Chapter 4. Energy and Potential
|
|
- Todd Lang
- 5 years ago
- Views:
Transcription
1 Chpte 4. Enegy nd Ptentil Hyt; 0/5/009; 4-4. Enegy Expended in Mving Pint Chge in n Electic Field The electic field intensity is defined s the fce n unit test chge. The fce exeted y the electic field n chge Q, F = QE E The diffeentil wk dne y the field t mve the chge in the diectin dl, dw = F dl E E The diffeentil wk equied y us t mve the chge y dl. We shuld pply fce ginst the fce y the electic field. dw = F dl dw = Q E dl ( E ) The wk equied t mve the chge finite distnce, W = Q E dl pth 4. The Line Integl Assume E = cnstnt f simplicity. The pth is divided int six segments. The wk equied t mve chge Q fm B t A, W Q E Δ L E ΔL ( 6 6) F unifm electic field, E = E = E =.. = E6, W QE Δ L ΔL ( 6) L BA, vect fm B t A The exct mthemticl expessin A A W = Q E dl W = QE dl B B E = cnstnt W = QE L Dependent n the initil nd the finl pints ut nt the pth. BA
2 Hyt; 0/5/009; 4-
3 Hyt; 0/5/009; 4-3 Diffeentil length vect dl = dx ˆ ˆ ˆ x + dy y + dz z dl = dˆ d ˆ + + dz ˆz dl = d ˆ + dθ ˆ + sinθdˆ θ : Rectngul Cdintes : Cylindicl Cdintes : Spheicl Cdintes Exmple An infinite line chge It is well knwn y nw L E = Eˆ = ˆ πε The diffeentil length vect n cicle f dius. dl = dˆ ˆ + d + dz ˆz =0 =0 The wk needed t mve chge Q ut the cicul pth. finl finl L L W = Q ˆ ˆ ˆ ˆ d Q d 0 initil initil πε πε Mve the chge fm = t = dl = d ˆ finl = L L d Q L W = Q ˆ d ˆ Q ln initil πε = πε πε Minus sign mens tht we eceive enegy fm the electic field. Mve the chge fm = t = When chge is mved in the diectin f decesing cdinte vlue, this is tken ce f y the limits f the integl nt y the sign f dl. dl = d ˆ finl = L L d Q L W = Q ˆ d ˆ Q ln initil πε + = πε πε
4 4.3 Definitin f Ptentil Diffeence nd Ptentil The ptentil diffeence V is the wk dne y mving unit psitive chge fm ne pint t nthe in the electic field. The ptentil diffeence etween tw pints A nd B. A V = E dl : B is ften tken t infinity AB B The unit f ptentil is jules pe culm vlt. Exmple The wk dne in tking chge Q fm Q L W = ln πε The ptentil diffeence etween W L V = = ln Q πε = t = Hyt; 0/5/009; 4-4 = t = in the pesence f line chge. Exmple The ptentil diffeence etween = A t = B in the pesence f pint chge. The electic field is Q E = E ˆ ˆ = The diffeentil length vect dl = d ˆ A = A Q Q VAB = E dl d B B = A B Ptentil slute ptentil is used when ze-efeence pint is defined. The ze efeence pint is tken t infinity in cnventin. V = V V AB A B Ptentil diffeence Ptentil t A Ptentil t B 4.4 The Ptentil Field f Pint Chge The ptentil diffeence etween A nd B A = A Q Q VAB = E dl d B B = A B Only -cmpnent. Independent f θ nd, nd theefe f the pth. = d ˆ + dθ ˆ + sinθ dˆ θ
5 Hyt; 0/5/009; 4-5 Ze efeence pint t infinity Let B, then the ptentil t A is V A Q = A Since A is ity, the ptentil t ny pint in spce is Q V = πε 4 Equiptentil sufce The sme vlue f ptentil n the equiptentil sufce. N wk is needed t mve chge n the equiptentil sufce. The equiptentil sufces in the ptentil field f pint chge e sphees. 4.5 The Ptentil Field f System f Chges: Cnsevtive Ppety The ptentil is defined s wk needed t ing unit chge fm the infinity t the pint. The ptentil is independent f the pth. This is why the ptentil is vey useful cncept. Ptentil fm system f chges () A pint chge Q lcted t The ptentil t pint Q V ( ) = () Pint chges Q nd Q lcted t nd, espectively. The ptentil t Q Q V ( ) = + (3) Similly, the ptentil ising fm N pint chges is N Q Qn Qm V ( ) = = n m = m (4) The ptentil fm vlume chge density v v ( ) Δv ( ) ( ) ( ) v n Δvn As Δv Δv n v dv V = + + (8) vlume 4 πε n (5) The ptentil fm line chge L ( ) dl V ( ) = (9) cuve 4 πε (6) The ptentil fm sufce chge S ( ) ds V ( ) = (0) sufce 4 πε
6 Exmple The ptentil fm line chge in the fm f ing dl = d ˆ, = zˆz, = ˆ, = + z Hyt; 0/5/009; 4-6 Using (9) π Ld L V = + z ε + z 0 Summy. The ptentil is the wk dne in cying unit psitive chge fm infinity t the pint. The wk is independent f the pth chsen.. The ptentil fm nume f pint chges is the sum f individul ptentils fm ech chge. The pth independence cn e stted s dl = 0 : Clsed pth integl f E is ze. E is cnsevtive field. C E P P Kichhff s vltge lw: The lgeic sum f vltge dps und ny clsed cicuit is ze.
7 4.6 Ptentil Gdient The ptentil cn e tined in tw methds. () Using line integl f the electic field. () Using vlume integl ve chge distiutin. (3) Find the ptentil fm the undy cnditin. Hyt; 0/5/009; 4-7 The mthemticl definitin f the ptentil V = E dl Fig. 4.6 Electic field stemlines Fig. 4.7 Equiptentil sufces F diffeentil length ΔL, ΔV E ΔL E ΔLcsθ ΔV ΔL =E csθ The mximum vlue is tined when csθ = ΔV = E ΔL mx Using vect nttin ΔV ΔV E = ˆ ˆ N ΔL ΔN mx N Since the mx. ccus lng ˆN ΔL is in ppsite diectin t E, ˆN : A unit vect nml t the equiptentil sufce in the diectin f highe ptentil Summy. E is the mximum vlue f the te f chnge f V with distnce.. E is ppsite t the diectin f the mst pidly incesing ptentil. E is pependicul t the equiptentil sufce. Δ V = E Δ L = 0 cnditin f equiptentil sufce Tw vects e pependicul
8 Hyt; 0/5/009; 4-8 Gdient f scl field The gdient f scl field T is defined s dt Gdient f T = gd T ˆN dn is unit vect nml the equiptentil sufce in the diectin f incesing T. ˆN The gdient f T is the mximum spce te f chnge f T in the diectin f incesing T. Reltin etween E nd V E =gdv Since V is functin f psitin, V V V dv = dx + dy + dz x y z Fm the definitin f the ptentil dv = E dl = E dx + E dy + E dz ( x y z ) = Eˆ ˆ ˆ x x + E y y + E z z = dx ˆ + dy ˆ + dz ˆ x y z Fm () nd (c) V x, V E = Ey =, E V z = x y z V V V E = ˆ ˆ ˆ x + y + z x y z () () (c) (d) Fm () nd (d) V ˆ V gd V = + ˆ + V ˆ x y z x y z del pet It is defined s = ˆ + ˆ + ˆ x y z T T T Its usge, T = ˆx + ˆy + ˆz x y z Theefe E = V x y z The gdient in the cdinte systems V ˆ V V = x + ˆ V y + ˆz x y z V V V V = ˆ ˆ + + ˆz z V V V V = ˆ ˆ ˆ + θ + θ sinθ gd T = T Rectngul cdintes Cylindicl cdintes Spheicl cdintes
9 Hyt; 0/5/009; The Diple An electic diple is tw pint chges f equl mgnitude nd ppsite sign septed y smll distnce. The electic field cn e fund y using Culm s lw y using the negtive gdient f the ptentil. The ttl ptentil fm tw chges, Q Q R R V = = R R 4 πε RR Using the ppximtin Qdcsθ V = πε 4 f R R d csθ (e) Nte tht z=0 plne θ = 90 plne is t ze ptentil Tking gdient in spheicl cdintes Qd E = 3 ( csθ ˆ + sinθ ˆ θ ) πε 4
10 Plt f equiptentil sufces Qd / =, then V = cs θ /. : Eq. f the equiptentil sufces Let ( ) Hyt; 0/5/009; 4-0 Blck, electic field stemlines; Red, equiptentil lines. Eq. f the electic field stemlines. In = cnstnt plne, Eθ d sin E = θ θ d csθ d = ct θ dθ = C sin θ Diple mment The diple mment p is vect quntity. p = Qd : d is the chge septin diected fm Q t +Q. The ptentil fm the electic diple. Using the definitin f diple mment in (e) p ˆ V = Genelize V = p 4 πε is field pint, is the cente f the diple. 3 We nte E ~/. It flls ff fste thn tht f pint chge. At gete distnces the electic diple lks t hve n chge t ll.
11 4.8 Enegy Density in the Electsttic Field Binging chge t ne nthe chge equies wk. This wk is sted s the ptentil enegy in the system f tw chges. Hyt; 0/5/009; 4- Binging chge Q fm infinity t ny psitin. N wk needed since thee is n electic field. Binging the secnd chge Q in the field f Q The wk dne = Q V,. The ptentil t the psitin f Q pduced y Q. Binging the thid chge Q3 in the field pduced y Q nd Q. The wk dne = Q3 V 3, + Q3 V 3, The ttl ptentil enegy f the field W E= Q V, +(Q3 V 3, + Q3 V 3,)+ (Q4 V 4, + Q4 V 4, + Q4 V 4,3 )+ () Nte, f exmple, Q Rewite Q3 QV = 3 3, Q3 Q QV R,3 3 is scl R R = () 3 3 Using () in () W E = Q V, +(Q V,3 + Q V,3 )+ (Q V,4 + Q V,4 + Q3 V 3,4 )+ (c) ()+(c) W E = Q (V, + V,3 + V,4 + ) + Q (V, + V,3 + V,4 + ) + Q3 (V 3, + V 3, + V 3,4 + ) + = V 3, Ptentil due t ll chges t the psitin f Q3 N W QV Q V Q V Q V = ( ) E m m m = The enegy sted in egin f cntinuus chge distiutin WE = vvdv vl Use the mthemticl identity, ( VD) V( D) + D ( V) W ( D ) Vdv E ( VD ) D = ( V ) Vdv vl vl Using the divegence theem W E ( ) ( ) VD ds C D = V dv vl /, /, E W E D Edv = = ε vl E dv vl (45)
12 Exmple Enegy sted in the electic field f cxil cle f length L. Fm sectin 3.3 S S D = ˆ nd E = ˆ ε Hyt; 0/5/009; 4- The sted enegy is L π S πl S WE = ln 0 0 d d dz ε = ε ε Find the ptentil diffeence etween the inne nd the ute cnduct. Let the ute cnduct e the ze-ptentil efeence. S S V = E d d ln = = ε ε The enegy sted is L π ln ( / S S ) WE = SVdS = ln π S 0 0 S d dz L ε ε : The sme esult Using the ttl chge, Q = π LS, WE = QV : The sme s the enegy sted in cpcit. Vlume enegy density Fm (45) WE = D Edv vl Diffeentil vlue dwe = D Edv dw E D E dv =
Electric Potential Energy
Electic Ptentil Enegy Ty Cnsevtive Fces n Enegy Cnsevtin Ttl enegy is cnstnt n is sum f kinetic n ptentil Electic Ptentil Enegy Electic Ptentil Cnsevtin f Enegy f pticle fm Phys 7 Kinetic Enegy (K) nn-eltivistic
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 informationCharge in a Cavity of Conductor
Tdy s Pln Electic Ptentil Enegy (mesued in Jules Electic Ptentil Ptentil Enegy pe unit Chge (mesued in Vlts). Recll tht the electic field E is fce F pe unit chge. Cpcitnce BB Chge in Cvity f Cnduct A pticle
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 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 informationChapter 4 Motion in Two and Three Dimensions
Chpte 4 Mtin in Tw nd Thee Dimensins In this chpte we will cntinue t stud the mtin f bjects withut the estictin we put in chpte t me ln stiht line. Insted we will cnside mtin in plne (tw dimensinl mtin)
More informationMAGNETIC FIELDS & UNIFORM PLANE WAVES
MAGNETIC FIELDS & UNIFORM PLANE WAVES Nme Sectin Multiple Chice 1. (8 Pts). (8 Pts) 3. (8 Pts) 4. (8 Pts) 5. (8 Pts) Ntes: 1. In the multiple chice questins, ech questin my hve me thn ne cect nswe; cicle
More informationInductance and Energy of B Maxwell s Equations Mon Potential Formulation HW8
Wed. Fi. 7..3-7..5 Inductnce nd Enegy f 7.3.-.3.3 Mxwell s Equtins Mn. 0. -.. Ptentil Fmultin HW8 Whee we ve been Sttiny Chges pducing nd intecting vi Electic Fields Stedy Cuents pducing nd intecting vi
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 informationElectric Charge. Electric charge is quantized. Electric charge is conserved
lectstatics lectic Chage lectic chage is uantized Chage cmes in incements f the elementay chage e = ne, whee n is an intege, and e =.6 x 0-9 C lectic chage is cnseved Chage (electns) can be mved fm ne
More informationA) 100 K B) 150 K C) 200 K D) 250 K E) 350 K
Phys10 Secnd Maj-09 Ze Vesin Cdinat: k Wednesday, May 05, 010 Page: 1 Q1. A ht bject and a cld bject ae placed in themal cntact and the cmbinatin is islated. They tansfe enegy until they each a final equilibium
More informationIntroduction. Electrostatics
UNIVESITY OF TECHNOLOGY, SYDNEY FACULTY OF ENGINEEING 4853 Electmechanical Systems Electstatics Tpics t cve:. Culmb's Law 5. Mateial Ppeties. Electic Field Stength 6. Gauss' Theem 3. Electic Ptential 7.
More informationME 236 Engineering Mechanics I Test #4 Solution
ME 36 Enineein Mechnics I est #4 Slutin Dte: id, M 14, 4 ie: 8:-1: inutes Instuctins: vein hptes 1-13 f the tetbk, clsed-bk test, clcults llwed. 1 (4% blck ves utwd ln the slt in the pltf with speed f
More informationSummary chapter 4. Electric field s can distort charge distributions in atoms and molecules by stretching and rotating:
Summa chapte 4. In chapte 4 dielectics ae discussed. In thse mateials the electns ae nded t the atms mlecules and cannt am fee thugh the mateial: the electns in insulats ae n a tight leash and all the
More informationCHAPTER 2 ELECTRIC FIELD
lecticity-mgnetim Tutil (QU PROJCT) 9 CHAPTR LCTRIC FILD.. Intductin If we plce tet chge in the pce ne chged d, n electttic fce will ct n the chge. In thi ce we pek f n electic field in thi pce ( nlgy
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 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 informationELECTRIC & MAGNETIC FIELDS I (STATIC FIELDS) ELC 205A
LCTRIC & MAGNTIC FILDS I (STATIC FILDS) LC 05A D. Hanna A. Kils Assciate Pfess lectnics & Cmmnicatins ngineeing Depatment Faclty f ngineeing Cai Univesity Fall 0 f Static lecticity lectic & Magnetic Fields
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 informationExample 11: The man shown in Figure (a) pulls on the cord with a force of 70
Chapte Tw ce System 35.4 α α 100 Rx cs 0.354 R 69.3 35.4 β β 100 Ry cs 0.354 R 111 Example 11: The man shwn in igue (a) pulls n the cd with a fce f 70 lb. Repesent this fce actin n the suppt A as Catesian
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 24 GAUSS LAW
CHAPTR 4 GAUSS LAW LCTRIC FLUX lectic flux is a measue f the numbe f electic filed lines penetating sme suface in a diectin pependicula t that suface. Φ = A = A csθ with θ is the angle between the and
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 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 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 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 informationChapter 6. Dielectrics and Capacitance
Chapter 6. Dielectrics and Capacitance Hayt; //009; 6- Dielectrics are insulating materials with n free charges. All charges are bund at mlecules by Culmb frce. An applied electric field displaces charges
More informationA) N B) 0.0 N C) N D) N E) N
Cdinat: H Bahluli Sunday, Nvembe, 015 Page: 1 Q1. Five identical pint chages each with chage =10 nc ae lcated at the cnes f a egula hexagn, as shwn in Figue 1. Find the magnitude f the net electic fce
More informationLecture 3. Electrostatics
Lecue lecsics In his lecue yu will len: Thee wys slve pblems in elecsics: ) Applicin f he Supepsiin Pinciple (SP) b) Applicin f Guss Lw in Inegl Fm (GLIF) c) Applicin f Guss Lw in Diffeenil Fm (GLDF) C
More informationSpecial Vector Calculus Session For Engineering Electromagnetics I. by Professor Robert A. Schill Jr.
pecil Vect Clculus essin Engineeing Electmgnetics I Pfess et. cill J. pecil Vect Clculus essin f Engineeing Electmgnetics I. imple cmputtin f cul diegence nd gdient f ect. [peicl Cdinte stem] Cul Diegence
More informationMeasurement of Residual Stress/Strain (Using Strain Gages and the Hole Drilling Method) Summary of Discussion in Section 8.9
Mesuement f Residul Stess/Stin (Using Stin Gges nd the Hle Dilling Methd) Summy f Discussin in Sectin 8.9 The Hle Dilling Methd Is Bsed On: () Stess tnsfmtin equtins τ x' x' y' y' x' y' xx xx cs sin sin
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 informationA) (0.46 î ) N B) (0.17 î ) N
Phys10 Secnd Maj-14 Ze Vesin Cdinat: xyz Thusday, Apil 3, 015 Page: 1 Q1. Thee chages, 1 = =.0 μc and Q = 4.0 μc, ae fixed in thei places as shwn in Figue 1. Find the net electstatic fce n Q due t 1 and.
More information5/20/2011. HITT An electron moves from point i to point f, in the direction of a uniform electric field. During this displacement:
5/0/011 Chapte 5 In the last lectue: CapacitanceII we calculated the capacitance C f a system f tw islated cnducts. We als calculated the capacitance f sme simple gemeties. In this chapte we will cve the
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 informationMagnetism. Chapter 21
1.1 Magnetic Fields Chapte 1 Magnetism The needle f a cmpass is pemanent magnet that has a nth magnetic ple (N) at ne end and a suth magnetic ple (S) at the the. 1.1 Magnetic Fields 1.1 Magnetic Fields
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 informationPH2200 Practice Exam I Summer 2003
PH00 Prctice Exm I Summer 003 INSTRUCTIONS. Write yur nme nd student identifictin number n the nswer sheet.. Plese cver yur nswer sheet t ll times. 3. This is clsed bk exm. Yu my use the PH00 frmul sheet
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 informationAnnouncements Candidates Visiting Next Monday 11 12:20 Class 4pm Research Talk Opportunity to learn a little about what physicists do
Wed., /11 Thus., /1 Fi., /13 Mn., /16 Tues., /17 Wed., /18 Thus., /19 Fi., / 17.7-9 Magnetic Field F Distibutins Lab 5: Bit-Savat B fields f mving chages (n quiz) 17.1-11 Pemanent Magnets 18.1-3 Mic. View
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 informationMarch 15. Induction and Inductance Chapter 31
Mach 15 Inductin and Inductance Chapte 31 > Fces due t B fields Lentz fce τ On a mving chage F B On a cuent F il B Cuent caying cil feels a tque = µ B Review > Cuents geneate B field Bit-Savat law = qv
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 information5.1 Moment of a Force Scalar Formation
Outline ment f a Cuple Equivalent System Resultants f a Fce and Cuple System ment f a fce abut a pint axis a measue f the tendency f the fce t cause a bdy t tate abut the pint axis Case 1 Cnside hizntal
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 informationWork, Energy, and Power. AP Physics C
k, Eneg, and Pwe AP Phsics C Thee ae man diffeent TYPES f Eneg. Eneg is expessed in JOULES (J) 4.19 J = 1 calie Eneg can be expessed me specificall b using the tem ORK() k = The Scala Dt Pduct between
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 informationThe Gradient and Applications This unit is based on Sections 9.5 and 9.6, Chapter 9. All assigned readings and exercises are from the textbook
The Gadient and Applicatins This unit is based n Sectins 9.5 and 9.6 Chapte 9. All assigned eadings and eecises ae fm the tetbk Objectives: Make cetain that u can define and use in cntet the tems cncepts
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 informationCHAPTER GAUSS'S LAW
lutins--ch 14 (Gauss's Law CHAPTE 14 -- GAU' LAW 141 This pblem is ticky An electic field line that flws int, then ut f the cap (see Figue I pduces a negative flux when enteing and an equal psitive flux
More information= ΔW a b. U 1 r m 1 + K 2
Chpite 3 Potentiel électiue [18 u 3 mi] DEVOIR : 31, 316, 354, 361, 35 Le potentiel électiue est le tvil p unité de chge (en J/C, ou volt) Ce concept est donc utile dns les polèmes de consevtion d énegie
More informationr = (0.250 m) + (0.250 m) r = m = = ( N m / C )
ELECTIC POTENTIAL IDENTIFY: Apply Eq() to clculte the wok The electic potentil enegy of pi of point chges is given y Eq(9) SET UP: Let the initil position of q e point nd the finl position e point, s shown
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 informationA 2 ab bc ca. Surface areas of basic solids Cube of side a. Sphere of radius r. Cuboid. Torus, with a circular cross section of radius r
Sufce e f ic lid Cue f ide R See f diu 6 Cuid c c Elliticl cectin c Cylinde, wit diu nd eigt Tu, wit cicul c ectin f diu R R Futum, ( tuncted ymid) f e eimete, t e eimete nd lnt eigt. nd e te eective e
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 informationPhysics 102. Final Examination. Spring Semester ( ) P M. Fundamental constants. n = 10P
ε µ0 N mp M G T Kuwit University hysics Deprtment hysics 0 Finl Exmintin Spring Semester (0-0) My, 0 Time: 5:00 M :00 M Nme.Student N Sectin N nstructrs: Drs. bdelkrim, frsheh, Dvis, Kkj, Ljk, Mrfi, ichler,
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 informationWYSE Academic Challenge Regional Physics 2008 SOLUTION SET
WYSE cdemic Chllenge eginl 008 SOLUTION SET. Crrect nswer: E. Since the blck is mving lng circulr rc when it is t pint Y, it hs centripetl ccelertin which is in the directin lbeled c. Hwever, the blck
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 informationn Power transmission, X rays, lightning protection n Solid-state Electronics: resistors, capacitors, FET n Computer peripherals: touch pads, LCD, CRT
.. Cu-Pl, INE 45- Electmagnetics I Electstatic fields anda Cu-Pl, Ph.. INE 45 ch 4 ECE UPM Maagüe, P me applicatins n Pwe tansmissin, X as, lightning ptectin n lid-state Electnics: esists, capacits, FET
More informationChapter 25: Current, Resistance and Electromotive Force. Charge carrier motion in a conductor in two parts
Chpte 5: Cuent, esistnce nd Electomotive Foce Chge cie motion in conducto in two pts Constnt Acceletion F m qe ndomizing Collisions (momentum, enegy) =>esulting Motion Avege motion = Dift elocity = v d
More informationModern Physics. Unit 6: Hydrogen Atom - Radiation Lecture 6.1: The Radial Probability Density. Ron Reifenberger Professor of Physics Purdue University
Mdern Physics Unit 6: Hydrgen Atm - Rditin Lecture 6.1: The Rdil Prbbility Density Rn Reifenberger Prfessr f Physics Purdue University 1 Prbbility Density Prbbility Density * ΨΨ = Ψ In 1-D, the prbbility
More informationFri. 10/23 (C14) Linear Dielectrics (read rest at your discretion) Mon. (C 17) , E to B; Lorentz Force Law: fields
Fi. 0/23 (C4) 4.4. Linea ielectics (ead est at yu discetin) Mn. (C 7) 2..-..2, 2.3. t B; 5..-..2 Lentz Fce Law: fields Wed. and fces Thus. (C 7) 5..3 Lentz Fce Law: cuents Fi. (C 7) 5.2 Bit-Savat Law HW6
More informationHomework: Study 6.2 #1, 3, 5, 7, 11, 15, 55, 57
Gols: 1. Undestnd volume s the sum of the es of n infinite nume of sufces. 2. Be le to identify: the ounded egion the efeence ectngle the sufce tht esults fom evolution of the ectngle ound n xis o foms
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 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 informationChapter 4 Motion in Two and Three Dimensions
Chapte 4 Mtin in Tw and Thee Dimensins In this chapte we will cntinue t stud the mtin f bjects withut the estictin we put in chapte t me aln a staiht line. Instead we will cnside mtin in a plane (tw dimensinal
More informationChapter 2. Coulomb s Law and Electric Field Intensity
Chapter. Culmb s Law and lectric Field Intensit Hat; 9/9/009; -1.1 The perimental Law f Culmb Frm the eperiment the frce between tw charged bjects is QQ F k : Frce in Newtn (N) where Q1 and Q : Charges
More informationElectromagnetic Waves
Chapte 3 lectmagnetic Waves 3.1 Maxwell s quatins and ectmagnetic Waves A. Gauss s Law: # clsed suface aea " da Q enc lectic fields may be geneated by electic chages. lectic field lines stat at psitive
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 informationELECTROSTATIC FIELDS IN MATERIAL MEDIA
MF LCTROSTATIC FILDS IN MATRIAL MDIA 3/4/07 LCTURS Materials media may be classified in terms f their cnductivity σ (S/m) as: Cnductrs The cnductivity usually depends n temperature and frequency A material
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 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 informationElectric potential energy Electrostatic force does work on a particle : Potential energy (: i initial state f : final state):
Electc ptental enegy Electstatc fce des wk n a patcle : v v v v W = F s = E s. Ptental enegy (: ntal state f : fnal state): Δ U = U U = W. f ΔU Electc ptental : Δ : ptental enegy pe unt chag e. J ( Jule)
More informationChapter 25: Current, Resistance and Electromotive Force. ~10-4 m/s Typical speeds ~ 10 6 m/s
Chpte 5: Cuent, esistnce nd lectomotive Foce Chge cie motion in conducto in two pts Constnt Acceletion F m q ndomizing Collisions (momentum, enegy) >esulting Motion http://phys3p.sl.psu.edu/phys_nim/m/ndom_wlk.vi
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 information11. HAFAT İş-Enerji Power of a force: Power in the ability of a force to do work
MÜHENDİSLİK MEKNİĞİ. HFT İş-Eneji Pwe f a fce: Pwe in he abiliy f a fce d wk F: The fce applied n paicle Q P = F v = Fv cs( θ ) F Q v θ Pah f Q v: The velciy f Q ÖRNEK: İŞ-ENERJİ ω µ k v Calculae he pwe
More informationWYSE Academic Challenge Sectional Mathematics 2006 Solution Set
WYSE Academic Challenge Sectinal 006 Slutin Set. Cect answe: e. mph is 76 feet pe minute, and 4 mph is 35 feet pe minute. The tip up the hill takes 600/76, 3.4 minutes, and the tip dwn takes 600/35,.70
More informationEdexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks
Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1
More informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
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 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 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 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 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 informationCHAPTER 25 ELECTRIC POTENTIAL
CHPTE 5 ELECTIC POTENTIL Potential Diffeence and Electic Potential Conside a chaged paticle of chage in a egion of an electic field E. This filed exets an electic foce on the paticle given by F=E. When
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 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 informationB.A. (PROGRAMME) 1 YEAR MATHEMATICS
Gdute Couse B.A. (PROGRAMME) YEAR MATHEMATICS ALGEBRA & CALCULUS PART B : CALCULUS SM 4 CONTENTS Lesson Lesson Lesson Lesson Lesson Lesson Lesson : Tngents nd Nomls : Tngents nd Nomls (Pol Co-odintes)
More informationPhysics Jonathan Dowling. Lecture 9 FIRST MIDTERM REVIEW
Physics 10 Jonthn Dowling Physics 10 ecture 9 FIRST MIDTERM REVIEW A few concepts: electric force, field nd potentil Electric force: Wht is the force on chrge produced by other chrges? Wht is the force
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 informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationThis chapter is about energy associated with electrical interactions. Every
23 ELECTRIC PTENTIAL whee d l is n infinitesiml displcement long the pticle s pth nd f is the ngle etween F nd d l t ech point long the pth. econd, if the foce F is consevtive, s we defined the tem in
More informationHomework Assignment 3 Solution Set
Homework Assignment 3 Solution Set PHYCS 44 6 Ferury, 4 Prolem 1 (Griffiths.5(c The potentil due to ny continuous chrge distriution is the sum of the contriutions from ech infinitesiml chrge in the distriution.
More informationHonors Physics Final Review Summary
Hnrs Physics Final Review Summary Wrk Dne By A Cnstant Frce: Wrk describes a frce s tendency t change the speed f an bject. Wrk is dne nly when an bject mves in respnse t a frce, and a cmpnent f the frce
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 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 information3. Vectors. Vectors: quantities which indicate both magnitude and direction. Examples: displacemement, velocity, acceleration
Rutgers University Deprtment of Physics & Astronomy 01:750:271 Honors Physics I Lecture 3 Pge 1 of 57 3. Vectors Vectors: quntities which indicte both mgnitude nd direction. Exmples: displcemement, velocity,
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 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 information