Jackson 3.3 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
|
|
- Kimberly Smith
- 5 years ago
- Views:
Transcription
1 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 maintained at a fixed potentia V. With the infomation that the chage density on a disc at fixed potentia is popotiona to ( 2 ρ 2 ) -/2, whee ρ is the distance out fom the cente of the disc, (a) show that fo > the potentia is,, = 2V (b) find the potentia fo <. 2 2 P 2 cos (c) What is the capacitance of the disc? SOLUTION: Waning! The soution that Jackson gives is wong. Let us sove the pobem the wong way (the way Jackson expects), then show why this soution is wong. Then et us sove the pobem the ight way and figue out whee Jackson went wong. The Wong Way: (a) The suface chage density was stated to be: = S 2 2 fo < and 0 othewise The thee-dimensiona chage density is then: =S / fo < and 0 othewise To find out what S is in tems of the potentia V, use Couomb's aw which integates ove a the chage density to find the potentia at the oigin and set it equa to V: = 4 0 x' x x' d x' V = 4 0 x ' x' d x'
2 V = V =S 2 π π 4 π ϵ S δ(θ' π/2) 2 ' 2 ' ' '2 sin θ' d ' d θ' d ϕ' 2 ' 2 d ' V =S 2 0[ sin ' ]0 S= 4 0 V Now knowing S, pugging it back in, we have the fina fom of the chage density: ρ= 4ϵ 0 V π δ(θ π /2) 2 2 fo < and 0 othewise Now appy Couomb's Law to find the potentia at any point on the z axis: = 4 0 x' x x' d x' < Use 0 = + (cos θ') to expand this (which is aowed because we ae on the z axis). > = V ' /2 2 ' 2 ' < > cos ' '2 sin ' d ' d ' d ' = 2V 0 < 2 ' 2 P 0 ' d ' > We have to teat the two egions sepaatey. Let us ook at the > ' egion because its intega is easie. Fo > we aso have > ' so that: = 2V ' 2 ' d ' Make a change of vaiabes u= 2 ' 2 and udu= ' d ' = 2V u 2 / 2 du Now (0) is zeo fo odd, so ony even tems contibute. Let us eabe to take account fo this fact:
3 = 2V 2 0 P u 2 du If we do the intega case by case fo = 0,, 2... the integation is tivia and we soon see a patten:,, = 2V 2 2 Now we must emembe that this is ony vaid on the z axis. We can make use of the handy theoem that fo pobems with azimutha symmety, the genea soution is just the soution on the z-axis, mutipied by P(cos θ): Φ(,θ,ϕ)= 2V π ( ) 2+( 2 ) P 2 fo > This is the soution (the wong one Jackson expects) to the potentia in the exteio egion. (b) To find the potentia in the nea egion ( < ), fist note that the pobem has azimutha symmety and no chage in the nea egion, so the genea soution to the Lapace equation hods: Φ(,θ, ϕ)= ( +B ) We need a finite soution at the oigin, so the soution must have the fom: Φ(,θ,ϕ)= A fo < This soution fo the potentia in the oute egion must match the soution fo the potentia in the inne egion at the inteface whee they touch, = : 2V π,even ( ) / 2 + P = A The Legende poynomias ae othogona, so the coefficients must match up sepaatey, eading to: = 2V π ( ) /2 + and = 0 fo odd The soution fo the nea egion is theefoe: Φ(,θ,ϕ)= 2V π ( ) 2+( ) 2 P 2 Now, the pate is hed at V, so the soution to the potentia shoud educe down to the constant V fo
4 θ=π/2 and <, independent of. It shoud be obvious that the soution above does not educe to V on the disc. The facto (0) is zeo ony fo odd, but this soution ony has even. The coect soution wi have ony odd vaues. The Coect Way: Note that thee ae eay fou egions that we need to teat, sepaatey, as indicted in the diagam. In each egion, thee is no chage, thee is azimutha symmety, and the poes ae incuded, so the soution to the potentia has the fom: Φ out,up z Φ(,θ,ϕ)= ( +B ) V x The inne egions incude the oigin, so they must have a B zeo to have a finite soution at the oigin. Simiay, the oute egions incude infinity, which we can assume to have zeo potentia, eading to a being zeo in these egions. Ou soutions in a egions theefoe become:,down Φ out,down Φ out,up (,θ,ϕ)= B (cos θ) fo > and θ < π/2 Φ out,down (,θ, ϕ)= B,down fo > and θ > π/2 (,θ,ϕ)= A fo < and θ < π/2,down (,θ,ϕ)=, down fo < and θ > π/2 Fist, due to symmety, the potentia at any point in an uppe egion must equa the potentia at the mio point acoss the x-y pane: Φ( z)=φ( z) =,down ( cos θ) and Φ out,up (cos θ)=φ out,down ( cosθ) =, down ( cosθ) and B = B,down ( cosθ), down = ( ) and B, down =B ( ) With these findings, ou soutions now become:
5 Φ out,up (,θ,ϕ)= B (cos θ) fo > and θ < π/2 Φ out,down (,θ, ϕ)= B ( ) fo > and θ > π/2 (,θ,ϕ)= fo < and θ < π/2,down (,θ,ϕ)= ( ) fo < and θ > π/2 Note that by focing the uppe and owe egion potentias to be mio images, we automaticay made them match up at thei inteface, and have aeady taken cae of this bounday condition. Next, the potentia in the inne egions must become V on the disc. V = eading to: A 0 =V, = (cos(π/2)) and V = ( ) (cos(π/2)) (0)=0, and = ( ) (0)=0 Note that (0) = 0 fo a odd, in which case the ast two equations ae automaticay satisfied. Fo even, (0) is not zeo, so: =0 fo even,>0 Ou soution so fa is: Φ out,up (,θ,ϕ)= B (cos θ) fo > and θ < π/2 Φ out,down (,θ, ϕ)= (,θ,ϕ)=v + B ( ) fo > and θ > π/2 =,3,5..,down (,θ,ϕ)=v + =,3,5... fo < and θ < π/2 ( ) (cos θ) fo < and θ > π/2 Note that now that is odd, (-) is aways just -. The potentia in the inne-down egion theefoe becomes,down (,θ,ϕ)=v =,3,5... combined into (,θ,ϕ)=v +sgn =,3,5.... The soutions in the two inne egions can now be (cos θ) whee sgn(cos θ) is + fo θ < π/2 and - fo θ > π/2. Aso note that because the potentia in the inne egions and oute egions must match at =, and due to othogonaity, ony the = odd tems wi contibute in the oute egions as we. The oute egion soutions can theefoe be combined in the same way. Ou soution so fa is thus:
6 Φ out (,θ, ϕ)= B 0 +sgn =,3,5... (,θ,ϕ)=v +sgn =,3,5... B (cos θ) Next, the potentias of the inne and oute egions shoud match at = : (,θ, ϕ)=φ out (, θ, ϕ) V +sgn(cos θ) = B 0 =,3,5... +sgn B (cos θ) =,3,5.. The Legende poynomias ae othogona, so we match up coefficients. Matching up a coefficients, we find: B 0 =V and B = 2 + The soution so fa becomes: Φ out (,θ,ϕ)=v +sgn(cos θ) =,3,5... (,θ,ϕ)=v +sgn =,3, fo > (cos θ) fo < A the at this point ae abitay, so et us edefine as / to make these equations symmetic, eading to: Φ out (,θ, ϕ)=v +sgn(cos θ) =,3,5... (,θ,ϕ)=v +sgn =,3,5... ( + ) (cos θ) fo > ( ) fo < The ast set of coefficients can be found by eating the eectic fied acoss the pate: (E 2 E ) n 2 = σ ϵ 0 (E in,down E in,up ) θ= σ ϵ 0 whee σ= S 2 2 fo a conducting pate To find out what S is in tems of the potentia V, use Couomb's aw to integate ove a the chage density and find the potentia at the oigin and set it equa to V: V = σ (x ') 4π ϵ 0 x ' d a
7 V = V =S 2π 4 π ϵ S 'sin θ' d ' d ϕ' 2 ' 2 ' 2 ' 2 d ' V =S 2 0[ sin ' ]0 S= 4ϵ 0 V π Using this vaue, the bounday condition on the eectic fied acoss the pate now becomes: (,down + ) θ= 4V π 2 2,down + θ θ = 4V π 2 2,down + θ θ = 4V π ( / ) 2 Pefom a binomia expansion on the ight side, using,down + θ,down θ,down θ θ = 4V π [ + 2( ) 3 + θ = 4V π [ =,3, ( ) ( 2)!! ( )!!( ) ] x =+ 2 2 x x x ( ) θ = 4V [ π ( ) 2 (0) =,3,5... ( ) ] +...] A Legende poynomia identity was used in the ast step to get the ight side in a fom that we anticipate wi be on the eft side. Now evauate the deivatives: 2 A =,3,5... ( ) 2 =,3,5... [ ( ) [ θ ]θ=π/ 2 x ( x) ]x=0 = 4V π [ = 4V π [ =,3,5... =,3,5... ( ) 2 (0) ( ) ] ( ) 2 (0) ( ) ]
8 2 2 =,3,5... =,3,5... ( ) = 2 V + π ( ) 2 [ x ( x) ( x) x 2 ]x=0 = 4V π [ =,3,5... ( ) [ P (0) ]= 4V [ π ( ) 2 (0) =,3,5... ( ) ] Ou fina soution is theefoe: Φ out (,θ, ϕ)=v +sgn(cos θ) 2V π (,θ,ϕ)=v +sgn 2V π =,3,5... =,3,5... ( ) + 2 ( ) ( ) + 2 ( ) + ( ) 2 (0) ( ) ] (cos θ) fo > fo < Note that this soution obeys a the bounday conditions that it shoud. On the pate, (cos θ) becomes (0), which is zeo fo odd, eaving just the constant V as it shoud. So whee did Jackson go wong? The symmety of the pobem equies Legende poynomias with odd, but Jackson's soution had even, indicating that he got the symmety wong. (c) The tota chage is: 2 Q= ', ', ' ' 2 sin ' d ' d ' d ' Q=8 0 V 0 2 ' 2 ' d ' Q=8 0 V [ 2 ' 2 ] 0 Q=8 0 V Q V =8 0 This is the capacitance: C=8 0
Jackson 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 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 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 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 informationMechanics Physics 151
Mechanics Physics 151 Lectue 6 Kepe Pobem (Chapte 3) What We Did Last Time Discussed enegy consevation Defined enegy function h Conseved if Conditions fo h = E Stated discussing Centa Foce Pobems Reduced
More informationJackson 4.10 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jackson 4.10 Homework Probem Soution Dr. Christopher S. Baird University of Massachusetts Lowe PROBLEM: Two concentric conducting spheres of inner and outer radii a and b, respectivey, carry charges ±.
More informationMechanics Physics 151
Mechanics Physics 5 Lectue 5 Centa Foce Pobem (Chapte 3) What We Did Last Time Intoduced Hamiton s Pincipe Action intega is stationay fo the actua path Deived Lagange s Equations Used cacuus of vaiation
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 informationMechanics Physics 151
Mechanics Physics 5 Lectue 5 Centa Foce Pobem (Chapte 3) What We Did Last Time Intoduced Hamiton s Pincipe Action intega is stationay fo the actua path Deived Lagange s Equations Used cacuus of vaiation
More informationThree-dimensional systems with spherical symmetry
Thee-dimensiona systems with spheica symmety Thee-dimensiona systems with spheica symmety 006 Quantum Mechanics Pof. Y. F. Chen Thee-dimensiona systems with spheica symmety We conside a patice moving in
More informationPHYS 705: Classical Mechanics. Central Force Problems I
1 PHYS 705: Cassica Mechanics Centa Foce Pobems I Two-Body Centa Foce Pobem Histoica Backgound: Kepe s Laws on ceestia bodies (~1605) - Based his 3 aws on obsevationa data fom Tycho Bahe - Fomuate his
More information2 E. on each of these two surfaces. r r r r. Q E E ε. 2 2 Qencl encl right left 0
Ch : 4, 9,, 9,,, 4, 9,, 4, 8 4 (a) Fom the diagam in the textbook, we see that the flux outwad though the hemispheical suface is the same as the flux inwad though the cicula suface base of the hemisphee
More informationSeidel s Trapezoidal Partitioning Algorithm
CS68: Geometic Agoithms Handout #6 Design and Anaysis Oigina Handout #6 Stanfod Univesity Tuesday, 5 Febuay 99 Oigina Lectue #7: 30 Januay 99 Topics: Seide s Tapezoida Patitioning Agoithm Scibe: Michae
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 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 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 informationHomework 1 Solutions CSE 101 Summer 2017
Homewok 1 Soutions CSE 101 Summe 2017 1 Waming Up 1.1 Pobem and Pobem Instance Find the smaest numbe in an aay of n integes a 1, a 2,..., a n. What is the input? What is the output? Is this a pobem o a
More informationPhysics 2212 GH Quiz #2 Solutions Spring 2016
Physics 2212 GH Quiz #2 Solutions Sping 216 I. 17 points) Thee point chages, each caying a chage Q = +6. nc, ae placed on an equilateal tiangle of side length = 3. mm. An additional point chage, caying
More informationLecture 8 - Gauss s Law
Lectue 8 - Gauss s Law A Puzzle... Example Calculate the potential enegy, pe ion, fo an infinite 1D ionic cystal with sepaation a; that is, a ow of equally spaced chages of magnitude e and altenating sign.
More informationFlux. Area Vector. Flux of Electric Field. Gauss s Law
Gauss s Law Flux Flux in Physics is used to two distinct ways. The fist meaning is the ate of flow, such as the amount of wate flowing in a ive, i.e. volume pe unit aea pe unit time. O, fo light, it is
More informationThe Solutions of the Classical Relativistic Two-Body Equation
T. J. of Physics (998), 07 4. c TÜBİTAK The Soutions of the Cassica Reativistic Two-Body Equation Coşkun ÖNEM Eciyes Univesity, Physics Depatment, 38039, Kaysei - TURKEY Received 3.08.996 Abstact With
More informationSeparation of Variables and a Spherical Shell with Surface Charge
Separation of Variabes and a Spherica She with Surface Charge In cass we worked out the eectrostatic potentia due to a spherica she of radius R with a surface charge density σθ = σ cos θ. This cacuation
More informationQuestion 1: The dipole
Septembe, 08 Conell Univesity, Depatment of Physics PHYS 337, Advance E&M, HW #, due: 9/5/08, :5 AM Question : The dipole Conside a system as discussed in class and shown in Fig.. in Heald & Maion.. Wite
More informationGreen s Identities and Green s Functions
LECTURE 7 Geen s Identities and Geen s Functions Let us ecall The ivegence Theoem in n-dimensions Theoem 7 Let F : R n R n be a vecto field ove R n that is of class C on some closed, connected, simply
More informationTHE LAPLACE EQUATION. The Laplace (or potential) equation is the equation. u = 0. = 2 x 2. x y 2 in R 2
THE LAPLACE EQUATION The Laplace (o potential) equation is the equation whee is the Laplace opeato = 2 x 2 u = 0. in R = 2 x 2 + 2 y 2 in R 2 = 2 x 2 + 2 y 2 + 2 z 2 in R 3 The solutions u of the Laplace
More informationarxiv: v1 [physics.pop-ph] 3 Jun 2013
A note on the electostatic enegy of two point chages axiv:1306.0401v1 [physics.pop-ph] 3 Jun 013 A C Tot Instituto de Física Univesidade Fedeal do io de Janeio Caixa Postal 68.58; CEP 1941-97 io de Janeio,
More informationThe Divergence Theorem
13.8 The ivegence Theoem Back in 13.5 we ewote Geen s Theoem in vecto fom as C F n ds= div F x, y da ( ) whee C is the positively-oiented bounday cuve of the plane egion (in the xy-plane). Notice this
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 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 informationI. CONSTRUCTION OF THE GREEN S FUNCTION
I. CONSTRUCTION OF THE GREEN S FUNCTION The Helmohltz equation in 4 dimensions is 4 + k G 4 x, x = δ 4 x x. In this equation, G is the Geen s function and 4 efes to the dimensionality. In the vey end,
More information2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum
2. Electostatics D. Rakhesh Singh Kshetimayum 1 2.1 Intoduction In this chapte, we will study how to find the electostatic fields fo vaious cases? fo symmetic known chage distibution fo un-symmetic known
More informationPOISSON S EQUATION 2 V 0
POISSON S EQUATION We have seen how to solve the equation but geneally we have V V4k We now look at a vey geneal way of attacking this poblem though Geen s Functions. It tuns out that this poblem has applications
More informationPHYS 2135 Exam I February 13, 2018
Exam Total /200 PHYS 2135 Exam I Febuay 13, 2018 Name: Recitation Section: Five multiple choice questions, 8 points each Choose the best o most nealy coect answe Fo questions 6-9, solutions must begin
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 information3. Electromagnetic Waves II
Lectue 3 - Electomagnetic Waves II 9 3. Electomagnetic Waves II Last time, we discussed the following. 1. The popagation of an EM wave though a macoscopic media: We discussed how the wave inteacts with
More informationRelating Scattering Amplitudes to Bound States
Reating Scatteing Ampitudes to Bound States Michae Fowe UVa. 1/17/8 Low Enegy Appoximations fo the S Matix In this section we examine the popeties of the patia-wave scatteing matix ( ) = 1+ ( ) S k ikf
More informationAdvanced Quantum Mechanics
Advanced Quantum Mechanics Rajdeep Sensama sensama@theoy.tif.es.in Scatteing Theoy Ref : Sakuai, Moden Quantum Mechanics Tayo, Quantum Theoy of Non-Reativistic Coisions Landau and Lifshitz, Quantum Mechanics
More informationWhat Form of Gravitation Ensures Weakened Kepler s Third Law?
Bulletin of Aichi Univ. of Education, 6(Natual Sciences, pp. - 6, Mach, 03 What Fom of Gavitation Ensues Weakened Keple s Thid Law? Kenzi ODANI Depatment of Mathematics Education, Aichi Univesity of Education,
More informationExam 3, vers Physics Spring, 2003
1 of 9 Exam 3, ves. 0001 - Physics 1120 - Sping, 2003 NAME Signatue Student ID # TA s Name(Cicle one): Michael Scheffestein, Chis Kelle, Paisa Seelungsawat Stating time of you Tues ecitation (wite time
More informationLegendre Polynomials - Lecture 8
Legendre Poynomias - Lecture 8 Introduction In spherica coordinates the separation of variabes for the function of the poar ange resuts in Legendre s equation when the soution is independent of the azimutha
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 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 informationMerging to ordered sequences. Efficient (Parallel) Sorting. Merging (cont.)
Efficient (Paae) Soting One of the most fequent opeations pefomed by computes is oganising (soting) data The access to soted data is moe convenient/faste Thee is a constant need fo good soting agoithms
More informationPhysics Courseware Physics II Electric Field and Force
Physics Cousewae Physics II lectic iel an oce Coulomb s law, whee k Nm /C test Definition of electic fiel. This is a vecto. test Q lectic fiel fo a point chage. This is a vecto. Poblem.- chage of µc is
More informationHowever, because the center-of-mass is at the co-ordinate origin, r1 and r2 are not independent, but are related by
PHYS60 Fa 08 Notes - 3 Centa foce motion The motion of two patices inteacting by a foce that has diection aong the ine joining the patices and stength that depends ony on the sepaation of the two patices
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 informationMagnetic field due to a current loop.
Example using spheical hamonics Sp 18 Magnetic field due to a cuent loop. A cicula loop of adius a caies cuent I. We place the oigin at the cente of the loop, with pola axis pependicula to the plane of
More information3.6 Applied Optimization
.6 Applied Optimization Section.6 Notes Page In this section we will be looking at wod poblems whee it asks us to maimize o minimize something. Fo all the poblems in this section you will be taking the
More informationEM-2. 1 Coulomb s law, electric field, potential field, superposition q. Electric field of a point charge (1)
EM- Coulomb s law, electic field, potential field, supeposition q ' Electic field of a point chage ( ') E( ) kq, whee k / 4 () ' Foce of q on a test chage e at position is ee( ) Electic potential O kq
More informationyou of a spring. The potential energy for a spring is given by the parabola U( x)
Small oscillations The theoy of small oscillations is an extemely impotant topic in mechanics. Conside a system that has a potential enegy diagam as below: U B C A x Thee ae thee points of stable equilibium,
More informationGauss s Law: Circuits
Gauss s Law: Cicuits Can we have excess chage inside in steady state? E suface nˆ A q inside E nˆ A E nˆ A left _ suface ight _ suface q inside 1 Gauss s Law: Junction Between two Wies n 2
More informationPhysics 505 Fall 2007 Homework Assignment #5 Solutions. Textbook problems: Ch. 3: 3.13, 3.17, 3.26, 3.27
Physics 55 Fa 7 Homework Assignment #5 Soutions Textook proems: Ch. 3: 3.3, 3.7, 3.6, 3.7 3.3 Sove for the potentia in Proem 3., using the appropriate Green function otained in the text, and verify that
More information7.2. Coulomb s Law. The Electric Force
Coulomb s aw Recall that chaged objects attact some objects and epel othes at a distance, without making any contact with those objects Electic foce,, o the foce acting between two chaged objects, is somewhat
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 informationEM Boundary Value Problems
EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do
More informationGauss Law. Physics 231 Lecture 2-1
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 informationPhysics 181. Assignment 4
Physics 181 Assignment 4 Solutions 1. A sphee has within it a gavitational field given by g = g, whee g is constant and is the position vecto of the field point elative to the cente of the sphee. This
More informationQuestion Bank. Section A. is skew-hermitian matrix. is diagonalizable. (, ) , Evaluate (, ) 12 about = 1 and = Find, if
Subject: Mathematics-I Question Bank Section A T T. Find the value of fo which the matix A = T T has ank one. T T i. Is the matix A = i is skew-hemitian matix. i. alculate the invese of the matix = 5 7
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 informationStress, Cauchy s equation and the Navier-Stokes equations
Chapte 3 Stess, Cauchy s equation and the Navie-Stokes equations 3. The concept of taction/stess Conside the volume of fluid shown in the left half of Fig. 3.. The volume of fluid is subjected to distibuted
More informationPHYS 705: Classical Mechanics. Central Force Problems II
PHYS 75: Cassica Mechanics Centa Foce Pobems II Obits in Centa Foce Pobem Sppose we e inteested moe in the shape of the obit, (not necessay the time evotion) Then, a sotion fo = () o = () wod be moe sef!
More informationDOING PHYSICS WITH MATLAB COMPUTATIONAL OPTICS
DOING PHYIC WITH MTLB COMPUTTIONL OPTIC FOUNDTION OF CLR DIFFRCTION THEORY Ian Coope chool of Physics, Univesity of ydney ian.coope@sydney.edu.au DOWNLOD DIRECTORY FOR MTLB CRIPT View document: Numeical
More information2.25 Advanced Fluid Mechanics
MIT Depatment of Mechanical Engineeing 2.25 Advanced Fluid Mechanics Poblem 4.27 This poblem is fom Advanced Fluid Mechanics Poblems by A.H. Shapio and A.A. Sonin u(,t) pg Gas Liquid, density Conside a
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 informationToday s Plan. Electric Dipoles. More on Gauss Law. Comment on PDF copies of Lectures. Final iclicker roll-call
Today s Plan lectic Dipoles Moe on Gauss Law Comment on PDF copies of Lectues Final iclicke oll-call lectic Dipoles A positive (q) and negative chage (-q) sepaated by a small distance d. lectic dipole
More informationMultipole Radiation. February 29, The electromagnetic field of an isolated, oscillating source
Multipole Radiation Febuay 29, 26 The electomagnetic field of an isolated, oscillating souce Conside a localized, oscillating souce, located in othewise empty space. We know that the solution fo the vecto
More informationSolutions to Problems : Chapter 19 Problems appeared on the end of chapter 19 of the Textbook
Solutions to Poblems Chapte 9 Poblems appeae on the en of chapte 9 of the Textbook 8. Pictue the Poblem Two point chages exet an electostatic foce on each othe. Stategy Solve Coulomb s law (equation 9-5)
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 information8 Separation of Variables in Other Coordinate Systems
8 Sepaation of Vaiables in Othe Coodinate Systems Fo the method of sepaation of vaiables to succeed you need to be able to expess the poblem at hand in a coodinate system in which the physical boundaies
More informationAn Exact Solution of Navier Stokes Equation
An Exact Solution of Navie Stokes Equation A. Salih Depatment of Aeospace Engineeing Indian Institute of Space Science and Technology, Thiuvananthapuam, Keala, India. July 20 The pincipal difficulty in
More informationPhysics 4A Chapter 8: Dynamics II Motion in a Plane
Physics 4A Chapte 8: Dynamics II Motion in a Plane Conceptual Questions and Example Poblems fom Chapte 8 Conceptual Question 8.5 The figue below shows two balls of equal mass moving in vetical cicles.
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 informationWhy Professor Richard Feynman was upset solving the Laplace equation for spherical waves? Anzor A. Khelashvili a)
Why Pofesso Richad Feynman was upset solving the Laplace equation fo spheical waves? Anzo A. Khelashvili a) Institute of High Enegy Physics, Iv. Javakhishvili Tbilisi State Univesity, Univesity St. 9,
More information( ) [ ] [ ] [ ] δf φ = F φ+δφ F. xdx.
9. LAGRANGIAN OF THE ELECTROMAGNETIC FIELD In the pevious section the Lagangian and Hamiltonian of an ensemble of point paticles was developed. This appoach is based on a qt. This discete fomulation can
More informationand Slater Sum Rule Method * M L = 0, M S = 0 block: L L+ L 2
5.7 Lectue #4 e / ij and Sate Sum Rue Method 4 - LAST TIME:. L,S method fo setting up NLM L SM S many-eecton basis states in tems of inea combination of Sate deteminants * M L = 0, M S = 0 boc: L L+ L
More informationf(k) e p 2 (k) e iax 2 (k a) r 2 e a x a a 2 + k 2 e a2 x 1 2 H(x) ik p (k) 4 r 3 cos Y 2 = 4
Fouie tansfom pais: f(x) 1 f(k) e p 2 (k) p e iax 2 (k a) 2 e a x a a 2 + k 2 e a2 x 1 2, a > 0 a p k2 /4a2 e 2 1 H(x) ik p 2 + 2 (k) The fist few Y m Y 0 0 = Y 0 1 = Y ±1 1 = l : 1 Y2 0 = 4 3 ±1 cos Y
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 informationChapter 2: Basic Physics and Math Supplements
Chapte 2: Basic Physics and Math Supplements Decembe 1, 215 1 Supplement 2.1: Centipetal Acceleation This supplement expands on a topic addessed on page 19 of the textbook. Ou task hee is to calculate
More informationElectromagnetism Physics 15b
lectomagnetism Physics 15b Lectue #20 Dielectics lectic Dipoles Pucell 10.1 10.6 What We Did Last Time Plane wave solutions of Maxwell s equations = 0 sin(k ωt) B = B 0 sin(k ωt) ω = kc, 0 = B, 0 ˆk =
More informationLecture 1. time, say t=0, to find the wavefunction at any subsequent time t. This can be carried out by
Lectue The Schödinge equation In quantum mechanics, the fundamenta quantity that descibes both the patice-ike and waveike chaacteistics of patices is wavefunction, Ψ(. The pobabiity of finding a patice
More informationPressure in the Average-Atom Model
Pessue in the Aveage-Atom Moe W.. Johnson Depatment of Physics, 225 Nieuwan Science Ha Note Dame Univesity, Note Dame, IN 46556 Febuay 28, 2002 Abstact The (we-known) quantum mechanica expession fo the
More informationTo Feel a Force Chapter 7 Static equilibrium - torque and friction
To eel a oce Chapte 7 Chapte 7: Static fiction, toque and static equilibium A. Review of foce vectos Between the eath and a small mass, gavitational foces of equal magnitude and opposite diection act on
More informationEELE 3331 Electromagnetic I Chapter 4. Electrostatic fields. Islamic University of Gaza Electrical Engineering Department Dr.
EELE 3331 Electomagnetic I Chapte 4 Electostatic fields Islamic Univesity of Gaza Electical Engineeing Depatment D. Talal Skaik 212 1 Electic Potential The Gavitational Analogy Moving an object upwad against
More informationProblem set 6. Solution. The problem of firm 3 is. The FOC is: 2 =0. The reaction function of firm 3 is: = 2
Pobem set 6 ) Thee oigopoists opeate in a maket with invese demand function given by = whee = + + and is the quantity poduced by fim i. Each fim has constant magina cost of poduction, c, and no fixed cost.
More informationAn o5en- confusing point:
An o5en- confusing point: Recall this example fom last lectue: E due to a unifom spheical suface chage, density = σ. Let s calculate the pessue on the suface. Due to the epulsive foces, thee is an outwad
More information7.2.1 Basic relations for Torsion of Circular Members
Section 7. 7. osion In this section, the geomety to be consideed is that of a long slende cicula ba and the load is one which twists the ba. Such poblems ae impotant in the analysis of twisting components,
More informationDo not turn over until you are told to do so by the Invigilator.
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Seies UG Examination 2015 16 FLUID DYNAMICS WITH ADVANCED TOPICS MTH-MD59 Time allowed: 3 Hous Attempt QUESTIONS 1 and 2, and THREE othe questions.
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 informationPhysics Courseware Electromagnetism
Pysics Cousewae lectomagnetism lectic field Poblem.- a) Find te electic field at point P poduced by te wie sown in te figue. Conside tat te wie as a unifom linea cage distibution of λ.5µ C / m b) Find
More informationPHY481: Electromagnetism
PHY48: Electomagnetism HW5 Lectue Cal Bombeg - Pof. of Physics Bounday condition ( ) = C n cos n + V x, y ( ) n= ( ) = V cos x V x,± a 5.3 V = V cos x a & ' at y = ± a Geneal solution (fo even bounday
More informationHopefully Helpful Hints for Gauss s Law
Hopefully Helpful Hints fo Gauss s Law As befoe, thee ae things you need to know about Gauss s Law. In no paticula ode, they ae: a.) In the context of Gauss s Law, at a diffeential level, the electic flux
More informationA Tutorial on Multiple Integrals (for Natural Sciences / Computer Sciences Tripos Part IA Maths)
A Tutoial on Multiple Integals (fo Natual Sciences / Compute Sciences Tipos Pat IA Maths) Coections to D Ian Rud (http://people.ds.cam.ac.uk/ia/contact.html) please. This tutoial gives some bief eamples
More information12th WSEAS Int. Conf. on APPLIED MATHEMATICS, Cairo, Egypt, December 29-31,
th WSEAS Int. Conf. on APPLIED MATHEMATICS, Caio, Egypt, Decembe 9-3, 7 5 Magnetostatic Field calculations associated with thick Solenoids in the Pesence of Ion using a Powe Seies expansion and the Complete
More informationElectrostatics (Electric Charges and Field) #2 2010
Electic Field: The concept of electic field explains the action at a distance foce between two chaged paticles. Evey chage poduces a field aound it so that any othe chaged paticle expeiences a foce when
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 informationPY208 Matter & Interactions Final Exam S2005
PY Matte & Inteactions Final Exam S2005 Name (pint) Please cicle you lectue section below: 003 (Ramakishnan 11:20 AM) 004 (Clake 1:30 PM) 005 (Chabay 2:35 PM) When you tun in the test, including the fomula
More informationPHYS 110B - HW #1 Fall 2005, Solutions by David Pace Equations referenced as Eq. # are from Griffiths Problem statements are paraphrased
PHYS 110B - HW #1 Fa 2005, Soutions by David Pace Equations referenced as Eq. # are from Griffiths Probem statements are paraphrased [1.] Probem 6.8 from Griffiths A ong cyinder has radius R and a magnetization
More informationMath 2263 Solutions for Spring 2003 Final Exam
Math 6 Solutions fo Sping Final Exam ) A staightfowad appoach to finding the tangent plane to a suface at a point ( x, y, z ) would be to expess the cuve as an explicit function z = f ( x, y ), calculate
More informationof Technology: MIT OpenCourseWare). (accessed MM DD, YYYY). License: Creative Commons Attribution- Noncommercial-Share Alike.
MIT OpenCouseWae http://ocw.mit.eu 6.013/ESD.013J Electomagnetics an Applications, Fall 005 Please use the following citation fomat: Makus Zahn, Eich Ippen, an Davi Staelin, 6.013/ESD.013J Electomagnetics
More informationCOORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT
COORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT Link to: phsicspages home page. To leave a comment o epot an eo, please use the auilia blog. Refeence: d Inveno, Ra, Intoducing Einstein s Relativit
More information