#6A&B Magnetic Field Mapping
|
|
- Richard Young
- 6 years ago
- Views:
Transcription
1 #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 vrious conductor geometries (different cses ech week). Reding Sources of mgnetic fields nd the iot-svrt lw re discussed in Young nd Freedmn, Sec nd 8.5 in the 1 th edition. Overview 3 bsics 1. Current in wire cretes mgnetic field in the vicinity of the wire. A long stright conductor is surrounded by concentric mgnetic field lines, whose mgnitude depends on r, the distnce from the wire. As indicted in Fig. 1(), the direction of is given by right-hnd rule.. The mgnetic field is vector quntity, so t ech point in spce the field hs mgnitude nd direction. As with ny vector, the mgnetic field cn be resolved into three mutully perpendiculr components. 3. We will mesure this field using probe coil. As we lerned from Frdy s Lw, the probe is sensitive to the field component prllel to the xis of the coil. The voltge signl seen on the oscilloscope is proportionl to the mgnitude of this mgnetic field component. (n our experiment n oscillting field is generted by n lternting current (c) signl from function genertor, becuse we cn mesure n oscillting field with this method bsed on Frdy s lw.) Theory A. Genertion of Mgnetic Fields The mgnetic field d produced t displcement r from smll current segment dl is given by the iot-svrt lw: d = μ 0 dl r 4π r 3. The totl mgnet field produced by complete current-crrying circuit is obtined by integrting this expression over the entire circuit. A long stright conductor is surrounded by mgnetic field lines of mgnitude = μ 0 /πr, where is the current nd r is the rdius of the circle, i.e., the distnce from the conductor. As result of the vector product in the iot- Svrt lw, the direction of is given by the right-hnd rule. Figure 1(b) shows the mgnetic field lines produced by circulr current loop, while Fig. 1(c) shows two identicl circulr coils rrnged in Helmholtz configurtion, where the seprtion of the coils is equl to their rdii. The Helmholtz configurtion is distinguished by the uniformity of the mgnetic field in the region between the two coils. The mgnetic fields generted by these nd other geometries will be mesured in this lb. Formule describing some properties of these fields re given in the Appendix. () Stright conductor (b) Circulr current loop (c) Helmholtz coils Fig. 1: Mgnetic field configurtions for three of the circuit geometries studied in this lb. Unlike electric field lines, which strt t positive chrges nd end t negtive chrges, mgnetic field lines must form closed loops. Note then tht most of the field lines indicted in (b) nd (c) re incomplete. 1
2 . Mgnetic Field Detection & Mesurement The fields will be mesured using Frdy probe consisting of coil mounted on non-mgnetic rod. The opertion of the probe relies on Frdy s lw of mgnetic induction, which sttes tht chnging mgnetic field produces n electric field. Specificlly, if the totl mgnetic flux Φ = da through circuit chnges, n electromotive force (emf) E will be produced. This emf is effectively equivlent to potentil difference. Formlly, E = n dφ, (1) dt where the mgnetic flux Φ is obtined by integrting the field over the cross sectionl re A of single coil loop, nd n is the number of loops or turns in the probe coil. According to the vector definition of re, A is directed norml to the coil plnes, i.e., prllel to the xis of the coil. Let this be the x-direction, s shown in Fig.. Thus the integrl Φ = da = x A, where x is the x-component of the mgnetic field, verged over re A. f sufficiently smll, the coil probes region of spce where x is effectively constnt. According to Eq. (1), n emf will only result if the mgnetic flux Φ chnges with time. This cn be ccomplished if the mgnetic field is produced by sinusoidl current, becuse, ccording to the iot-svrt lw the mgnetic field is proportionl to A x the energizing current, i: x = β x i, where the constnt β x depends only on geometry. Fig. : Projection of field f the current is sinusoidl, i.e., i = 0 cos(ωt), then the field nd flux oscillte onto re vector A in sinusoidlly, so tht definition of mgnetic flux. E() t = na d x di = naβ x dt dt = naωβ x 0 sin ωt The induced sinusoidl emf E hs mplitude naωβ x 0 = naω 0 x, reltion tht enbles bsolute, not just reltive, mesurements of the field component 0 x. For mximum experimentl sensitivity to 0 x the number of turns n, ngulr frequency ω, nd current mgnitude 0 should ll be lrge. Apprtus Vrious conductor geometries re vilble for study: solenoid, long stright wire, circulr loop, circulr coil, nd pir of Helmholtz coils, toroid. Use sinusoidl current derived from the function genertor to produce oscillting mgnetic fields. The Frdy probe on your work bench hs the coil xis prllel to the rod. t is therefore sensitive only to the mgnetic field component directed long the rod. Also vilble for your use re probes tht hve the coil xis perpendiculr to the rod xis. Fig. 3. Circuit for ll mesurements to be mde in this lb. R = 10 Ω
3 Mesurement Overview (lso see report sheet) 1. First fmilirize yourself with the pprtus. Wire the circuit bove with R = 10 Ω, nd the solenoid s the source coil. Adjust the function genertor to produce 5 khz sine wve with mximum output mplitude. nspect the voltge cross the 10 Ω resistor to ensure tht it forms good sinusoidl oscilltion. f not, djusting the offset control or reducing the function genertor output mplitude my improve it.. t is prticulrly importnt to keep the unshielded wires wy from the probe coil, especilly with the long stright wire, where the mgnetic field is smll. (Why?) 3. We will spend some time setting up the oscilloscopes s group for these mesurements. We will use verging to minimize the effect of rndom high frequency electronic noise in the mesurements. 4. Fmilirize yourself with the Frdy probe by moving it both inside nd outside the solenoid, nd chnging its orienttion. Observe the emf induced in the probe coil. Develop strtegies for positioning nd orienting the probe. Do this crefully for ech new cse before mking serious mesurements. 5. Next mke mgnetic field mesurements for severl different mgnetic source circuits, s indicted below. n some cses you re sked to mesure nd plot. n other cses you re sked to explore nd then to comment on, or to compre, your observtions. Frdy s lw implies tht, t constnt genertor current nd frequency, the induced emf in the probe coil is directly proportionl to the mgnetic field component long the probe coil xis. Thus, you cn record the pek-to-pek mplitude of the induced emf s mesure of the mgnetic field. 6. Although we try to minimize this effect by signl verging, the trce of the induced emf signl on the oscilloscope screen my look fuzzy becuse of rndom electronic noise, especilly noticeble when the signl is smll. The mesurement of pek-to-pek mplitude includes the rndom noise, but mesurements fr from the source circuit provide bseline for determintion of the emf corresponding to zero field. Reltive Field Mesurements for week 1: 1) Solenoid (r is the distnce from the xis, nd x is the distnce long the solenoid xis, mesured from the geometric center.) Mesure the emf induced in the probe coil by the xil field t the geometric center of the solenoid. Mesure nd plot x vs x, for x > 0. On your grph, indicte () the physicl end of the solenoid, (b) the point t which x flls to 90% of the vlue t the center, nd (c) the point where x is equl to 50% of the vlue t the center. Explore x (x =0) vs r both inside nd outside the coil. Wht do you find? Explore r outside the coil. Where do you find the lrgest vlue of r? Explore r inside the coil. Wht do you find? How does the mgnitude compre with x (x =0)? ) "nfinitely long stright conductor (r is the distnce from the center of the wire.) The induced signl is smll for this cse, but the sensitivity my be improved by incresing the frequency. Mesure the tngentil field vs r nd plot s function of 1/r. This is expected to yield stright line. (Use the vernier cliper or micrometer to mesure the rdius of the wire nd of the probe coil.) Explore the other components of, i.e., the rdil component nd the component prllel to the conductor. Wht do you find? s this wht you expect? Reltive Field Mesurements for week : nstructions for dditionl mesurements of the mgnetic fields produced by single multi-turn coil, nd by Helmholtz coil pir, nd other configurtions will be provided with the report sheet for week. 3
4 Appendix: Mgnetic field formule Note tht in ll these equtions the current does not necessrily hve to be constnt in time. t could, for exmple, be oscillting sinusoidlly, s in the experiment. nfinitely long stright conductor Tngentil field t rdius r from xis: r ()= μ 0 π r. f the current nd mgnetic field vry in time, s in this experiment, we cn write this s: μ ( ) 0 ( t) r, t =. π r For simplicity, we will not explicitly indicte the time dependence. Circulr coil Field t distnce x long xis of coil with rdius nd N turns: x ()= μ 0 N [ ] 3/ x + ( ) x 1 = 0 1+ x 3/ where 0 = x= ( 0)= μ 0 N. Solenoid coil Field long xis of solenoid tht hs rdius, length L, nd N turns: x ()= μ 0 N x + L x L x= ( 0)= L ( x + L /) + ( x L /) + Helmholtz coils μ 0 N L + 4. Field long xis of Helmholtz coil pir, ech with N turns nd rdius. The coils re lso seprted by distnce, nd x is the distnce of the point on the xis, midwy between the two coils: x ()= μ 0 N x 3/ + x x 3/ x + 5 x= ( 0)= μ 0 N 4 4 L / / x x O x circulr coil solenoid Helmholtz coil 4
5 #6A Lbortory Report Sheet Mgnetic Field Mpping - A Nme: Prtner: Lb Section: Dte: Mke mgnetic field mesurements for the circuits, referring to the lb hndout for the detils. Use the pek-to-pek induced emf s mesurement proportionl to the mgnetic field component. Two grphs re required. n ddition, quntittive nd qulittive observtions re required. n ech cse crefully describe exctly wht you re mesuring. A smll sketch of the geometry might help in some cses. Mke sure you mesure the mgnetic field component tht is requested! The mesured mgnetic field component depends on the Frdy probe nd on how you hold it. 1) Solenoid Attch your grph of x vs x, for x > 0, where x is the distnce long the solenoid xis from the center. On your grph, indicte () the physicl end of the solenoid, (b) the point t which x flls to 90% of the vlue t the center, nd (c) the point where x is equl to 50% of the vlue t the center. Comments: Comment on your observtions of x (x =0) vs r both inside nd outside the coil. Where do you find the lrgest vlue of r? (Use sketch if convenient.) Wht do you find for r inside the coil? (Use sketch if convenient.) How does the mgnitude compre with x (x = 0)? How did you mke this comprison? 1
6 ) nfinitely long stright conductor e creful to keep the unshielded wires wy from the probe coil. Mesure the tngentil field vs r nd plot s function of 1/r. R (wire) = R (probe) = r (min) = R (wire) + R (probe) = Trget r (cm) Actul r (cm) r (min) /r (cm -1 ) E (mv) Comment on your grph of s function of 1/r. Wht did you lern bout the other components of, i.e., the rdil component nd the component tht is prllel to the conductor? s this wht you expect? Why were you cutioned to keep the unshielded wires wy from the probe coil?
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 informationPHYSICS ASSIGNMENT-9
MPS/PHY-XII-11/A9 PHYSICS ASSIGNMENT-9 *********************************************************************************************************** 1. A wire kept long the north-south direction is llowed
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 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? Biot-Svrt s lw: Coulomb s lw of Mgnetic field Stedy current: current flow is independent
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 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 non-conducting
More informationProblem 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 informationPotential Formulation Lunch with UCR Engr 12:20 1:00
Wed. Fri., Mon., Tues. Wed. 7.1.3-7.2.2 Emf & Induction 7.2.3-7.2.5 Inductnce nd Energy of 7.3.1-.3.3 Mxwell s Equtions 10.1 -.2.1 Potentil Formultion Lunch with UCR Engr 12:20 1:00 HW10 Generliztion of
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 informationHomework Assignment 9 Solution Set
Homework Assignment 9 Solution Set PHYCS 44 3 Mrch, 4 Problem (Griffiths 77) The mgnitude of the current in the loop is loop = ε induced = Φ B = A B = π = π µ n (µ n) = π µ nk According to Lense s Lw this
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 Engineering-Physics, 845-7737 Print your nme netly: Lst nme: First nme: Sign your nme: Plese
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 informationPhysics 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 informationThe 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 informationPhysics 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 informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
More informationMagnetic forces on a moving charge. EE Lecture 26. Lorentz Force Law and forces on currents. Laws of magnetostatics
Mgnetic forces on moving chrge o fr we ve studied electric forces between chrges t rest, nd the currents tht cn result in conducting medium 1. Mgnetic forces on chrge 2. Lws of mgnetosttics 3. Mgnetic
More informationVersion 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 informationMotion of Electrons in Electric and Magnetic Fields & Measurement of the Charge to Mass Ratio of Electrons
n eperiment of the Electron topic Motion of Electrons in Electric nd Mgnetic Fields & Mesurement of the Chrge to Mss Rtio of Electrons Instructor: 梁生 Office: 7-318 Emil: shling@bjtu.edu.cn Purposes 1.
More informationPOLYPHASE CIRCUITS. Introduction:
POLYPHASE CIRCUITS Introduction: Three-phse systems re commonly used in genertion, trnsmission nd distribution of electric power. Power in three-phse system is constnt rther thn pulsting nd three-phse
More informationProf. 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 informationMeasuring Electron Work Function in Metal
n experiment of the Electron topic Mesuring Electron Work Function in Metl Instructor: 梁生 Office: 7-318 Emil: shling@bjtu.edu.cn Purposes 1. To understnd the concept of electron work function in metl nd
More informationSummary 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 informationKINEMATICS 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 informationHomework 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 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 information( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that
Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we
More informationSample 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 informationLecture 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 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 informationUnique Solutions R. All about Electromagnetism. C h a p t e r. G l a n c e
5. C h p t e r t G l n c e When electric current is pssed through conductor, it produces mgnetic field round it. The first discovery of the connection between electricity nd mgnetism ws mde by H. C. Oersted.
More informationProblem Set 4: Mostly Magnetic
University of Albm Deprtment of Physics nd Astronomy PH 102 / LeClir Summer 2012 nstructions: Problem Set 4: Mostly Mgnetic 1. Answer ll questions below. Show your work for full credit. 2. All problems
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 informationEMF 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 informationVersion 001 HW#6 - Electromagnetic Induction arts (00224) 1 3 T
Version 001 HW#6 - lectromgnetic Induction rts (00224) 1 This print-out should hve 12 questions. Multiple-choice questions my continue on the next column or pge find ll choices efore nswering. AP 1998
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 informationragsdale (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 informationPhysics 202H - Introductory Quantum Physics I Homework #08 - Solutions Fall 2004 Due 5:01 PM, Monday 2004/11/15
Physics H - Introductory Quntum Physics I Homework #8 - Solutions Fll 4 Due 5:1 PM, Mondy 4/11/15 [55 points totl] Journl questions. Briefly shre your thoughts on the following questions: Of the mteril
More informationSimple 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 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 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 2009-2010 ASTRONOMY DEPARTMENT
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
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 cross-section or slice. In this section, we restrict
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
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 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 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 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 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 informationSection 4.8. D v(t j 1 ) t. (4.8.1) j=1
Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions
More informationin 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 informationThe 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 informationPhysics 212. Faraday s Law
Phsics 1 Lecture 17 Frd s Lw Phsics 1 Lecture 17, Slide 1 Motionl EMF Chnge Are of loop Chnge mgnetic field through loop Chnge orienttion of loop reltive to In ech cse the flu of the mgnetic field through
More informationSection 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
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 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 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 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 informationPhysics 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 informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
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 cross-section in plne pssing through
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 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 informationDIFFRACTION OF LIGHT
DIFFRACTION OF LIGHT The phenomenon of bending of light round the edges of obstcles or nrrow slits nd hence its encrochment into the region of geometricl shdow is known s diffrction. P Frunhofer versus
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 informationPhysics 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 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 informationTHREE-DIMENSIONAL KINEMATICS OF RIGID BODIES
THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES 1. TRANSLATION Figure shows rigid body trnslting in three-dimensionl spce. Any two points in the body, such s A nd B, will move long prllel stright lines if
More informationMath 116 Final Exam April 26, 2013
Mth 6 Finl Exm April 26, 23 Nme: EXAM SOLUTIONS Instructor: Section:. Do not open this exm until you re told to do so. 2. This exm hs 5 pges including this cover. There re problems. Note tht the problems
More informationMATH 144: Business Calculus Final Review
MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives
More informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
More information5.7 Improper Integrals
458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the
More informationSources of the Magnetic Field
2.2 This is the Nerest One Hed 937 P U Z Z L E R All three of these commonplce items use mgnetism to store informtion. The cssette cn store more thn n hour of music, the floppy disk cn hold the equivlent
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 informationMath 113 Exam 1-Review
Mth 113 Exm 1-Review September 26, 2016 Exm 1 covers 6.1-7.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between
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 informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
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 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 hlf-spce defined by, with Dirichlet boundry conditions on the plne
More informationinteratomic distance
Dissocition energy of Iodine molecule using constnt devition spectrometer Tbish Qureshi September 2003 Aim: To verify the Hrtmnn Dispersion Formul nd to determine the dissocition energy of I 2 molecule
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationHeat flux and total heat
Het flux nd totl het John McCun Mrch 14, 2017 1 Introduction Yesterdy (if I remember correctly) Ms. Prsd sked me question bout the condition of insulted boundry for the 1D het eqution, nd (bsed on glnce
More informationDuality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.
Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we
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 informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
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 two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero
More informationJim Lambers MAT 169 Fall Semester Lecture 4 Notes
Jim Lmbers MAT 169 Fll Semester 2009-10 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of
More informationPhysics 202, Lecture 10. Basic Circuit Components
Physics 202, Lecture 10 Tody s Topics DC Circuits (Chpter 26) Circuit components Kirchhoff s Rules RC Circuits Bsic Circuit Components Component del ttery, emf Resistor Relistic Bttery (del) wire Cpcitor
More informationResistors. 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 informationMATH 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 informationRiemann is the Mann! (But Lebesgue may besgue to differ.)
Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >
More information2. 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 informationFlow in porous media
Red: Ch 2. nd 2.2 PART 4 Flow in porous medi Drcy s lw Imgine point (A) in column of wter (figure below); the point hs following chrcteristics: () elevtion z (2) pressure p (3) velocity v (4) density ρ
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 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 informationConservation Law. Chapter Goal. 5.2 Theory
Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very
More informationFig. 1. Open-Loop and Closed-Loop Systems with Plant Variations
ME 3600 Control ystems Chrcteristics of Open-Loop nd Closed-Loop ystems Importnt Control ystem Chrcteristics o ensitivity of system response to prmetric vritions cn be reduced o rnsient nd stedy-stte responses
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 informationFEM ANALYSIS OF ROGOWSKI COILS COUPLED WITH BAR CONDUCTORS
XIX IMEKO orld Congress Fundmentl nd Applied Metrology September 6 11, 2009, Lisbon, Portugl FEM ANALYSIS OF ROGOSKI COILS COUPLED ITH BAR CONDUCTORS Mirko Mrrcci, Bernrdo Tellini, Crmine Zppcost University
More information8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.
8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the
More information