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


 Suzan Booth
 1 years ago
 Views:
Transcription
1 Prof. Anchordoui Problems set # 4 Physics 169 Mrch 3, (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 infinitely fr wy. (ii) Four point chrges re fixed t corners of sure centered t origin, s shown in Fig. 1. The length of ech side of sure is. The chrges re locted s follows: is t (, ), is t (, ), 3 is t (, ), nd 6 is t (, ). A fifth prticle tht hs mss m nd chrge is plced t origin nd relesed from rest. Find its speed when it is very fr from origin.. Five identicl point chrges re rrnged in two different mnners s shown in Fig. : in once cse s fcecentered sure, in or s regulr pentgon. Find potentil energy of ech system of chrges, tking zero of potentil energy to be infinitely fr wy. Express your nswer in terms of constnt times energy of two chrges seprted by distnce. 3. Consider system of two chrges shown in Fig. 3. Find electric potentil t n rbitrry point on x xis nd mke plot of electric potentil s function of x/. 4. A point prticle tht hs chrge of 11.1 nc is t origin. (i) Wht is (re) shpes of euipotentil surfces in region round this chrge? (ii) Assuming potentil to be zero t r =, clculte rdii of five surfces tht hve potentils eul to. V, 4. V, 6. V, 8. V nd 1. V, nd sketch m to scle centered on chrge. (iii) Are se surfces eully spced? Explin your nswer. (iv) Estimte electric field strength between 4.V nd 6.V euipotentil surfces by dividing difference between two potentils by difference between two rdii. Compre this estimte to exct vlue t loction midwy between se two surfces. 5. Two coxil conducting cylindricl shells hve eul nd opposite chrges. The inner shell hs chrge nd n outer rdius, nd outer shell hs chrge nd n inner rdius b. The length of ech cylindricl shell is L, nd L is very long compred with b. Find potentil difference, V V b between shells. 6. An electric potentil V (z) is described by function V m 1 z 4 V, z >. m, 1. m < z <. m V (z) = 3 V 3 V m 3 z 3, m < z < 1. m 3 V 3 V m 3 z 3, 1. m < z < m,. m < z < 1. m V m 1 z 4 V, z <. m The grph in Fig. 4 shows vrition of n electric potentil V (z) s function of z. (i) Give electric field vector E for ech of six regions. (ii) Mke plot of zcomponent of electric field, E z, s function of z. Mke sure you lbel xes to indicte numeric mgnitude of field. (iii) Qulittively describe distribution of chrges tht gives rise to this potentil lndscpe nd hence electric fields you clculted. Tht is, where re chrges, wht sign re y, wht shpe re y (plne, slb...)?
2 7. Two conducting, concentric spheres hve rdii nd b. The outer sphere is given chrge Q. Wht is chrge on inner sphere if it is erd?. 8. Consider two nested, sphericl conducting shells. The first hs inner rdius nd outer rdius b. The second hs inner rdius c nd outer rdius d. The system is shown in Fig. 5. In following four situtions, determine totl chrge on ech of fces of conducting spheres (inner nd outer for ech), s well s electric field nd potentil everywhere in spce (s function of distnce r from center of sphericl shells). In ll cses shells begin unchrged, nd chrge is n instntly introduced somewhere. (i) Both shells re not connected to ny or conductors (floting) tht is, ir net chrge will remin fixed. A positive chrge Q is introduced into center of inner sphericl shell. Tke zero of potentil to be t infinity. (ii) The inner shell is not connected to ground (floting) but outer shell is grounded tht is, it is fixed t V = nd hs whtever chrge is necessry on it to mintin this potentil. A negtive chrge Q is introduced into center of inner sphericl shell. (iii) The inner shell is grounded but outer shell is floting. A positive chrge Q is introduced into center of inner sphericl shell. (iv) Finlly, outer shell is grounded nd inner shell is floting. This time positive chrge Q is introduced into region in between two shells. In this cse uestion Wht re E(r) nd V (r)? cnnot be nswered nlyticlly in some regions of spce. In regions where se uestions cn be nswered nlyticlly, give nswers. In regions where y cnnot be nswered nlyticlly, explin why, but try to drw wht you think electric field should look like nd give s much informtion bout potentil s possible. 9. The hydrogen tom in its ground stte cn be modeled s positive point chrge of mgnitude e ( proton) surrounded by negtive chrge distribution tht hs chrge density ( electron) tht vries with distnce from center of proton r s: ρ(r) = ρ e r/ ( result obtined from untum mechnics), where =.53 nm is most probble distnce of electron from proton. (i) Clculte vlue of ρ needed for hydrogen tom to be neutrl. (ii) Clculte electrosttic potentil (reltive to infinity) of this system s function of distnce r from proton. 1. A prticle tht hs mss m nd positive chrge is constrined to move long xxis. At x = L nd x = L re two ring chrges of rdius L. Ech ring is centered on xxis nd lies in plne perpendiculr to it. Ech ring hs totl positive chrge Q uniformly distributed on it. (i) Obtin n expression for potentil V (x) on x xis due to chrge on rings. (ii) Show tht V (x) hs minimum t x =. (iii) Show tht for x << L, potentil pproches form V (x) = V () αx. (iv) Use result of Prt (iii) to derive n expression for ngulr freuency of oscilltion of mss m if it is displced slightly from origin nd relesed. (Assume potentil euls zero t points fr from rings.)
3 ving L. A perugh negred )zk. Q is n if. C. C. C. C 4. m 4. m 69. Eight point chrges, ech of mgnitude, re locted on. C. C. C. C corners of cube of edge s, s shown in Figure P m () Determine x, y, nd z components of resultnt 4. m force exerted by or chrges on chrge locted 3. mj t point A. (b) Wht re mgnitude nd direction of this resultnt force? 69 [SSM] Four point chrges re fixed t corners of sure centered t origin. The length of ech side of sure is. The chrges re locted s follows: is t (, ), is t (, ), 3 is t (, ), nd 6 z is t (, ). A fifth prticle tht hs mss m nd chrge is plced t origin nd relesed from rest. Find its speed when it is very fr from origin. Picture Problem y The digrm shows four point chrges fixed t corners of sure nd fifth chrged prticle Pointtht is relesed from m, rest t Aorigin. We cn use s conservtion of energy to relte s initil potentil energy of prticle to y its kinetic energy when it is t gret s distnce from origin nd x 6 electrosttic potentil t origin to express Ui. 69 nd 7. Figure P3.69 Problems Figure 1: Problem 1. K U Use conservtion of energy to relte 7. Consider chrge distribution shownenergy in Figure initil potentil of P3.69. or, becuse Ki = Uf =, () Show tht mgnitude electric prticle toof its kinetic energyfield whentit K f U i center of ny fce of is cube hs vlue of.18k e /s. t gret distnce from origin: (b) Wht is direction of electric field t center of top fce of cube? U V Express initil potentil energy C is d of ced nduion. prt e. fricd t De point chrges re rrnged in two differe sure, in or s regulr pentgon. Fi e zero of potentil energy to be infinitely fr e energy of two chrges seprted by dis chrged prticle # is 71. Review problem. A negtively of prticle to its chrge nd plced t center of uniformly chrged ring, where electrosttic potentil t origin: ring hs totl positive chrge Q s shown in Exmple 3.8. The prticle, confined to for move long x xis, is Ui to obtin: Substitute Kf nd displced smll distnce x long xis (where x '' ) nd relesed. Show tht prticle oscilltes in simple hrmonic motion with freuency given by f$ 1 ( " ke Q m 3 x 3 i 1 mv V v V m # 1/ 7. A line of chrge with uniform density 35. nc/m lies long line y $ # 15. cm, between points with coordintes x $ nd x $ 4. cm. Find electric field it cretes t origin. 73. Review problem. An electric dipole in uniform electric field is displced slightly from its euilibrium position, s shown in Figure P3.73, where! is smll. The seprtion of chrges is, nd moment of inerti of dipole is I. Assuming dipole is relesed from this Figure : Problem. erposition, we know tht potentil energy of
4 tric Potentil Due to System of Two Chrges MASSACHUSETTS INSTITUTE OF TECHNOLOGY Deprtment of Physics system of two chrges shown in Figure Exm Prctice Problems Prt 1 Solutions Problem 1 Electric Field nd Chrge Distributions from Electric Potentil An electric potentil V ( z ) is described by function )(!V "m 1 )z 4V ; z >. m ; 1. m < z <. m 3 V! # 3 V & $ % "m3 ' ( z3 ; m < z < 1. m V (z) = * 3 V # 3 V & $ % "m3 ' ( z3 ;!1. m < z < m Figure ; 3:! The. m electric < z < dipole!1. mof problem 3.,(V "m 1 )z 4V ; z <!. m Figure Electric dipole The grph below shows vrition of n electric potentil V ( z ) s function of z. lectric potentil t n rbitrry point on x xis nd mke plot. ric potentil cn be found by superposition principle. At poi ve V( x) 1 1 ( ) x 4 x 4 x x expression my be rewritten s ) Give electric field vector E! for ech of six regions in (i) to (vi) below? V( x) =!" V 1 1 V x/ 1 x / 1 Solution: We shll fct tht E!!. Since electric potentil only depends on vrible z, we hve tht z component of electric field is given by Figure 4: Problem 6. dv
5 Figure 5: The Frdy cge of problem 8.
ragsdale (zdr82) HW2 ditmire (58335) 1
rgsdle (zdr82) HW2 ditmire (58335) This printout should hve 22 questions. Multiplechoice questions my continue on the next column or pge find ll choices before nswering. 00 0.0 points A chrge of 8. µc
More informationPhysics 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 informationLecture 13  Linking E, ϕ, and ρ
Lecture 13  Linking E, ϕ, nd ρ A Puzzle... InnerSurfce Chrge Density A positive point chrge q is locted offcenter inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on
More informationPhysics 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 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 informationProblems 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 information200 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 informationPhysics 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 informationExam 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 informationwest (mrw3223) HW 24 lyle (16001) 1
west (mrw3223) HW 24 lyle (16001) 1 This printout should hve 30 questions. Multiplechoice questions my continue on the next column or pge find ll choices before nswering. Reding ssignment: Hecht, sections
More informationPhysics 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 informationPhysics 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 = 8.85 112 N m 2 e =1.62 119 c = 2.99792458 1 8 m/s (speed of light)
More informationPhysics 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 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 nonconducting
More informationCandidates 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 informationCAPACITORS 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 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 informationPhysics 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 informationQuantum Mechanics Qualifying Exam  August 2016 Notes and Instructions
Quntum Mechnics Qulifying Exm  August 016 Notes nd Instructions There re 6 problems. Attempt them ll s prtil credit will be given. Write on only one side of the pper for your solutions. Write your lis
More informationVersion 001 Exam 1 shih (57480) 1
Version 001 Exm 1 shih 57480) 1 This printout should hve 6 questions. Multiplechoice 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 informationSample Exam 5  Skip Problems 13
Smple Exm 5  Skip Problems 13 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 informationJURONG JUNIOR COLLEGE
JURONG JUNIOR COLLEGE 2010 JC1 H1 8866 hysics utoril : Dynmics Lerning Outcomes Subtopic utoril Questions Newton's lws of motion 1 1 st Lw, b, e f 2 nd Lw, including drwing FBDs nd solving problems by
More informationThis 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 information5.2 Volumes: Disks and Washers
4 pplictions of definite integrls 5. Volumes: Disks nd Wshers In the previous section, we computed volumes of solids for which we could determine the re of crosssection or slice. In this section, we restrict
More informationElectric Potential. Concepts and Principles. An Alternative Approach. A Gravitational Analogy
. Electric Potentil Concepts nd Principles An Alterntive Approch The electric field surrounding electric chrges nd the mgnetic field surrounding moving electric chrges cn both be conceptulized s informtion
More informationIMPORTANT. Read these directions carefully:
Physics 208: Electricity nd Mgnetism Finl Exm, Secs. 506 510. 7 My. 2004 Instructor: Dr. George R. Welch, 415 EngineeringPhysics, 8457737 Print your nme netly: Lst nme: First nme: Sign your nme: Plese
More informationInClass Problems 2 and 3: Projectile Motion Solutions. InClass Problem 2: Throwing a Stone Down a Hill
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Deprtment of Physics Physics 8T Fll Term 4 InClss Problems nd 3: Projectile Motion Solutions We would like ech group to pply the problem solving strtegy with the
More informationInstructor(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 informationPhysics 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 informationJackson 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 twodimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero
More informationPhys 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 informationReading 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 informationConducting 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 informationAP Physics C: Electricity & Magnetism 1999 FreeResponse Questions
AP Physics C: Electricity & Mgnetism 1999 Freeesponse Questions The mterils included in these files re intended for noncommercil use by AP techers for course nd exm preprtion; permission for ny other
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 informationName Solutions to Test 3 November 8, 2017
Nme Solutions to Test 3 November 8, 07 This test consists of three prts. Plese note tht in prts II nd III, you cn skip one question of those offered. Some possibly useful formuls cn be found below. Brrier
More information#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 informationAPPLICATIONS OF THE DEFINITE INTEGRAL
APPLICATIONS OF THE DEFINITE INTEGRAL. Volume: Slicing, disks nd wshers.. Volumes by Slicing. Suppose solid object hs boundries extending from x =, to x = b, nd tht its crosssection in plne pssing through
More informationTheoretische 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 FriedrichAlexnderUniversität ErlngenNürnberg Theoretische
More informationHomework 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 informationMethod 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 informationThe Velocity Factor of an Insulated TwoWire Transmission Line
The Velocity Fctor of n Insulted TwoWire 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 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 informationToday 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 informationMath 120 Answers for Homework 13
Mth 12 Answers for Homework 13 1. In this problem we will use the fct tht if m f(x M on n intervl [, b] (nd if f is integrble on [, b] then (* m(b f dx M(b. ( The function f(x = 1 + x 3 is n incresing
More informationChapter 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 information13.4 Work done by Constant Forces
13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push
More informationDIRECT CURRENT CIRCUITS
DRECT CURRENT CUTS ELECTRC POWER Consider the circuit shown in the Figure where bttery is connected to resistor R. A positive chrge dq will gin potentil energy s it moves from point to point b through
More informationTest , 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 information7.1 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 informationDensity of Energy Stored in the Electric Field
Density of Energy Stored in the Electric Field Deprtment of Physics, Cornell University c Tomás A. Aris October 14, 01 Figure 1: Digrm of Crtesin vortices from René Descrtes Principi philosophie, published
More informationMathematics for Physicists and Astronomers
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 20092010 ASTRONOMY DEPARTMENT
More informationWeek 10: Line Integrals
Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.
More informationJackson 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 hlfspce defined by, with Dirichlet boundry conditions on the plne
More informationElectromagnetism 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 informationThat 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, 5:006:00 PM Test rooms: Instructor Sections Room Dr. Hle F, H 104 Physics Dr. Kurter
More informationProblem 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
Prolem Consider disc of chrge density r r nd rdius R tht lies within the xyplne. 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,
More informationSummary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
More informationLecture 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 informationThe Wave Equation I. MA 436 Kurt Bryan
1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string
More informationMath 0230 Calculus 2 Lectures
Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two
More informationUniversity of Alabama Department of Physics and Astronomy. PH126: Exam 1
University of Albm Deprtment of Physics nd Astronomy PH 16 LeClir Fll 011 Instructions: PH16: Exm 1 1. Answer four of the five questions below. All problems hve equl weight.. You must show your work for
More informationPhys 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 informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More informationMathematics of Motion II Projectiles
Chmp+ Fll 2001 Dn Stump 1 Mthemtics of Motion II Projectiles Tble of vribles t time v velocity, v 0 initil velocity ccelertion D distnce x position coordinte, x 0 initil position x horizontl coordinte
More informationSolutions to Physics: Principles with Applications, 5/E, Giancoli Chapter 16 CHAPTER 16
CHAPTER 16 1. The number of electrons is N = Q/e = ( 30.0 10 6 C)/( 1.60 10 19 C/electrons) = 1.88 10 14 electrons.. The mgnitude of the Coulomb force is Q /r. If we divide the epressions for the two forces,
More informationHIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time)
HIGHER SCHOOL CERTIFICATE EXAMINATION 999 MATHEMATICS UNIT (ADDITIONAL) Time llowed Three hours (Plus 5 minutes reding time) DIRECTIONS TO CANDIDATES Attempt ALL questions ALL questions re of equl vlue
More informationchapter Figure 24.1 Field lines representing a uniform electric field penetrating a plane of area A perpendicular to the field. 24.
chpter 24 Guss s Lw 24.1 lectric Flux 24.2 Guss s Lw 24.3 Appliction of Guss s Lw to Vrious Chrge Distributions 24.4 Conductors in lectrosttic quilibrium In Chpter 23, we showed how to clculte the electric
More informationProblem 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 informationUniversity of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10
University of Wshington Deprtment of Chemistry Chemistry 45 Winter Qurter Homework Assignment 4; Due t 5p.m. on // We lerned tht the Hmiltonin for the quntized hrmonic oscilltor is ˆ d κ H. You cn obtin
More informationMath 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions
Mth 1102: Clculus I (Mth/Sci mjors) MWF 3pm, Fulton Hll 230 Homework 2 solutions Plese write netly, nd show ll work. Cution: An nswer with no work is wrong! Do the following problems from Chpter III: 6,
More informationMath 113 Exam 2 Practice
Mth 3 Exm Prctice Februry 8, 03 Exm will cover 7.4, 7.5, 7.7, 7.8, 8.3 nd 8.5. Plese note tht integrtion skills lerned in erlier sections will still be needed for the mteril in 7.5, 7.8 nd chpter 8. This
More information1 Which of the following summarises the change in wave characteristics on going from infrared to ultraviolet in the electromagnetic spectrum?
Which of the following summrises the chnge in wve chrcteristics on going from infrred to ultrviolet in the electromgnetic spectrum? frequency speed (in vcuum) decreses decreses decreses remins constnt
More informationStrategy: Use the Gibbs phase rule (Equation 5.3). How many components are present?
University Chemistry Quiz 4 2014/12/11 1. (5%) Wht is the dimensionlity of the threephse coexistence region in mixture of Al, Ni, nd Cu? Wht type of geometricl region dose this define? Strtegy: Use the
More informationThe area under the graph of f and above the xaxis 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 xxis etween nd is denoted y f(x) dx nd clled the
More informationBME 207 Introduction to Biomechanics Spring 2018
April 6, 28 UNIVERSITY O RHODE ISAND Deprtment of Electricl, Computer nd Biomedicl Engineering BME 27 Introduction to Biomechnics Spring 28 Homework 8 Prolem 14.6 in the textook. In ddition to prts e,
More informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
More informationEnergy creation in a moving solenoid? Abstract
Energy cretion in moving solenoid? Nelson R. F. Brg nd Rnieri V. Nery Instituto de Físic, Universidde Federl do Rio de Jneiro, Cix Postl 68528, RJ 21941972 Brzil Abstrct The electromgnetic energy U em
More information7.6 The Use of Definite Integrals in Physics and Engineering
Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationElectricity 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 20092010 ASTRONOMY DEPARTMENT
More informationThe Properties of Stars
10/11/010 The Properties of Strs sses Using Newton s Lw of Grvity to Determine the ss of Celestil ody ny two prticles in the universe ttrct ech other with force tht is directly proportionl to the product
More informationF is on a moving charged particle. F = 0, if B v. (sin " = 0)
F is on moving chrged prticle. Chpter 29 Mgnetic Fields Ech mgnet hs two poles, north pole nd south pole, regrdless the size nd shpe of the mgnet. Like poles repel ech other, unlike poles ttrct ech other.
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationChapter 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? BiotSvrt s lw: Coulomb s lw of Mgnetic field Stedy current: current flow is independent
More informationl 2 p2 n 4n 2, the total surface area of the
Week 6 Lectures Sections 7.5, 7.6 Section 7.5: Surfce re of Revolution Surfce re of Cone: Let C be circle of rdius r. Let P n be n nsided regulr polygon of perimeter p n with vertices on C. Form cone
More informationKey for Chem 130 Second Exam
Nme Key for Chem 130 Second Exm On the following pges you will find questions tht cover the structure of molecules, ions, nd solids, nd the different models we use to explin the nture of chemicl bonding.
More informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More informationHung 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 informationPhysics Honors. Final Exam Review Free Response Problems
Physics Honors inl Exm Review ree Response Problems m t m h 1. A 40 kg mss is pulled cross frictionless tble by string which goes over the pulley nd is connected to 20 kg mss.. Drw free body digrm, indicting
More informationSpace Curves. Recall the parametric equations of a curve in xyplane and compare them with parametric equations of a curve in space.
Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xyplne 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 informationMath 32B Discussion Session Session 7 Notes August 28, 2018
Mth 32B iscussion ession ession 7 Notes August 28, 28 In tody s discussion we ll tlk bout surfce integrls both of sclr functions nd of vector fields nd we ll try to relte these to the mny other integrls
More informationPractice Problem Set 3
Prctice Problem Set 3 #1. A dipole nd chrge A dipole with chrges ±q nd seprtion 2 is locted distnce from point chrge Q, oriented s shown in Figure 20.32 of the tetbook (reproduced here for convenience.
More informationReference. 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 informationREVIEW SHEET FOR PRECALCULUS MIDTERM
. If A, nd B 8, REVIEW SHEET FOR PRECALCULUS 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 information7.1 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 informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More information