Problem 1. Solution: a) The coordinate of a point on the disc is given by r r cos,sin,0. The potential at P is then given by. r z 2 rcos 2 rsin 2

Size: px
Start display at page:

Download "Problem 1. Solution: a) The coordinate of a point on the disc is given by r r cos,sin,0. The potential at P is then given by. r z 2 rcos 2 rsin 2"

Transcription

1 Prolem Consider disc of chrge density r r nd rdius R tht lies within the xy-plne. The origin of the coordinte systems is locted t the center of the ring. ) Give the potentil t the point P,,z in terms of,r, ndz. ) We next put conducting plne into the z d plne. The potentil of the conducting plne is fixed t V. Compute the totl potentil t point P,,z. c) If the totl chrge, Q, on the disc is fixed, find the chrge density in terms of Q nd use it to otin the form of tot P in terms of Q,R,z in the limit R z,d up to leding order in z/r? d) Give n explicit form of the induced chrge density t P,,d in the limit R d using the results of prt c). Solution: ) The coordinte of point on the disc is given y r rcos,sin, The potentil t P is then given y P 4 dq P r 4 d r r P r R 4 d rdr R 4 d dr r R z r dr r z r R z R z ln R z R z r z rcos rsin where I used dr r z r r z r z ln r z r ) Using the method of imges, the imge chrge is disc of chrge density r r nd rdius R tht is prllel to the conducting plne t distnce of d from the first disc. The resulting totl potentil is then

2 tot P R z R z ln R z R z R d z R d z ln R d z R d z c) If the totl chrge, Q, on the disc is fixed, wht is the form of tot P in terms of Q,R,z in the limit R z,d up to leding order in z/r? Ifthechrgeisfixed,wehve or nd thus Q dq d R rdr r R Q R tot P Q R R z R z ln R z R z Q R R d z R d z ln R d z R d z Q 8 R R z R z ln R z R z Q 8 R R d z R d z ln R d z R d z We first consider the first two terms which yield Q 8 R z R d z R Q z z d 6 R The second set yields Q 6 R 4zd 4d Qd z d 4 R

3 Q 8 R z ln R z R z d z ln Q 8 R z ln R z d z ln R d z This is the leding contriution. R d z R d z d) Give n explicit form of the induced chrge density t P,,d in the limit R d using the results of prt c). The chrge density follows from tot z nd thus tot Q z z zd 8R z ln Q zln z z 8R R z d zln d z R Q zln d d d dln d d 8R R R 6Qd 4R ln d R zd z d z ln d z R R d z d d zd d z zd

4 Prolem Consider sphere of rdius R. The potentil on the surfce of the sphere vries s (see figure elow) (free region inside nd outside) cos ) Compute the potentil inside nd outside of the sphere. ) Compute the electric field inside the sphere. c) Using Guss; lw, show tht while the electric field inside the sphere is non-zero, no chrge is contined inside the sphere. Solution:

5 ) Since the prolem hs n ziumuthl symmetry, we hve r, A l r l B l r l P l cos l Using tht we otin P cos P cos cos And thus cos P cos P cos P cos dxr,xpm x dx P cos P cosp m x m m, m m, 4 5 m, m, If we wnt to evlute the potentil inside of the sphere, we need to set B l nd otin nd thus for m nd for m nd thus dxr,xpm x dx l A m R m m 4 5 A R 5 A R A A A l R l P l x P m x r, P cos r R P cos

6 For the potentil outside of the sphere, we set A l nd otin nd thus for m dxr,xpm x dx l B m R m m B l R l P l cos P m x nd for m nd thus 4 5 B R 5 B R B R B R r, R r P cos R r P cos ) The electric field inside the sphere is then given y E r, r r, r, r r r 4 r R P cos r r sincos R 4 r R P cosr r R sincos c) Using Guss lw inside the shere E da 4 Thus, no chrges re contined inside the sphere. r R r d dcosp cos Prolem ) Consider the two conducting spheres with rdii nd ( ) s shown in the figure

7 elow. The volume etween the two spheres (region II) is filled with mteril of permitivity. The permitivity in regions I nd III is tht of free spce,. The two spheres re uniformly chrged with totl chrge Q. (i) Compute the mgnitude nd direction of the electric field in regions I, II, nd III. (ii) Compute the cpcitnce of the two spheres. ) Consider next two infinitely long concentric cylinders, s shown in the figure elow. The inner cylinder of rdius is conductor with liner chrge density. The second cylinder with inner rdius nd outer rdius c consists of mteril with permitivity nd is uniformly chrged with line chrge density ( ). The spce etween the two cylinders (i.e., r ) is filled with medium of permitivity. The medium outside the outer cylinder possesses the permitivity. Compute the potentil difference etween point t r c (mesured from the center of the inner cylinders) nd the center of the inner cylinder.

8 Solutions: ) (i) Using Guss lw, we cn compute the electric fields of the two sphere system. Region I: The electric field in reggion I for r,with r mesured from the center of the sphere, is given y E ds E4r q encl E the electric field is zero inside the inner sphere. Region II: The electric displcement field for r,with r mesured from the center of the sphere is given y D ds D4r Q D Q 4r r E D Q 4r r The electric field points rdilly inwrd. Region III: The electric field in reggion III for r,with r mesured from the center of the sphere, is given y E ds E4r q encl E the electric field is zero inside the outside the outer sphere.

9 (ii) The potentil difference etween the two sphere is nd the cpcitnce is thus Δ E tot dr Q 4 C Q Δ Q 4 4 r dr )We need to compute the electric field in the different regions of the prolem. i) r. WehveE inside the inner cylinder is zero, since it is conductor. ii) r. Weuse nd thus D d l rld l D E D r Note tht since the electric field points rdilly outwrds. iii) r c. Note tht the mteril in this region is insulting nd uniformly chrged. The volume chrge density is r c nd thus D d r c D r l rld r c r c l nd thus E D r r c iv) c r. Herewehve E r

10 We cn now compute the potentil difference Δ c c E tot dr c Edr c Edr Edr Edr where nd c c c Edr c r dr ln c Edr c c r r c r c c dr c ln c 4 r c dr nd nd Edr r dr ln Edr Thus I otin Δ ln c ln c ln Prolem 4 A solenoid of finite length nd redius hs N turns per unit length nd crries current I, with ciculr cross section s shown in the figure elow.

11 ) Compute the mgnetic induction on the solenoid xis in the limit N in terms of the ngles nd. ) For, how does the mgnetic induction scle with? Solutions: ) We first compute the mgnetic induction due to single loop. Using Biot-Svrt where db 4 I dl r r r z By symmetry, integrtion over the loop yields mgnetic induction long the z-xis. We then hve B 4 I dl r r 4 I z / dlr sinẑ where nd thus sin z /

12 B 4 I z / z / I z / ẑ ẑ z / Next, we consider the entire solenoid. B ẑ NI z dz z / z dz z / ẑ NI z z / ẑ NI cos cos The limit N is necessry to perform the integrtion over z. z z / ) From the ove expression, we hve limb limẑ NI z ẑ NI z z / z ẑ I N z z / Prolem 5 An electromgnetic plne wve is incident perpendiculr to lyered interfce, s shown in the figure elow. The indices of refrction of the three medi is n,n n nd n 4n while the permeility of ll three regions is the sme,. The thickness of the intermedite lyer is d. Ech of the other medi is semi-infinite.

13 d E B k n n n z= z=d ) Stte the oundry conditions t oth interfces in terms of the electric fields. ) Compute the rtio etween the incident electric field in medium nd the trnsmitted electric field in medium, i..e, compute E i /E t. c) If the thickness d is vried, the rtio E i /E t oscilltes. Wht is the period of the oscilltion? For which vlues of d is E i /E t the smllest? Solutions: ) Stte the oundry conditions t oth interfces in terms of the electric fields. The EM wve contins only components tht re perpendiculr to the interfce. In region, there is n incoming nd reflected wve, in region there is right-moving nd left-moving wve, nd in region, there is only trnsmitted wve. Thus the oundry conditions t z re E i E r E E E i E r c nd t z d we hve E E c E i E r n n E E E E E e ik d E e ik d E t e ik d E e ik d E e ik d c E te ik d c E e ik d E e ik d n n E t e ik d E t e ik d ) Compute the rtio etween the incident electric field in medium nd the trnsmitted electric field in medium, i..e, compute E i /E t. From the lst two equtions, I otin E n n E n n E t e ik d e ik d E te ik d e ik d E t e ik d e ik d E te ik d e ik d nd from the first two equtions

14 E i n n E n n E E E n n n n E t e ik d e ik d n n n n 9 E te ik d e ik d E te ik d e ik d E t e ik d e ik d nd thus 4 E i e E ikd e ik d n t n n n n n n n e ik d e ik d 9 e ik d e ik d nd hence 6 E i E t n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n 4 n n n n 4 n n 4 4 4sin k d 6sin k d e ik d n n n n sin k d e ik d cosk d n n sin k d or lterntively E i E t 4 n n n n n n n n sin k d sin k d 4 5 9sin k d c) If the thickness d is vried, the rtio E i /E t oscilltes. Wht is the period of the oscilltion? Assuming n n n, for which vlues of d is E i /E t the smllest? The period of the oscilltion is,nd E i /E t is the smllest for d m / with m eing n integer.

15 Mthemticl Formule Definitions r 4 d r r r r E r r B r 4 d r J r r r r r A r 4 d r J r r r Δ E r dr C Q Δ ; E ; B n In sphericl coordintes E B t ; E r,, r r,, r B J r r,, rsin r,, Integrls, Series, Expnsions nd Identities d cos K where K is the complete elliptic integrl

16 x x dx / / dx x x / x / dr r z r r z r z ln r z r c dx x x / x c c c dx Pl x l for even l for l ll!! l l for odd l dxpl x l dx Pl x dxpl xp m x l lm dx Pl x l P cos P cos cos P cos cos P cos 5cos cos r, A n r n B n r n P ncos n r dr r r dr lnr x x...

Homework Assignment 3 Solution Set

Homework 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 information

ragsdale (zdr82) HW2 ditmire (58335) 1

ragsdale (zdr82) HW2 ditmire (58335) 1 rgsdle (zdr82) HW2 ditmire (58335) This print-out should hve 22 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. 00 0.0 points A chrge of 8. µc

More information

Physics 3323, Fall 2016 Problem Set 7 due Oct 14, 2016

Physics 3323, Fall 2016 Problem Set 7 due Oct 14, 2016 Physics 333, Fll 16 Problem Set 7 due Oct 14, 16 Reding: Griffiths 4.1 through 4.4.1 1. Electric dipole An electric dipole with p = p ẑ is locted t the origin nd is sitting in n otherwise uniform electric

More information

This final is a three hour open book, open notes exam. Do all four problems.

This final is a three hour open book, open notes exam. Do all four problems. Physics 55 Fll 27 Finl Exm Solutions This finl is three hour open book, open notes exm. Do ll four problems. [25 pts] 1. A point electric dipole with dipole moment p is locted in vcuum pointing wy from

More information

Candidates must show on each answer book the type of calculator used.

Candidates must show on each answer book the type of calculator used. UNIVERSITY OF EAST ANGLIA School of Mthemtics My/June UG Exmintion 2007 2008 ELECTRICITY AND MAGNETISM Time llowed: 3 hours Attempt FIVE questions. Cndidtes must show on ech nswer book the type of clcultor

More information

Physics 2135 Exam 1 February 14, 2017

Physics 2135 Exam 1 February 14, 2017 Exm Totl / 200 Physics 215 Exm 1 Ferury 14, 2017 Printed Nme: Rec. Sec. Letter: Five multiple choice questions, 8 points ech. Choose the est or most nerly correct nswer. 1. Two chrges 1 nd 2 re seprted

More information

Physics Jonathan Dowling. Lecture 9 FIRST MIDTERM REVIEW

Physics 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 information

Problems for HW X. C. Gwinn. November 30, 2009

Problems for HW X. C. Gwinn. November 30, 2009 Problems for HW X C. Gwinn November 30, 2009 These problems will not be grded. 1 HWX Problem 1 Suppose thn n object is composed of liner dielectric mteril, with constnt reltive permittivity ɛ r. The object

More information

Problem Set 3 Solutions

Problem Set 3 Solutions Msschusetts Institute of Technology Deprtment of Physics Physics 8.07 Fll 2005 Problem Set 3 Solutions Problem 1: Cylindricl Cpcitor Griffiths Problems 2.39: Let the totl chrge per unit length on the inner

More information

Lecture 1: Electrostatic Fields

Lecture 1: Electrostatic Fields Lecture 1: Electrosttic Fields Instructor: Dr. Vhid Nyyeri Contct: nyyeri@iust.c.ir Clss web site: http://webpges.iust.c. ir/nyyeri/courses/bee 1.1. Coulomb s Lw Something known from the ncient time (here

More information

IMPORTANT. Read these directions carefully:

IMPORTANT. Read these directions carefully: Physics 208: Electricity nd Mgnetism Finl Exm, Secs. 506 510. 7 My. 2004 Instructor: Dr. George R. Welch, 415 Engineering-Physics, 845-7737 Print your nme netly: Lst nme: First nme: Sign your nme: Plese

More information

Chapter 7 Steady Magnetic Field. september 2016 Microwave Laboratory Sogang University

Chapter 7 Steady Magnetic Field. september 2016 Microwave Laboratory Sogang University Chpter 7 Stedy Mgnetic Field september 2016 Microwve Lbortory Sogng University Teching point Wht is the mgnetic field? Biot-Svrt s lw: Coulomb s lw of Mgnetic field Stedy current: current flow is independent

More information

200 points 5 Problems on 4 Pages and 20 Multiple Choice/Short Answer Questions on 5 pages 1 hour, 48 minutes

200 points 5 Problems on 4 Pages and 20 Multiple Choice/Short Answer Questions on 5 pages 1 hour, 48 minutes PHYSICS 132 Smple Finl 200 points 5 Problems on 4 Pges nd 20 Multiple Choice/Short Answer Questions on 5 pges 1 hour, 48 minutes Student Nme: Recittion Instructor (circle one): nme1 nme2 nme3 nme4 Write

More information

Homework Assignment 6 Solution Set

Homework Assignment 6 Solution Set Homework Assignment 6 Solution Set PHYCS 440 Mrch, 004 Prolem (Griffiths 4.6 One wy to find the energy is to find the E nd D fields everywhere nd then integrte the energy density for those fields. We know

More information

Electromagnetism Answers to Problem Set 10 Spring 2006

Electromagnetism Answers to Problem Set 10 Spring 2006 Electromgnetism 76 Answers to Problem Set 1 Spring 6 1. Jckson Prob. 5.15: Shielded Bifilr Circuit: Two wires crrying oppositely directed currents re surrounded by cylindricl shell of inner rdius, outer

More information

Chapter 6 Electrostatic Boundary Value Problems. Dr. Talal Skaik

Chapter 6 Electrostatic Boundary Value Problems. Dr. Talal Skaik Chpter 6 Electrosttic Boundry lue Problems Dr. Tll Skik 1 1 Introduction In previous chpters, E ws determined by coulombs lw or Guss lw when chrge distribution is known, or potentil is known throughout

More information

CAPACITORS AND DIELECTRICS

CAPACITORS AND DIELECTRICS Importnt Definitions nd Units Cpcitnce: CAPACITORS AND DIELECTRICS The property of system of electricl conductors nd insultors which enbles it to store electric chrge when potentil difference exists between

More information

Physics 1402: Lecture 7 Today s Agenda

Physics 1402: Lecture 7 Today s Agenda 1 Physics 1402: Lecture 7 Tody s gend nnouncements: Lectures posted on: www.phys.uconn.edu/~rcote/ HW ssignments, solutions etc. Homework #2: On Msterphysics tody: due Fridy Go to msteringphysics.com Ls:

More information

Phys 4321 Final Exam December 14, 2009

Phys 4321 Final Exam December 14, 2009 Phys 4321 Finl Exm December 14, 2009 You my NOT use the text book or notes to complete this exm. You nd my not receive ny id from nyone other tht the instructor. You will hve 3 hours to finish. DO YOUR

More information

Physics 24 Exam 1 February 18, 2014

Physics 24 Exam 1 February 18, 2014 Exm Totl / 200 Physics 24 Exm 1 Februry 18, 2014 Printed Nme: Rec. Sec. Letter: Five multiple choice questions, 8 points ech. Choose the best or most nerly correct nswer. 1. The totl electric flux pssing

More information

Exam 1 Solutions (1) C, D, A, B (2) C, A, D, B (3) C, B, D, A (4) A, C, D, B (5) D, C, A, B

Exam 1 Solutions (1) C, D, A, B (2) C, A, D, B (3) C, B, D, A (4) A, C, D, B (5) D, C, A, B PHY 249, Fll 216 Exm 1 Solutions nswer 1 is correct for ll problems. 1. Two uniformly chrged spheres, nd B, re plced t lrge distnce from ech other, with their centers on the x xis. The chrge on sphere

More information

That reminds me must download the test prep HW. adapted from (nz118.jpg)

That reminds me must download the test prep HW. adapted from   (nz118.jpg) Tht reminds me must downlod the test prep HW. dpted from http://www.neringzero.net (nz118.jpg) Em 1: Tuesdy, Feb 14, 5:00-6:00 PM Test rooms: Instructor Sections Room Dr. Hle F, H 104 Physics Dr. Kurter

More information

Lecture 13 - Linking E, ϕ, and ρ

Lecture 13 - Linking E, ϕ, and ρ Lecture 13 - Linking E, ϕ, nd ρ A Puzzle... Inner-Surfce Chrge Density A positive point chrge q is locted off-center inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on

More information

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus 7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e

More information

Conducting Ellipsoid and Circular Disk

Conducting Ellipsoid and Circular Disk 1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid,

More information

Theoretische Physik 2: Elektrodynamik (Prof. A.-S. Smith) Home assignment 4

Theoretische Physik 2: Elektrodynamik (Prof. A.-S. Smith) Home assignment 4 WiSe 1 8.1.1 Prof. Dr. A.-S. Smith Dipl.-Phys. Ellen Fischermeier Dipl.-Phys. Mtthis Sb m Lehrstuhl für Theoretische Physik I Deprtment für Physik Friedrich-Alexnder-Universität Erlngen-Nürnberg Theoretische

More information

Reference. Vector Analysis Chapter 2

Reference. Vector Analysis Chapter 2 Reference Vector nlsis Chpter Sttic Electric Fields (3 Weeks) Chpter 3.3 Coulomb s Lw Chpter 3.4 Guss s Lw nd pplictions Chpter 3.5 Electric Potentil Chpter 3.6 Mteril Medi in Sttic Electric Field Chpter

More information

Problem Solving 7: Faraday s Law Solution

Problem Solving 7: Faraday s Law Solution MASSACHUSETTS NSTTUTE OF TECHNOLOGY Deprtment of Physics: 8.02 Prolem Solving 7: Frdy s Lw Solution Ojectives 1. To explore prticulr sitution tht cn led to chnging mgnetic flux through the open surfce

More information

Physics Graduate Prelim exam

Physics Graduate Prelim exam Physics Grdute Prelim exm Fll 2008 Instructions: This exm hs 3 sections: Mechnics, EM nd Quntum. There re 3 problems in ech section You re required to solve 2 from ech section. Show ll work. This exm is

More information

Physics 2135 Exam 3 April 21, 2015

Physics 2135 Exam 3 April 21, 2015 Em Totl hysics 2135 Em 3 April 21, 2015 Key rinted Nme: 200 / 200 N/A Rec. Sec. Letter: Five multiple choice questions, 8 points ech. Choose the best or most nerly correct nswer. 1. C Two long stright

More information

Reading from Young & Freedman: For this topic, read the introduction to chapter 24 and sections 24.1 to 24.5.

Reading from Young & Freedman: For this topic, read the introduction to chapter 24 and sections 24.1 to 24.5. PHY1 Electricity Topic 5 (Lectures 7 & 8) pcitors nd Dielectrics In this topic, we will cover: 1) pcitors nd pcitnce ) omintions of pcitors Series nd Prllel 3) The energy stored in cpcitor 4) Dielectrics

More information

Waveguide Guide: A and V. Ross L. Spencer

Waveguide Guide: A and V. Ross L. Spencer Wveguide Guide: A nd V Ross L. Spencer I relly think tht wveguide fields re esier to understnd using the potentils A nd V thn they re using the electric nd mgnetic fields. Since Griffiths doesn t do it

More information

MATH 253 WORKSHEET 24 MORE INTEGRATION IN POLAR COORDINATES. r dr = = 4 = Here we used: (1) The half-angle formula cos 2 θ = 1 2

MATH 253 WORKSHEET 24 MORE INTEGRATION IN POLAR COORDINATES. r dr = = 4 = Here we used: (1) The half-angle formula cos 2 θ = 1 2 MATH 53 WORKSHEET MORE INTEGRATION IN POLAR COORDINATES ) Find the volume of the solid lying bove the xy-plne, below the prboloid x + y nd inside the cylinder x ) + y. ) We found lst time the set of points

More information

P812 Midterm Examination February Solutions

P812 Midterm Examination February Solutions P8 Midterm Exmintion Februry s. A one dimensionl chin of chrges consist of e nd e lterntively plced with neighbouring distnce. Show tht the potentil energy of ech chrge is given by U = ln. 4πε Explin qulittively

More information

Prof. Anchordoqui. Problems set # 4 Physics 169 March 3, 2015

Prof. Anchordoqui. Problems set # 4 Physics 169 March 3, 2015 Prof. Anchordoui Problems set # 4 Physics 169 Mrch 3, 15 1. (i) Eight eul chrges re locted t corners of cube of side s, s shown in Fig. 1. Find electric potentil t one corner, tking zero potentil to be

More information

Today in Physics 122: work, energy and potential in electrostatics

Today in Physics 122: work, energy and potential in electrostatics Tody in Physics 1: work, energy nd potentil in electrosttics Leftovers Perfect conductors Fields from chrges distriuted on perfect conductors Guss s lw for grvity Work nd energy Electrosttic potentil energy,

More information

Physics 712 Electricity and Magnetism Solutions to Final Exam, Spring 2016

Physics 712 Electricity and Magnetism Solutions to Final Exam, Spring 2016 Physics 7 Electricity nd Mgnetism Solutions to Finl Em, Spring 6 Plese note tht some possibly helpful formuls pper on the second pge The number of points on ech problem nd prt is mrked in squre brckets

More information

R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of

R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of

More information

- 5 - TEST 2. This test is on the final sections of this session's syllabus and. should be attempted by all students.

- 5 - TEST 2. This test is on the final sections of this session's syllabus and. should be attempted by all students. - 5 - TEST 2 This test is on the finl sections of this session's syllbus nd should be ttempted by ll students. Anything written here will not be mrked. - 6 - QUESTION 1 [Mrks 22] A thin non-conducting

More information

The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Factor of an Insulated Two-Wire Transmission Line The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the

More information

Method of Localisation and Controlled Ejection of Swarms of Likely Charged Particles

Method of Localisation and Controlled Ejection of Swarms of Likely Charged Particles Method of Loclistion nd Controlled Ejection of Swrms of Likely Chrged Prticles I. N. Tukev July 3, 17 Astrct This work considers Coulom forces cting on chrged point prticle locted etween the two coxil,

More information

Physics 202, Lecture 14

Physics 202, Lecture 14 Physics 202, Lecture 14 Tody s Topics Sources of the Mgnetic Field (Ch. 28) Biot-Svrt Lw Ampere s Lw Mgnetism in Mtter Mxwell s Equtions Homework #7: due Tues 3/11 t 11 PM (4th problem optionl) Mgnetic

More information

Jackson 2.7 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jackson 2.7 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell Jckson.7 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: Consider potentil problem in the hlf-spce defined by, with Dirichlet boundry conditions on the plne

More information

Hung problem # 3 April 10, 2011 () [4 pts.] The electric field points rdilly inwrd [1 pt.]. Since the chrge distribution is cylindriclly symmetric, we pick cylinder of rdius r for our Gussin surfce S.

More information

Phys 6321 Final Exam - Solutions May 3, 2013

Phys 6321 Final Exam - Solutions May 3, 2013 Phys 6321 Finl Exm - Solutions My 3, 2013 You my NOT use ny book or notes other thn tht supplied with this test. You will hve 3 hours to finish. DO YOUR OWN WORK. Express your nswers clerly nd concisely

More information

Version 001 HW#6 - Electromagnetism arts (00224) 1

Version 001 HW#6 - Electromagnetism arts (00224) 1 Version 001 HW#6 - Electromgnetism rts (00224) 1 This print-out should hve 11 questions. Multiple-choice questions my continue on the next column or pge find ll choices efore nswering. rightest Light ul

More information

AP Physics C: Electricity & Magnetism 1999 Free-Response Questions

AP Physics C: Electricity & Magnetism 1999 Free-Response Questions AP Physics C: Electricity & Mgnetism 1999 Free-esponse Questions The mterils included in these files re intended for non-commercil use by AP techers for course nd exm preprtion; permission for ny other

More information

Chapter 1 VECTOR ALGEBRA

Chapter 1 VECTOR ALGEBRA Chpter 1 VECTOR LGEBR INTRODUCTION: Electromgnetics (EM) m be regrded s the stud of the interctions between electric chrges t rest nd in motion. Electromgnetics is brnch of phsics or electricl engineering

More information

Thomas Whitham Sixth Form

Thomas Whitham Sixth Form Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos

More information

Resistors. Consider a uniform cylinder of material with mediocre to poor to pathetic conductivity ( )

Resistors. Consider a uniform cylinder of material with mediocre to poor to pathetic conductivity ( ) 10/25/2005 Resistors.doc 1/7 Resistors Consider uniform cylinder of mteril with mediocre to poor to r. pthetic conductivity ( ) ˆ This cylinder is centered on the -xis, nd hs length. The surfce re of the

More information

in a uniform magnetic flux density B = Boa z. (a) Show that the electron moves in a circular path. (b) Find the radius r o

in a uniform magnetic flux density B = Boa z. (a) Show that the electron moves in a circular path. (b) Find the radius r o 6. THE TATC MAGNETC FELD 6- LOENTZ FOCE EQUATON Lorent force eqution F = Fe + Fm = q ( E + v B ) Exmple 6- An electron hs n initil velocity vo = vo y in uniform mgnetic flux density B = Bo. () how tht

More information

Physics 1B: Review for Final Exam Solutions

Physics 1B: Review for Final Exam Solutions Physics B: eview for Finl Exm s Andrew Forrester June 6, 2008 In this worksheet we review mteril from the following chpters of Young nd Freedmn plus some dditionl concepts): Chpter 3: Periodic Motion Chpter

More information

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero

More information

#6A&B Magnetic Field Mapping

#6A&B Magnetic Field Mapping #6A& Mgnetic Field Mpping Gol y performing this lb experiment, you will: 1. use mgnetic field mesurement technique bsed on Frdy s Lw (see the previous experiment),. study the mgnetic fields generted by

More information

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Higher 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 information

Physics 202, Lecture 13. Today s Topics

Physics 202, Lecture 13. Today s Topics Physics 202, Lecture 13 Tody s Topics Sources of the Mgnetic Field (Ch. 30) Clculting the B field due to currents Biot-Svrt Lw Emples: ring, stright wire Force between prllel wires Ampere s Lw: infinite

More information

Partial Derivatives. Limits. For a single variable function f (x), the limit lim

Partial Derivatives. Limits. For a single variable function f (x), the limit lim Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles

More information

Version 001 Exam 1 shih (57480) 1

Version 001 Exam 1 shih (57480) 1 Version 001 Exm 1 shih 57480) 1 This print-out should hve 6 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. Holt SF 17Rev 1 001 prt 1 of ) 10.0

More information

Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.

Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space. Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)

More information

Homework Assignment #1 Solutions

Homework Assignment #1 Solutions Physics 56 Winter 8 Textook prolems: h. 8: 8., 8.4 Homework Assignment # Solutions 8. A trnsmission line consisting of two concentric circulr cylinders of metl with conductivity σ nd skin depth δ, s shown,

More information

Polynomials and Division Theory

Polynomials and Division Theory Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the

More information

Electricity and Magnetism

Electricity and Magnetism PHY472 Dt Provided: Formul sheet nd physicl constnts Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS & Autumn Semester 2009-2010 ASTRONOMY DEPARTMENT

More information

7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?

7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement? 7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge

More information

Phys102 General Physics II

Phys102 General Physics II Phys1 Generl Physics II pcitnce pcitnce pcitnce definition nd exmples. Dischrge cpcitor irculr prllel plte cpcitior ylindricl cpcitor oncentric sphericl cpcitor Dielectric Sls 1 pcitnce Definition of cpcitnce

More information

Light and Optics Propagation of light Electromagnetic waves (light) in vacuum and matter Reflection and refraction of light Huygens principle

Light and Optics Propagation of light Electromagnetic waves (light) in vacuum and matter Reflection and refraction of light Huygens principle Light nd Optics Propgtion of light Electromgnetic wves (light) in vcuum nd mtter Reflection nd refrction of light Huygens principle Polristion of light Geometric optics Plne nd curved mirrors Thin lenses

More information

Edexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks

Edexcel 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 information

2A1A Vector Algebra and Calculus I

2A1A Vector Algebra and Calculus I Vector Algebr nd Clculus I (23) 2AA 2AA Vector Algebr nd Clculus I Bugs/queries to sjrob@robots.ox.c.uk Michelms 23. The tetrhedron in the figure hs vertices A, B, C, D t positions, b, c, d, respectively.

More information

Physics 241 Exam 1 February 19, 2004

Physics 241 Exam 1 February 19, 2004 Phsics 241 Em 1 Februr 19, 24 One (both sides) 8 1/2 11 crib sheet is llowed. It must be of our own cretion. k = 1 = 9 1 9 N m2 4p 2 2 = 8.85 1-12 N m 2 e =1.62 1-19 c = 2.99792458 1 8 m/s (speed of light)

More information

Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.

Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q. 1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples

More information

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O 1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

More information

REVIEW SHEET FOR PRE-CALCULUS MIDTERM

REVIEW SHEET FOR PRE-CALCULUS MIDTERM . If A, nd B 8, REVIEW SHEET FOR PRE-CALCULUS MIDTERM. For the following figure, wht is the eqution of the line?, write n eqution of the line tht psses through these points.. Given the following lines,

More information

KINEMATICS OF RIGID BODIES

KINEMATICS OF RIGID BODIES KINEMTICS OF RIGID ODIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. Description

More information

Instructor(s): Acosta/Woodard PHYSICS DEPARTMENT PHY 2049, Fall 2015 Midterm 1 September 29, 2015

Instructor(s): Acosta/Woodard PHYSICS DEPARTMENT PHY 2049, Fall 2015 Midterm 1 September 29, 2015 Instructor(s): Acost/Woodrd PHYSICS DEPATMENT PHY 049, Fll 015 Midterm 1 September 9, 015 Nme (print): Signture: On m honor, I hve neither given nor received unuthorized id on this emintion. YOU TEST NUMBE

More information

Mathematics. Area under Curve.

Mathematics. Area under Curve. Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding

More information

Sample Exam 5 - Skip Problems 1-3

Sample Exam 5 - Skip Problems 1-3 Smple Exm 5 - Skip Problems 1-3 Physics 121 Common Exm 2: Fll 2010 Nme (Print): 4 igit I: Section: Honors Code Pledge: As n NJIT student I, pledge to comply with the provisions of the NJIT Acdemic Honor

More information

Review of Gaussian Quadrature method

Review of Gaussian Quadrature method Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge

More information

Final Exam Solutions, MAC 3474 Calculus 3 Honors, Fall 2018

Final Exam Solutions, MAC 3474 Calculus 3 Honors, Fall 2018 Finl xm olutions, MA 3474 lculus 3 Honors, Fll 28. Find the re of the prt of the sddle surfce z xy/ tht lies inside the cylinder x 2 + y 2 2 in the first positive) octnt; is positive constnt. olution:

More information

Physics 9 Fall 2011 Homework 2 - Solutions Friday September 2, 2011

Physics 9 Fall 2011 Homework 2 - Solutions Friday September 2, 2011 Physics 9 Fll 0 Homework - s Fridy September, 0 Mke sure your nme is on your homework, nd plese box your finl nswer. Becuse we will be giving prtil credit, be sure to ttempt ll the problems, even if you

More information

2. VECTORS AND MATRICES IN 3 DIMENSIONS

2. VECTORS AND MATRICES IN 3 DIMENSIONS 2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

More information

Summary of equations chapters 7. To make current flow you have to push on the charges. For most materials:

Summary of equations chapters 7. To make current flow you have to push on the charges. For most materials: Summry of equtions chpters 7. To mke current flow you hve to push on the chrges. For most mterils: J E E [] The resistivity is prmeter tht vries more thn 4 orders of mgnitude between silver (.6E-8 Ohm.m)

More information

KEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a

KEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a KEY CONCEPTS THINGS TO REMEMBER :. The re ounded y the curve y = f(), the -is nd the ordintes t = & = is given y, A = f () d = y d.. If the re is elow the is then A is negtive. The convention is to consider

More information

Phys. 506 Electricity and Magnetism Winter 2004 Prof. G. Raithel Problem Set 1 Total 30 Points. 1. Jackson Points

Phys. 506 Electricity and Magnetism Winter 2004 Prof. G. Raithel Problem Set 1 Total 30 Points. 1. Jackson Points Phys. 56 Electricity nd Mgnetism Winter 4 Prof. G. Rithel Prolem Set Totl 3 Points. Jckson 8. Points : The electric field is the sme s in the -dimensionl electrosttic prolem of two concentric cylinders,

More information

Homework Assignment 5 Solution Set

Homework Assignment 5 Solution Set Homework Assignment 5 Solution Set PHYCS 44 3 Februry, 4 Problem Griffiths 3.8 The first imge chrge gurntees potentil of zero on the surfce. The secon imge chrge won t chnge the contribution to the potentil

More information

Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution

Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: olumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge

More information

EMF Notes 9; Electromagnetic Induction ELECTROMAGNETIC INDUCTION

EMF Notes 9; Electromagnetic Induction ELECTROMAGNETIC INDUCTION EMF Notes 9; Electromgnetic nduction EECTOMAGNETC NDUCTON (Y&F Chpters 3, 3; Ohnin Chpter 3) These notes cover: Motionl emf nd the electric genertor Electromgnetic nduction nd Frdy s w enz s w nduced electric

More information

Physics 121 Sample Common Exam 1 NOTE: ANSWERS ARE ON PAGE 8. Instructions:

Physics 121 Sample Common Exam 1 NOTE: ANSWERS ARE ON PAGE 8. Instructions: Physics 121 Smple Common Exm 1 NOTE: ANSWERS ARE ON PAGE 8 Nme (Print): 4 Digit ID: Section: Instructions: Answer ll questions. uestions 1 through 16 re multiple choice questions worth 5 points ech. You

More information

Chapter E - Problems

Chapter E - Problems Chpter E - Prolems Blinn College - Physics 2426 - Terry Honn Prolem E.1 A wire with dimeter d feeds current to cpcitor. The chrge on the cpcitor vries with time s QHtL = Q 0 sin w t. Wht re the current

More information

Log1 Contest Round 3 Theta Individual. 4 points each 1 What is the sum of the first 5 Fibonacci numbers if the first two are 1, 1?

Log1 Contest Round 3 Theta Individual. 4 points each 1 What is the sum of the first 5 Fibonacci numbers if the first two are 1, 1? 008 009 Log1 Contest Round Thet Individul Nme: points ech 1 Wht is the sum of the first Fiboncci numbers if the first two re 1, 1? If two crds re drwn from stndrd crd deck, wht is the probbility of drwing

More information

Eigen Values and Eigen Vectors of a given matrix

Eigen Values and Eigen Vectors of a given matrix Engineering Mthemtics 0 SUBJECT NAME SUBJECT CODE MATERIAL NAME MATERIAL CODE : Engineering Mthemtics I : 80/MA : Prolem Mteril : JM08AM00 (Scn the ove QR code for the direct downlod of this mteril) Nme

More information

NORMALS. a y a y. Therefore, the slope of the normal is. a y1. b x1. b x. a b. x y a b. x y

NORMALS. a y a y. Therefore, the slope of the normal is. a y1. b x1. b x. a b. x y a b. x y LOCUS 50 Section - 4 NORMALS Consider n ellipse. We need to find the eqution of the norml to this ellipse t given point P on it. In generl, we lso need to find wht condition must e stisfied if m c is to

More information

Test , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes

Test , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes Test 2 8., 8.2, 8.4 (density only), 8.5 (work only), 9., 9.2 nd 9.3 relted test mteril nd mteril from prior clsses Locl to Globl Perspectives Anlyze smll pieces to understnd the big picture. Exmples: numericl

More information

( ) 2. ( ) is the Fourier transform of! ( x). ( ) ( ) ( ) = Ae i kx"#t ( ) = 1 2" ( )"( x,t) PC 3101 Quantum Mechanics Section 1

( ) 2. ( ) is the Fourier transform of! ( x). ( ) ( ) ( ) = Ae i kx#t ( ) = 1 2 ( )( x,t) PC 3101 Quantum Mechanics Section 1 1. 1D Schrödinger Eqution G chpters 3-4. 1.1 the Free Prticle V 0 "( x,t) i = 2 t 2m x,t = Ae i kxt "( x,t) x 2 where = k 2 2m. Normliztion must hppen: 2 x,t = 1 Here, however: " A 2 dx " " As this integrl

More information

Chapter 9 Definite Integrals

Chapter 9 Definite Integrals Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

More information

APPLICATIONS OF DEFINITE INTEGRALS

APPLICATIONS OF DEFINITE INTEGRALS Chpter 6 APPICATIONS OF DEFINITE INTEGRAS OVERVIEW In Chpter 5 we discovered the connection etween Riemnn sums ssocited with prtition P of the finite closed intervl [, ] nd the process of integrtion. We

More information

[ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves

[ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves Gols: 1. To find the re etween two curves Section 6.1 Are of Regions etween two Curves I. Are of Region Between Two Curves A. Grphicl Represention = _ B. Integrl Represention [ ( ) ( )] f x g x dx = C.

More information

MATH 260 Final Exam April 30, 2013

MATH 260 Final Exam April 30, 2013 MATH 60 Finl Exm April 30, 03 Let Mpn,Rq e the spce of n-y-n mtrices with rel entries () We know tht (with the opertions of mtrix ddition nd sclr multipliction), M pn, Rq is vector spce Wht is the dimension

More information

(b) Let S 1 : f(x, y, z) = (x a) 2 + (y b) 2 + (z c) 2 = 1, this is a level set in 3D, hence

(b) Let S 1 : f(x, y, z) = (x a) 2 + (y b) 2 + (z c) 2 = 1, this is a level set in 3D, hence Problem ( points) Find the vector eqution of the line tht joins points on the two lines L : r ( + t) i t j ( + t) k L : r t i + (t ) j ( + t) k nd is perpendiculr to both those lines. Find the set of ll

More information

CONIC SECTIONS. Chapter 11

CONIC SECTIONS. Chapter 11 CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round

More information

Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution

Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: Volumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge

More information

Simple Harmonic Motion I Sem

Simple Harmonic Motion I Sem Simple Hrmonic Motion I Sem Sllus: Differentil eqution of liner SHM. Energ of prticle, potentil energ nd kinetic energ (derivtion), Composition of two rectngulr SHM s hving sme periods, Lissjous figures.

More information