 Toque on a Cuent Loop in a Unifom Magnetic Field 4 Poblem: Cuent Loop Place ectangula
More informationCSU ATS601 Fall Other reading: Vallis 2.1, 2.2; Marshall and Plumb Ch. 6; Holton Ch. 2; Schubert Ch r or v i = v r + r (3.
3 Eath s Rotaton 3.1 Rotatng Famewok Othe eadng: Valls 2.1, 2.2; Mashall and Plumb Ch. 6; Holton Ch. 2; Schubet Ch. 3 Consde the poston vecto (the same as C n the fgue above) otatng at angula velocty.
More informationChapter 23: GAUSS LAW 343
Chapte 23: GAUSS LAW 1 A total chage of 63 10 8 C is distibuted unifomly thoughout a 27cm adius sphee The volume chage density is: A 37 10 7 C/m 3 B 69 10 6 C/m 3 C 69 10 6 C/m 2 D 25 10 4 C/m 3 76 10
More informationConcept of Game Equilibrium. Game theory. Normal Form Representation. Game definition. Lecture Notes II1 Static Games of Complete Information
Game theoy he study of multeson decsons Fou tyes of games Statc games of comlete nfomaton ynamc games of comlete nfomaton Statc games of ncomlete nfomaton ynamc games of ncomlete nfomaton Statc v. dynamc
More informationPhysics 11 Chapter 20: Electric Fields and Forces
Physics Chapte 0: Electic Fields and Foces Yesteday is not ous to ecove, but tomoow is ous to win o lose. Lyndon B. Johnson When I am anxious it is because I am living in the futue. When I am depessed
More informationChapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23,
hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, 07 47 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, 07 48 I. ETOR AND TENOR ANALYI I... Tenso functon th Let A n n
More information4. Electrodynamic fields
4. Electodynamic fields D. Rakhesh Singh Kshetimayum 1 4.1 Intoduction Electodynamics Faaday s law Maxwell s equations Wave equations Lenz s law Integal fom Diffeential fom Phaso fom Bounday conditions
More information7/1/2008. Adhi Harmoko S. a c = v 2 /r. F c = m x a c = m x v 2 /r. Ontang Anting Moment of Inertia. Energy
7//008 Adh Haoko S Ontang Antng Moent of neta Enegy Passenge undego unfo ccula oton (ccula path at constant speed) Theefoe, thee ust be a: centpetal acceleaton, a c. Theefoe thee ust be a centpetal foce,
More informationConsequences of Long Term Transients in Large Area High Density Plasma Processing: A 3Dimensional Computational Investigation*
ISPC 2003 June 2227, 2003 Consequences of Long Tem Tansents n Lage Aea Hgh Densty Plasma Pocessng: A 3Dmensonal Computatonal Investgaton* Pamod Subamonum** and Mak J Kushne*** **Dept of Chemcal and Bomolecula
More informationOn the Sun s ElectricField
On the Sun s ElecticField D. E. Scott, Ph.D. (EE) Intoduction Most investigatos who ae sympathetic to the Electic Sun Model have come to agee that the Sun is a body that acts much like a esisto with a
More informationAnalysis of the magnetic field, force, and torque for twodimensional Halbach cylinders
Downloaded fom obt.dtu.dk on: Feb 19, 218 Analyss of the magnetc feld, foce, and toque fo twodmensonal Halbach cylndes Bjøk, Rasmus; Smth, Andes; Bahl, Chstan Publshed n: Jounal of Magnetsm and Magnetc
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 cuentcaying wie 165 Cuent loops and
More informationInteratomic Forces. Overview
Inteatomic Foces Oveview an de Walls (shot ange ~1/ 6, weak ~0.010.1 e) Ionic (long ange, ~1/, stong ~510 e) Metallic (no simple dependence, ~0.1e) Covalent (no simple dependence, diectional,~3 e) Hydogen
More informationThe Law of BiotSavart & RHR P θ
The Law of iotsavat & RHR P R dx x Jeanaptiste iot élix Savat Phys 122 Lectue 19 G. Rybka Recall: Potential Enegy of Dipole Wok equied to otate a cuentcaying loop in a magnetic field Potential enegy
More informationMultistage Median Ranked Set Sampling for Estimating the Population Median
Jounal of Mathematcs and Statstcs 3 (: 5864 007 ISSN 5493644 007 Scence Publcatons Multstage Medan Ranked Set Samplng fo Estmatng the Populaton Medan Abdul Azz Jeman Ame AlOma and Kamaulzaman Ibahm
More informationCh 30  Sources of Magnetic Field! The BiotSavart Law! = k m. r 2. Example 1! Example 2!
Ch 30  Souces of Magnetic Field 1.) Example 1 Detemine the magnitude and diection of the magnetic field at the point O in the diagam. (Cuent flows fom top to bottom, adius of cuvatue.) Fo staight segments,
More informationPHYSICS 272 Electric & Magnetic Interactions
PHYS 7: Matte and Inteactions II  Electic And Magnetic Inteactions http://www.physics.pudue.edu/academic_pogams/couses/phys7/ PHYSICS 7 Electic & Magnetic Inteactions Lectue 3 Chaged Objects; Polaization
More informationYour Comments. Conductors and Insulators with Gauss's law please...so basically everything!
You Comments I feel like I watch a pelectue, and agee with eveything said, but feel like it doesn't click until lectue. Conductos and Insulatos with Gauss's law please...so basically eveything! I don't
More informationPHY121 Formula Sheet
HY Foula Sheet One Denson t t Equatons o oton l Δ t Δ d d d d a d + at t + at a + t + ½at² + a(  ) ojectle oton y cos θ sn θ gt ( cos θ) t y ( sn θ) t ½ gt y a a sn θ g sn θ g otatonal a a a + a t Ccula
More informationCurrent, Resistance and
Cuent, Resistance and Electomotive Foce Chapte 25 Octobe 2, 2012 Octobe 2, 2012 Physics 208 1 Leaning Goals The meaning of electic cuent, and how chages move in a conducto. What is meant by esistivity
More informationPotential Theory. Copyright 2004
Copyght 004 4 Potental Theoy We have seen how the soluton of any classcal echancs poble s fst one of detenng the equatons of oton. These then ust be solved n ode to fnd the oton of the patcles that copse
More informationDescription Linear Angular position x displacement x rate of change of position v x x v average rate of change of position
Chapte 5 Ccula Moton The language used to descbe otatonal moton s ey smla to the language used to descbe lnea moton. The symbols ae deent. Descpton Lnea Angula poston dsplacement ate o change o poston
More informationPage 1 of 6 Physics II Exam 1 155 points Name Discussion day/time Pat I. Questions 110. 8 points each. Multiple choice: Fo full cedit, cicle only the coect answe. Fo half cedit, cicle the coect answe and
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 informationMULTIPOLE FIELDS. Multipoles, 2 l poles. Monopoles, dipoles, quadrupoles, octupoles... Electric Dipole R 1 R 2. P(r,θ,φ) e r
MULTIPOLE FIELDS Mutpoes poes. Monopoes dpoes quadupoes octupoes... 4 8 6 Eectc Dpoe +q O θ e R R P(θφ) q e The potenta at the fed pont P(θφ) s ( θϕ )= q R R Bo E. Seneus : Now R = ( e) = + cosθ R = (
More informationPhysics 107 TUTORIAL ASSIGNMENT #8
Physics 07 TUTORIAL ASSIGNMENT #8 Cutnell & Johnson, 7 th edition Chapte 8: Poblems 5,, 3, 39, 76 Chapte 9: Poblems 9, 0, 4, 5, 6 Chapte 8 5 Inteactive Solution 8.5 povides a model fo solving this type
More informationClass #16 Monday, March 20, 2017
D. Pogo Class #16 Monday, Mach 0, 017 D NonCatesian Coodinate Systems A point in space can be specified by thee numbes:, y, and z. O, it can be specified by 3 diffeent numbes:,, and z, whee = cos, y =
More informationChapter 4 Gauss s Law
Chapte 4 Gauss s Law 4.1 lectic Flux... 1 4. Gauss s Law... xample 4.1: Infinitely Long Rod of Unifom Chage Density... 7 xample 4.: Infinite Plane of Chage... 9 xample 4.3: Spheical Shell... 11 xample
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 informationChapter 3 Waves in an Elastic Whole Space. Equation of Motion of a Solid
Chapte 3 Waves n an Elastc Whole Space Equaton of Moton of a Sold Hopefully, many of the topcs n ths chapte ae evew. Howeve, I fnd t useful to dscuss some of the key chaactestcs of elastc contnuous meda.
More informationSlide 1. Quantum Mechanics: the Practice
Slde Quantum Mecancs: te Pactce Slde Remnde: Electons As Waves Wavelengt momentum = Planck? λ p = = 6.6 x 034 J s Te wave s an exctaton a vbaton: We need to know te ampltude of te exctaton at evey pont
More informationPhysics 202, Lecture 2
Physics 202, Lectue 2 Todays Topics Electic Foce and Electic Fields Electic Chages and Electic Foces Coulomb's Law Physical Field The Electic Field Electic Field Lines Motion of Chaged Paticle in Electic
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 informationElectrostatics. 3) positive object: lack of electrons negative object: excess of electrons
Electostatics IB 12 1) electic chage: 2 types of electic chage: positive and negative 2) chaging by fiction: tansfe of electons fom one object to anothe 3) positive object: lack of electons negative object:
More informationPhys272 Lecture 18. Mutual Inductance SelfInductance RL Circuits
Phys7 ectue 8 Mutual nductance Selfnductance  Cicuits Mutual nductance f we have a constant cuent i in coil, a constant magnetic field is ceated and this poduces a constant magnetic flux in coil. Since
More information209 ELECTRIC FIELD LINES 209 ELECTRIC POTENTIAL. Answers to the Conceptual Questions. Chapter 20 Electricity 241
Chapte 0 Electicity 41 09 ELECTRIC IELD LINES Goals Illustate the concept of electic field lines. Content The electic field can be symbolized by lines of foce thoughout space. The electic field is stonge
More informationCelso José Faria de Araújo, M.Sc.
Celso José Faa de Aaújo, M.c. he deal dode: (a) dode ccut symbol; (b)  chaactestc; (c) equalent ccut n the eese decton; (d) equalent ccut n the fowad decton. odes (a) Rectfe ccut. (b) nput waefom. (c)
More informationLecture Note 3. Eshelby s Inclusion II
ME340B Elastcty of Mcroscopc Structures Stanford Unversty Wnter 004 Lecture Note 3. Eshelby s Incluson II Chrs Wenberger and We Ca c All rghts reserved January 6, 004 Contents 1 Incluson energy n an nfnte
More informationChapter 2. A Brief Review of Electron Diffraction Theory
Chapte. A Bef Revew of Electon Dffacton Theoy 8 Chapte. A Bef Revew of Electon Dffacton Theoy The theoy of gas phase electon dffacton s hadly a new topc. t s well establshed fo decades and has been thooughly
More informationCapacitors and Capacitance
Capacitos and Capacitance Capacitos ae devices that can stoe a chage Q at some voltage V. The geate the capacitance, the moe chage that can be stoed. The equation fo capacitance, C, is vey simple: C Q
More informationAnalysis of Arithmetic. Analysis of Arithmetic. Analysis of Arithmetic RoundOff Errors. Analysis of Arithmetic. Analysis of Arithmetic
In the fixedoint imlementation of a digital filte only the esult of the multilication oeation is quantied The eesentation of a actical multilie with the quantie at its outut is shown below u v Q ^v The
More information22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 4: Toroidal Equilibrium and Radial Pressure Balance
.615, MHD Theoy of Fusion Systems Pof. Feidbeg Lectue 4: Tooidal Equilibium and Radial Pessue Balance Basic Poblem of Tooidal Equilibium 1. Radial pessue balance. Tooidal foce balance Radial Pessue Balance
More informationTarget Boards, JEE Main & Advanced (IIT), NEET Physics Gauss Law. H. O. D. Physics, Concept Bokaro Centre P. K. Bharti
Page 1 CONCPT: JB, Nea Jitenda Cinema, City Cente, Bokao www.vidyadishti.og Gauss Law Autho: Panjal K. Bhati (IIT Khaagpu) Mb: 74884484 Taget Boads, J Main & Advanced (IIT), NT 15 Physics Gauss Law Autho:
More informationPhysics Electricity and Magnetism Lecture 3  Electric Field. Y&F Chapter 21 Sec. 4 7
Physics  lecticity and Magnetism Lectue 3  lectic Field Y&F Chapte Sec. 4 7 Recap & Definition of lectic Field lectic Field Lines Chages in xtenal lectic Fields Field due to a Point Chage Field Lines
More informationINTERVAL ESTIMATION FOR THE QUANTILE OF A TWOPARAMETER EXPONENTIAL DISTRIBUTION
Intenatonal Jounal of Innovatve Management, Infomaton & Poducton ISME Intenatonalc0 ISSN 855439 Volume, Numbe, June 0 PP. 788 INTERVAL ESTIMATION FOR THE QUANTILE OF A TWOPARAMETER EXPONENTIAL DISTRIBUTION
More informationPHYS 705: Classical Mechanics. Derivation of Lagrange Equations from D Alembert s Principle
1 PHYS 705: Classcal Mechancs Devaton of Lagange Equatons fom D Alembet s Pncple 2 D Alembet s Pncple Followng a smla agument fo the vtual dsplacement to be consstent wth constants,.e, (no vtual wok fo
More informationKepler s problem gravitational attraction
Kele s oblem gavitational attaction Summay of fomulas deived fo twobody motion Let the two masses be m and m. The total mass is M = m + m, the educed mass is µ = m m /(m + m ). The gavitational otential
More informationThis article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and
Ths atcle aeaed n a jounal ublshed by Elseve. The attached coy s funshed to the autho fo ntenal noncommecal eseach and educaton use, ncludng fo nstucton at the authos nsttuton and shang wth colleagues.
More informationSPH4UI 28/02/2011. Total energy = K + U is constant! Electric Potential Mr. Burns. GMm
8//11 Electicity has Enegy SPH4I Electic Potential M. Buns To sepaate negative an positive chages fom each othe, wok must be one against the foce of attaction. Theefoe sepeate chages ae in a higheenegy
More informationClassical Models of the Interface between an Electrode and an Electrolyte
Except fom the Poceedngs of the OMSOL onfeence 9 Mlan lasscal Models of the Inteface between an Electode and an Electolyte E. Gongadze *, S. Petesen, U. Beck, U. van Renen Insttute of Geneal Electcal Engneeng,
More informationAN ALGORITHM FOR CALCULATING THE CYCLETIME AND GREENTIMES FOR A SIGNALIZED INTERSECTION
AN AGORITHM OR CACUATING THE CYCETIME AND GREENTIMES OR A SIGNAIZED INTERSECTION Henk Taale 1. Intoducton o a snalzed ntesecton wth a fedte contol state the cclete and eentes ae the vaables that nfluence
More information(8) Gain Stage and Simple Output Stage
EEEB23 Electoncs Analyss & Desgn (8) Gan Stage and Smple Output Stage Leanng Outcome Able to: Analyze an example of a gan stage and output stage of a multstage amplfe. efeence: Neamen, Chapte 11 8.0) ntoducton
More informationB. Spherical Wave Propagation
11/8/007 Spheical Wave Popagation notes 1/1 B. Spheical Wave Popagation Evey antenna launches a spheical wave, thus its powe density educes as a function of 1, whee is the distance fom the antenna. We
More informationObjectives. Chapter 6. Learning Outcome. Newton's Laws in Action. Reflection: Reflection: 6.2 Gravitational Field
Chapte 6 Gataton Objectes 6. Newton's Law o nesal Gataton 6. Gatatonal Feld 6. Gatatonal Potental 6. Satellte oton n Ccula Obts 6.5 scape Velocty Leanng Outcoe (a and use the oula / (b explan the eanng
More informationNumerical Integration
MCEN 473/573 Chapte 0 Numeical Integation Fall, 2006 Textbook, 0.4 and 0.5 Isopaametic Fomula Numeical Integation [] e [ ] T k = h B [ D][ B] e B Jdsdt In pactice, the element stiffness is calculated numeically.
More informationCHAPTER 3 SYSTEMS OF PARTICLES
HAPTER 3 SYSTEMS O PARTILES 3. Intoducton By systes of patcles I ean such thngs as a swa of bees, a sta cluste, a cloud of gas, an ato, a bck. A bck s ndeed coposed of a syste of patcles atos whch ae constaned
More informationSOME NEW SELFDUAL [96, 48, 16] CODES WITH AN AUTOMORPHISM OF ORDER 15. KEYWORDS: automorphisms, construction, selfdual codes
Факултет по математика и информатика, том ХVІ С, 014 SOME NEW SELFDUAL [96, 48, 16] CODES WITH AN AUTOMORPHISM OF ORDER 15 NIKOLAY I. YANKOV ABSTRACT: A new method fo constuctng bnay selfdual codes wth
More informationSupersymmetry in Disorder and Chaos (Random matrices, physics of compound nuclei, mathematics of random processes)
Supesymmety n Dsoe an Chaos Ranom matces physcs of compoun nucle mathematcs of anom pocesses Lteatue: K.B. Efetov Supesymmety n Dsoe an Chaos Cambge Unvesty Pess 997999 Supesymmety an Tace Fomulae I.V.
More informationis needed and this can be established by multiplying A, obtained in step 3, by, resulting V = A x y =. = x, located in 1 st quadrant rotated about 2
Ct Cllege f New Yk MATH (Calculus Ntes) Page 1 f 1 Essental Calculus, nd edtn (Stewat) Chapte 7 Sectn, and 6 auth: M. Pak Chapte 7 sectn : Vlume Suface f evlutn (Dsc methd) 1) Estalsh the tatn as and the
More informationNewton s Laws, Kepler s Laws, and Planetary Orbits
Newton s Laws, Keple s Laws, and Planetay Obits PROBLEM SET 4 DUE TUESDAY AT START OF LECTURE 28 Septembe 2017 ASTRONOMY 111 FALL 2017 1 Newton s & Keple s laws and planetay obits Unifom cicula motion
More informationCross section dependence on ski pole sti ness
Coss section deendence on ski ole sti ness Johan Bystöm and Leonid Kuzmin Abstact Ski equiment oduce SWIX has ecently esented a new ai of ski oles, called SWIX Tiac, which di es fom conventional (ound)
More information6. Introduction to Transistor Amplifiers: Concepts and SmallSignal Model
6. ntoucton to anssto mples: oncepts an SmallSgnal Moel Lectue notes: Sec. 5 Sea & Smth 6 th E: Sec. 5.4, 5.6 & 6.36.4 Sea & Smth 5 th E: Sec. 4.4, 4.6 & 5.35.4 EE 65, Wnte203, F. Najmaba Founaton o
More informationECE 3318 Applied Electricity and Magnetism. Spring Prof. David R. Jackson ECE Dept. Notes 13
ECE 338 Applied Electicity and Magnetism ping 07 Pof. David R. Jackson ECE Dept. Notes 3 Divegence The Physical Concept Find the flux going outwad though a sphee of adius. x ρ v0 z a y ψ = D nˆ d = D ˆ
More informationDESIGN OF BEAMS FOR MOMENTS
CHAPTER Stuctual Steel Design RFD ethod Thid Edition DESIGN OF BEAS FOR OENTS A. J. Clak School of Engineeing Deatment of Civil and Envionmental Engineeing Pat II Stuctual Steel Design and Analysis 9 FA
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 informationComparative Study on Electrical Discharge and Operational Characteristics of Needle and WireCylinder Corona Chargers
50 Jounal of Electcal Engneeng & Technology, Vol. 1, No. 4, pp. 50~57, 006 Compaatve Study on Electcal Dschage and Opeatonal Chaactestcs of Needle and WeCylnde Coona Chages Panch Inta* and Nakon Tppayawong**
More informationCHAPTER IV RADIATION BY SIMPLE ACOUSTIC SOURCE. or by vibratory forces acting directly on the fluid, or by the violent motion of the fluid itself.
CHAPTER IV RADIATION BY SIMPLE ACOUSTIC SOURCE 4.1 POINT SOURCE Sound waves ae geneated by the vibation of any solid body in contact with the fluid medium o by vibatoy foces acting diectly on the fluid,
More informationSection 26 The Laws of Rotational Motion
Physics 24A Class Notes Section 26 The Laws of otational Motion What do objects do and why do they do it? They otate and we have established the quantities needed to descibe this motion. We now need to
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Electromagnetism II September 15, 2012 Prof. Alan Guth PROBLEM SET 2
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Depatment Physics 8.07: Electomagnetism II Septembe 5, 202 Pof. Alan Guth PROBLEM SET 2 DUE DATE: Monday, Septembe 24, 202. Eithe hand it in at the lectue,
More information( ) ( )( ) ˆ. Homework #8. Chapter 27 Magnetic Fields II.
Homewok #8. hapte 7 Magnetic ields. 6 Eplain how ou would modif Gauss s law if scientists discoveed that single, isolated magnetic poles actuall eisted. Detemine the oncept Gauss law fo magnetism now eads
More informationLecture 8. Light and Electromagnetic Spectrum
Letue 8 Optal oesses 7. henomenologal Theoy 7. Optal popetes of Cystals 7.. Maxwell s quatons 7.. Delet Funtons 7..3 Kames Kong Relatons 7.3 Optal oesses n semondutos  Det and Indet Tanstons Refeenes:
More informationAnyone who can contemplate quantum mechanics without getting dizzy hasn t understood it. Niels Bohr. Lecture 17, p 1
Anyone who can contemplate quantum mechanics without getting dizzy hasn t undestood it. Niels Boh Lectue 17, p 1 Special (Optional) Lectue Quantum Infomation One of the most moden applications of QM
More informationTRANSILVANIA UNIVERSITY OF BRASOV MECHANICAL ENGINEERING FACULTY DEPARTMENT OF MECHANICAL ENGINEERING ONLY FOR STUDENTS
TNSILVNI UNIVSITY OF BSOV CHNICL NGINING FCULTY DPTNT OF CHNICL NGINING Couse 9 Cuved as 9.. Intoduction The eams with plane o spatial cuved longitudinal axes ae called cuved as. Thee ae consideed two
More informationb Ψ Ψ Principles of Organic Chemistry lecture 22, page 1
Pinciples of Oganic Chemisty lectue, page. Basis fo LCAO and Hückel MO Theoy.. Souces... Hypephysics online. http://hypephysics.phyast.gsu.edu/hbase/quantum/qm.html#c... Zimmeman, H. E., Quantum Mechanics
More informationCOMPUTATIONAL METHODS AND ALGORITHMS Vol. I  Methods of Potential Theory  V.I. Agoshkov, P.B. Dubovski
METHODS OF POTENTIAL THEORY.I. Agoshkov and P.B. Dubovsk Insttute of Numecal Mathematcs, Russan Academy of Scences, Moscow, Russa Keywods: Potental, volume potental, Newton s potental, smple laye potental,
More informationMesoscale Simulation for Polymer Migration in Confined Uniform Shear Flow
CHEM. ES. CHINESE UNIVESITIES 00, 6(), 7 Mesoscale Smulaton fo Polyme Mgaton n Confned Unfom Shea Flow HE Yandong, WANG Yongle, LÜ Zhongyuan * and LI Zesheng State Key Laboatoy of Theetcal and Computatonal
More informationOptimization Methods: Linear Programming Revised Simplex Method. Module 3 Lecture Notes 5. Revised Simplex Method, Duality and Sensitivity analysis
Optmzaton Meods: Lnea Pogammng Revsed Smple Meod Module Lectue Notes Revsed Smple Meod, Dualty and Senstvty analyss Intoducton In e pevous class, e smple meod was dscussed whee e smple tableau at each
More informationd 2 x 0a d d =0. Relative to an arbitrary (accelerating frame) specified by x a = x a (x 0b ), the latter becomes: d 2 x a d 2 + a dx b dx c
Chapte 6 Geneal Relativity 6.1 Towads the Einstein equations Thee ae seveal ways of motivating the Einstein equations. The most natual is pehaps though consideations involving the Equivalence Pinciple.
More informationCorrelation between System (Lagrangian) concept Controlvolume (Eulerian) concept for comprehensive understanding of fluid motion?
The Reynolds Tanspot Theoem Coelation between System (Lagangian) concept Contolvolume (Euleian) concept fo compehensive undestanding of fluid motion? Reynolds Tanspot Theoem Let s set a fundamental equation
More informationMotion in a Plane Uniform Circular Motion
Lectue 11 Chapte 8 Physics I Motion in a Plane Unifom Cicula Motion Couse website: http://faculty.uml.edu/andiy_danylo/teaching/physicsi PHYS.1410 Lectue 11 Danylo Depatment of Physics and Applied Physics
More informationQuiz 6Work, Gravitation, Circular Motion, Torque. (60 pts available, 50 points possible)
Name: Class: Date: ID: A Quiz 6Wok, Gavitation, Cicula Motion, Toque. (60 pts available, 50 points possible) Multiple Choice, 2 point each Identify the choice that best completes the statement o answes
More informationrt () is constant. We know how to find the length of the radius vector by r( t) r( t) r( t)
Cicula Motion Fom ancient times cicula tajectoies hae occupied a special place in ou model of the Uniese. Although these obits hae been eplaced by the moe geneal elliptical geomety, cicula motion is still
More informationConventional Current B = In some materials current moving charges are positive: Ionic solution Holes in some materials (same charge as electron but +)
Conventional Cuent In some mateials cuent moving chages ae positive: Ionic solution Holes in some mateials (same chage as electon but +) Obseving magnetic field aound coppe wie: Can we tell whethe the
More informationkq r 2 2kQ 2kQ (A) (B) (C) (D)
PHYS 1202W MULTIPL CHOIC QUSTIONS QUIZ #1 Answer the followng multple choce questons on the bubble sheet. Choose the best answer, 5 pts each. MC1 An uncharged metal sphere wll (A) be repelled by a charged
More information