Week 11  Inductance


 Barnard Ellis
 5 years ago
 Views:
Transcription
1 Week  Inductnce November 6, 202 Exercise.: Discussion Questions ) A trnsformer consists bsiclly of two coils in close proximity but not in electricl contct. A current in one coil mgneticlly induces n emf in the second coil, with properties tht cn be controlled by djusting the geometry of the two coils. Such device will work only with lternting current, however, nd not with direct current. Explin. A trnsformer only works with lternting current becuse it s dependent upon the phenomenon of mutul inductnce. Here the emf in the second coil due to the first is given by ε 2 = M di dt, () nd thus, for there to be n emf, it hs to be chnging current in the first coil. How much emf for given current is determined by the mutul inductnce M, which is only reflection of the geometry of the coils. b) Suppose there is stedy current in n inductor. If you ttemt to reduce the current to zero instntneously by quickly opening switch, n rc cn pper t the switch contcts. Why? Is it physiclly possible to stop the current instntneously? Explin. The rc is formed becuse of the self inductnce in the circuit. When you open the switch the rte of chnge of the current become extremely high nd therefore the emf generted become enormous. This emf forces the current cross the cross the gp between the switch. Mthemticlly this is seen with the eqution ε = L di dt (2) where di/dt becomes very lrge nd negtive s you open the switch, nd consequently ε becomes huges nd forces the current cross the gp. Since ll circuits hve some self conductnce it is not possible to stop ny current instntneously. An electric rc is luminous dischrge of current tht is formed when current jumps gp in circuit between two electrodes.
2 Exercise.2: Inductnce of Solenoid A long, stright solenoid hs N turns, uniform crosssectionl re A, nd length l. Show tht the inductnce of this solenoid is given by the eqution L = µ 0 AN 2 /l. Assume tht the mgnetic field is uniform inside the solenoid nd zero outside. (Your nswer is pproximte becuse B is ctully smller t the ends thn t the center. For this reson your nswer is ctully n upper limit on the inductnce.) For the cse of n infinitely long solenoid the mgnetic field is is B = µ 0 ni where n is the number of turns per unit length. Now we use this field to pproximte the inductnce of finite solenoid. A finte solenoid hs finite length l nd finite number of turns N so we cn write B = µ 0 ni = µ 0NI. (3) l For self inductnce the flux trough the re enclosed by the turns of wire in the solenoid is φ B = B(NA) = µ 0AN 2 I (4) l nd here we see the clim tht the flux φ B is lwys proportionl to the current I. The proportionlity constnt is the self inductnce L L = µ 0AN 2. (5) l Figure Exercise.3: A RLCircuit In the circuit shown in figure, E = 50.0 V, R = 40.0 Ω,R 2 = 30.0 Ω nd L = 0.70 H. Switch S is closed t t = 0. Just fter the switch is closed.. 2
3 ) Wht is the potentil difference v b cross the resistor R? Just fter the switch is closed the current trough the inductor is zero becuse it hs nt hd ny time to increse (Recll how the current grows in simple RLcircuit). Therefore by pplying the loop rule to the upper loop E v b = 0 v b = E. (6) v b = E. (7) b) Which point, or b, is t higher potentil. As the current trvels trough R the voltge drops by IR, therefore b is t lower potentil thn. b is t the lower potentil. c) Wht is the potentil difference v cd cross the inductor L? There is no current trough R 2 so by the loop rule E i 2 R 2 v cd = E v cd = 0 v cd = E. (8) v cd = E. (9) d) Which point, c or d, is t higher potentil? The voltge increses when we go from b to trough the emf, so it must therefore drop s we go trough the inductor from c to d. Thus c is t the higher potentil. 3
4 c is t the higher potentil. The switch is left closed long time nd then opened. Just fter the switch is opened... Repet () to (d). () Since the switch hs been open for long time, i 2 is not chnging nymore (di 2 /dt = 0), so v cd = 0. Therefore by the loop rule E = v R2 = i 2 R 2 i 2 = E R 2. (0) (b) Now when the switch suddenly opens the current trough the inductor hs not hd time to chnge so the current trough the lower prt of the circuit (with S open) is i 2. This current trvels from b to, so b is therefore t higher potentil. i 2 = E R 2 () is t the higher potentil. Exercise.4: An Electromgnetic Cr Alrm Your ltes invention is cr lrm tht produces sound prticulrly nnoying frequency of 3500 Hz. To do this, the crlrm circuitry must produce n lternting electric current of the sme frequency. Tht s why your design includes n inductor nd cpcitor in series. The mximum voltge cross the cpcitor is to be 2.0 V (the sme voltge s tht of the cr bttery). To produce sufficiently loud sound, the cpcitor must store J of energy. Wht vlues of cpcitnce nd inductnce should you chose for your crlrm circuit? Here one is fter the reltion between frequency of LCcircuit nd it s cpcitnce nd inductne ω = /LC. If your not very good t remembering fromuls I would recommend remembering the method for the result. The volge round the loop hs to be zero, for cpcitor nd n inductor of respective voltges q/c nd Ldi/dt this mens tht L di dt + q C = 0. (2) Now the chrge on the cpcitor is relted to the current by I = dq/dt, therefore in terms of q we cn write d 2 q dt 2 + q = 0. (3) LC 4
5 A good guess for this differentil eqution is q(t) = Q cos (ωt + φ). Substitution yields ( Q ω 2 cos (ωt + φ) + ) cos (ωt + φ) = 0, (4) LC our guess is only solution ω = LC which is our desired result. Now for our crlrm frequency; ω is the ngulr frequency nd this is relted to the ctul frequency ω = 2πf. Therefore we get f = ω 2π = 2π LC. (5) We hve to find the pproprite vlues of C nd L. We know the mximum voltge cross the cpcitor s well s it s energy. Therefore we cn use the reltion for the energy of cpcitor U E = Q 2 /2C = V 2 C/2 to find it s cpcitnce. We get. C = 2U E V 2 = (2.0) 2 F = 222 µf (6) Now we cn determine the necessry inductnce by our eqution for the frequency. Rerrnging we get L = ( ) 2 ( 2πf C = 2π 3500 ) 2 H = 9.3 µh. (7) ( L = 2π 3500 ) 2 H = 9.3 µh. (8) Exercise.5: Solr Mgnetic Energy Mgnetic fields within sunspot cn be s strong s 0.4 T. (By comprison, the erth s mgnetic field is bout /0000 s strong.) Sunspots cn be s lrge s km in rdius. The mteril in sunspot hs density of bout kg/m 3. Assume µ for the sunspot mteril to be µ 0. If 00% of the mgneticfield energy stored in sunspot could be used to eject the sunspot s mteril wy from the sun s surfce, t wht speed would tht mteril be ejected? Compre to the sun s escpe speed, which is bout m/s. Hint: Clculte the kinetic energ the mgnetic field could supply to m 3 if sunspot mteril. Under the ssumption tht the permebility is µ 0 the mgnetic field energy density is given by u B = B2 (9) 5
6 nd therefore energy in m 3 is U = B2 m 3. (20) Now if this energy is 00% converted to kinetic energy of the sunspot mteril, we cn sett up conservtion of energy eqution for m 3 of this mteril. When ll the energy is converted to kinetic energy we hve or equivlently B 2 m 3 = 2 mv2 (2) v = B µ0 m. (22) m 3 of sunspot mteril hve mss m = kg/m 3 m 3. We then get velocity v = B µ0 m = 0.4 m/s = m/s. (23) 4π This is only 3.4% of the escpe velocity of the sun, so the mgnetic field energy lone is not enough cuse coronl mss ejections which is mssive burst of prticles being ejected into spce. It remins mystery exctly how the prticles get tht much energy. 2 v == m/s. (24) This is only 3.4% of the escpe velocity of the sun, so the mgnetic field energy lone is not enough cuse coronl mss ejections which is mssive burst of prticles being ejected into spce. It remins mystery exctly how the prticles get tht much energy. 3 Exercise.6: Inductnce of Coxil Cble A smll solid conductor of with rdius is supported by insulting, nonmgnetic disks on the xis of thinwlled tube with inner rdius b. The inner nd outer conductors crry equl currents i in opposite directions. ) Use Ampere s lw to find the mgnetic field t n point in the volume between the two conductors. 2 Solr flre Wikipedi. 3 Solr flre Wikipedi. 6
7 Figure 2 We construct circulr integrtion pth with rdius r centered t the xis of the cble. Then by Ampere s lw B dl = B2πr = µ 0 i (25) or equivlently B = µ 0i 2πr. (26) B = µ 0i 2πr. (27) b) Write the expression for the flux dφ B through nrrow strip of length l prllel to the xis, of width dr, t sitnce r from the xis of the cble nd lying in plne contining the xis. The nrrow strip is shown in figure 3. The flux trough this strip will be the mgnetic field (which is perpendiculr to the strip t every point) times the re of the strip. I.e. it will be dφ B = B (ldr) = l µ 0i dr. (28) 2πr dφ B = B (ldr) = l µ 0i dr. (29) 2πr 7
8 Figure 3 c) Integrte your expression from prt (b) over the volue between the two conductors to find the totl flux produced by the current i in the centrl conductor. To clculte the entire flux we hve to integrte this expression from to b. φ B = b dφ = l µ 0i 2π b dr r = l µ ( ) 0i b 2π ln. (30) φ B = b dφ = l µ ( ) 0i b 2π ln. (3) d) Show tht the inductnce of length l of the cble is L = l µ ( ) 0 b 2π ln. (32) The inductnce is the constnt of proportionlity between φ B nd i. Looking t our expression for the flux we see tht φ B = l µ ( ) 0 b 2π ln i = Li (33) 8
9 L = l µ ( ) 0 b 2π ln. (34) e) Use eqution 32 to clculte the energy stored in the mgnetic field for lentgh l of the cble. We know tht the energy stored in the mgnetic field of n inductor is 2 Li2. Therefore we get U = l µ 0i 2 ( ) b 4π ln. (35) U = l µ 0i 2 ( ) b 4π ln. (36) f) Verify tht this is the correct expression energy by lso clculting th e totl stored energy through the energy density formul u B = B2. Hint: Clculte the energy of thin, cylindricl, shell of length l nd rdius r plced in between the two conductors. We found the mgnetic field in between the two conductors in (). The result ws The corresponding energy density is therefore given by B = µ 0i 2πr. (37) u B = B2 = ( µ0i ) 2 2πr = µ 0i 2 8π 2 r 2. (38) Now to find the energy of thin, cylindricl shell of length l nd rdius r we must multiply the energy density with the volume of the cylidricl shell. The volume of such shell is given by dv = l2πrdr nd therefore du = u B dv = µ 0i 2 8π 2 r 2 2πrdr = l µ 0i 2 dr. (39) 4πr such tht the totl energy contined in the mgnetic field within the conductors is l µ 0i 2 4π b dr r = l µ 0i 2 4π ln b (40) which indeed grees with the expression from (e). 9
Lesson 13 Inductance, Magnetic energy /force /torque
Lesson 3 nductnce, Mgnetic energy /force /torque 楊 尚 達 ShngD Yng nstitute of Photonics Technologies Deprtment of Electricl Engineering Ntionl Tsing Hu Uniersity, Tiwn Outline nductnce Mgnetic energy Mgnetic
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 informationAnswer, Key Homework 10 David McIntyre 1
Answer, Key Homework 10 Dvid McIntyre 1 This printout should hve 22 questions, check tht it is complete. Multiplechoice questions my continue on the next column or pge: find ll choices efore mking your
More information, and the number of electrons is 19. e e 1.60 10 C. The negatively charged electrons move in the direction opposite to the conventional current flow.
Prolem 1. f current of 80.0 ma exists in metl wire, how mny electrons flow pst given cross section of the wire in 10.0 min? Sketch the directions of the current nd the electrons motion. Solution: The chrge
More informationRotating DC Motors Part II
Rotting Motors rt II II.1 Motor Equivlent Circuit The next step in our consiertion of motors is to evelop n equivlent circuit which cn be use to better unerstn motor opertion. The rmtures in rel motors
More informationPHY 222 Lab 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS
PHY 222 Lb 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS Nme: Prtners: INTRODUCTION Before coming to lb, plese red this pcket nd do the prelb on pge 13 of this hndout. From previous experiments,
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More informationExperiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
More informationReview Problems for the Final of Math 121, Fall 2014
Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since
More informationAAPT UNITED STATES PHYSICS TEAM AIP 2010
2010 F = m Exm 1 AAPT UNITED STATES PHYSICS TEAM AIP 2010 Enti non multiplicnd sunt preter necessittem 2010 F = m Contest 25 QUESTIONS  75 MINUTES INSTRUCTIONS DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments  they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationAREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
More information1. In the Bohr model, compare the magnitudes of the electron s kinetic and potential energies in orbit. What does this imply?
Assignment 3: Bohr s model nd lser fundmentls 1. In the Bohr model, compre the mgnitudes of the electron s kinetic nd potentil energies in orit. Wht does this imply? When n electron moves in n orit, the
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationPhysics 2102 Lecture 2. Physics 2102
Physics 10 Jonthn Dowling Physics 10 Lecture Electric Fields ChrlesAugustin de Coulomb (17361806) Jnury 17, 07 Version: 1/17/07 Wht re we going to lern? A rod mp Electric chrge Electric force on other
More informationVersion 001 Summer Review #03 tubman (IBII20142015) 1
Version 001 Summer Reiew #03 tubmn (IBII20142015) 1 This printout should he 35 questions. Multiplechoice questions my continue on the next column or pge find ll choices before nswering. Concept 20 P03
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationPhysics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nmwide region t x
More information4. DC MOTORS. Understand the basic principles of operation of a DC motor. Understand the operation and basic characteristics of simple DC motors.
4. DC MOTORS Almost every mechnicl movement tht we see round us is ccomplished by n electric motor. Electric mchines re mens o converting energy. Motors tke electricl energy nd produce mechnicl energy.
More information2 DIODE CLIPPING and CLAMPING CIRCUITS
2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of
More informationt 3 t 4 Part A: Multiple Choice Canadian Association of Physicists 1999 Prize Exam
Cndin Assocition of Physicists 1999 Prize Exm This is three hour exm. Ntionl rnking nd prizes will be bsed on student s performnce on both sections A nd B of the exm. However, performnce on the multiple
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationCOMPONENTS: COMBINED LOADING
LECTURE COMPONENTS: COMBINED LOADING Third Edition A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering 24 Chpter 8.4 by Dr. Ibrhim A. Asskkf SPRING 2003 ENES 220 Mechnics of
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationSolution Derivations for Capa #11
Solution Derivations for Capa #11 Caution: The symbol E is used interchangeably for energy and EMF. 1) DATA: V b = 5.0 V, = 155 Ω, L = 8.400 10 2 H. In the diagram above, what is the voltage across the
More informationIncreasing Q of Waveguide PulseCompression Cavities
Circuit nd Electromgnetic System Design Notes Note 61 3 July 009 Incresing Q of Wveguide PulseCompression Cvities Crl E. Bum University of New Mexico Deprtment of Electricl nd Computer Engineering Albuquerque
More informationPROBLEMS 13  APPLICATIONS OF DERIVATIVES Page 1
PROBLEMS  APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.
More informationApplications to Physics and Engineering
Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics
More information15.6. The mean value and the rootmeansquare value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style
The men vlue nd the rootmensqure vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives
More information9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is soclled becuse when the sclr product of two vectors
More informationCypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period:
Nme: SOLUTIONS Dte: Period: Directions: Solve ny 5 problems. You my ttempt dditionl problems for extr credit. 1. Two blocks re sliding to the right cross horizontl surfce, s the drwing shows. In Cse A
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationAll pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 12015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More informationwww.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)
www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input
More informationScalar and Vector Quantities. A scalar is a quantity having only magnitude (and possibly phase). LECTURE 2a: VECTOR ANALYSIS Vector Algebra
Sclr nd Vector Quntities : VECTO NLYSIS Vector lgebr sclr is quntit hving onl mgnitude (nd possibl phse). Emples: voltge, current, chrge, energ, temperture vector is quntit hving direction in ddition to
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationReview guide for the final exam in Math 233
Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered
More informationBinary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse rel number to its binry representtion,. convert binry number to n equivlent bse number. In everydy
More informationPure C4. Revision Notes
Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd
More informationSection 54 Trigonometric Functions
5 Trigonometric Functions Section 5 Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationHarvard College. Math 21a: Multivariable Calculus Formula and Theorem Review
Hrvrd College Mth 21: Multivrible Clculus Formul nd Theorem Review Tommy McWillim, 13 tmcwillim@college.hrvrd.edu December 15, 2009 1 Contents Tble of Contents 4 9 Vectors nd the Geometry of Spce 5 9.1
More informationEuler Euler Everywhere Using the EulerLagrange Equation to Solve Calculus of Variation Problems
Euler Euler Everywhere Using the EulerLgrnge Eqution to Solve Clculus of Vrition Problems Jenine Smllwood Principles of Anlysis Professor Flschk My 12, 1998 1 1. Introduction Clculus of vritions is brnch
More informationChapter 30 Inductance
Chapter 30 Inductance  Mutual Inductance  SelfInductance and Inductors  MagneticField Energy  The R Circuit  The C Circuit  The RC Series Circuit . Mutual Inductance  A changing current in
More informationHomework #11 20311721 Physics 2 for Students of Mechanical Engineering
Homework #11 20311721 Physics 2 for Students of Mechanical Engineering 2. A circular coil has a 10.3 cm radius and consists of 34 closely wound turns of wire. An externally produced magnetic field of
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine soclled volumes of
More informationDistributions. (corresponding to the cumulative distribution function for the discrete case).
Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive
More informationMATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
More informationPHY114 S11 Term Exam 3
PHY4 S Term Exam S. G. Rajeev Mar 2 20 2:0 pm to :45 pm PLEASE write your workshop number and your workshop leader s name at the top of your book, so that you can collect your graded exams at the workshop.
More informationSOLUTIONS TO CONCEPTS CHAPTER 5
1. m k S 10m Let, ccelertion, Initil velocity u 0. S ut + 1/ t 10 ½ ( ) 10 5 m/s orce: m 5 10N (ns) 40000. u 40 km/hr 11.11 m/s. 3600 m 000 k ; v 0 ; s 4m v u ccelertion s SOLUIONS O CONCEPS CHPE 5 0 11.11
More informationDlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report
DlNBVRGH + + THE CITY OF EDINBURGH COUNCIL Sickness Absence Monitoring Report Executive of the Council 8fh My 4 I.I...3 Purpose of report This report quntifies the mount of working time lost s result of
More information** Dpt. Chemical Engineering, Kasetsart University, Bangkok 10900, Thailand
Modelling nd Simultion of hemicl Processes in Multi Pulse TP Experiment P. Phnwdee* S.O. Shekhtmn +. Jrungmnorom** J.T. Gleves ++ * Dpt. hemicl Engineering, Ksetsrt University, Bngkok 10900, Thilnd + Dpt.hemicl
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationWarmup for Differential Calculus
Summer Assignment Wrmup for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
More informationTITLE THE PRINCIPLES OF COINTAP METHOD OF NONDESTRUCTIVE TESTING
TITLE THE PRINCIPLES OF COINTAP METHOD OF NONDESTRUCTIVE TESTING Sung Joon Kim*, DongChul Che Kore Aerospce Reserch Institute, 45 EoeunDong, YouseongGu, Dejeon, 35333, Kore Phone : 824286231 FAX
More informationLecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
More information12. The current in an inductor is changing at the rate of 100 A/s, and the inductor emf is 40 V. What is its selfinductance?
12. The current in an inductor is changing at the rate of 100 A/s, and the inductor emf is 40 V. What is its selfinductance? From Equation 325, L = E=(dI =dt) = 40 V=(100 A/s) = 0.4 H. 15. A cardboard
More informationTreatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the ttest is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the ttest cn be used to compre the mens of only
More informationTutorial on How to Create Electric Machine Models
PSIM Sotwre Tutoril on How to Crete Electric Mchine Models Powersi Inc. Septber 2009 www.powersitech.co Tutoril on Creting Electric Mchine Models Users cn crete electric chine odels using the bsic unction
More information6 Energy Methods And The Energy of Waves MATH 22C
6 Energy Methods And The Energy of Wves MATH 22C. Conservtion of Energy We discuss the principle of conservtion of energy for ODE s, derive the energy ssocited with the hrmonic oscilltor, nd then use this
More informationEcon 4721 Money and Banking Problem Set 2 Answer Key
Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in
More informationRadius of the Earth  Radii Used in Geodesy James R. Clynch February 2006
dius of the Erth  dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.
More informationCOMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT
COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE Skndz, Stockholm ABSTRACT Three methods for fitting multiplictive models to observed, crossclssified
More informationProject 6 Aircraft static stability and control
Project 6 Aircrft sttic stbility nd control The min objective of the project No. 6 is to compute the chrcteristics of the ircrft sttic stbility nd control chrcteristics in the pitch nd roll chnnel. The
More informationLecture 22. Inductance. Magnetic Field Energy. Outline:
Lecture 22. Inductance. Magnetic Field Energy. Outline: Selfinduction and selfinductance. Inductance of a solenoid. The energy of a magnetic field. Alternative definition of inductance. Mutual Inductance.
More informationCHAPTER 6 MAGNETIC EFFECT OF AN ELECTRIC CURRENT
CHAPTER 6 MAGNETIC EFFECT OF AN ELECTRIC CURRENT 6. Introduction Most of us re fmilir with the more obvious properties of mgnets nd compss needles. A mgnet, often in the form of short iron br, will ttrct
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
More informationBasic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
More informationHaus, Hermann A., and James R. Melcher. Electromagnetic Fields and Energy. Englewood Cliffs, NJ: PrenticeHall, 1989. ISBN: 9780132490207.
MIT OpenCourseWre http://ocw.mit.edu Hus, Hermnn A., nd Jmes R. Melcher. Electromgnetic Fields nd Energy. Englewood Cliffs, NJ: PrenticeHll, 1989. ISBN: 9780132490207. Plese use the following cittion
More informationEdmund Li. Where is defined as the mutual inductance between and and has the SI units of Henries (H).
INDUCTANCE MUTUAL INDUCTANCE If we consider two neighbouring closed loops and with bounding surfaces respectively then a current through will create a magnetic field which will link with as the flux passes
More informationSpace Vector Pulse Width Modulation Based Induction Motor with V/F Control
Interntionl Journl of Science nd Reserch (IJSR) Spce Vector Pulse Width Modultion Bsed Induction Motor with V/F Control Vikrmrjn Jmbulingm Electricl nd Electronics Engineering, VIT University, Indi Abstrct:
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationUnderstanding Basic Analog Ideal Op Amps
Appliction Report SLAA068A  April 2000 Understnding Bsic Anlog Idel Op Amps Ron Mncini Mixed Signl Products ABSTRACT This ppliction report develops the equtions for the idel opertionl mplifier (op mp).
More informationaddition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix.
APPENDIX A: The ellipse August 15, 1997 Becuse of its importnce in both pproximting the erth s shpe nd describing stellite orbits, n informl discussion of the ellipse is presented in this ppendix. The
More informationLECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.
LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 6483.
More informationPhysics 6010, Fall 2010 Symmetries and Conservation Laws: Energy, Momentum and Angular Momentum Relevant Sections in Text: 2.6, 2.
Physics 6010, Fll 2010 Symmetries nd Conservtion Lws: Energy, Momentum nd Angulr Momentum Relevnt Sections in Text: 2.6, 2.7 Symmetries nd Conservtion Lws By conservtion lw we men quntity constructed from
More informationRegular Sets and Expressions
Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite
More informationLectures 8 and 9 1 Rectangular waveguides
1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves
More information200506 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration
Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 256 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting
More informationPHY 140A: Solid State Physics. Solution to Homework #2
PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.
More informationMath 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.
Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose
More informationHelicopter Theme and Variations
Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the
More informationSection 74 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 74 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More informationUnit 6: Exponents and Radicals
Eponents nd Rdicls : The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N):  counting numers. {,,,,, } Whole Numers (W):  counting numers with 0. {0,,,,,, } Integers (I): 
More information6.5  Areas of Surfaces of Revolution and the Theorems of Pappus
Lecture_06_05.n 1 6.5  Ares of Surfces of Revolution n the Theorems of Pppus Introuction Suppose we rotte some curve out line to otin surfce, we cn use efinite integrl to clculte the re of the surfce.
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationbody.allowsidebar OR.nosidebar.homepage (if this is the home page).hascustombanner OR.nocustombanner .IR OR.noIR
body.llowsidebr OR.nosidebr.homepge (if this is the home pge).hscustombnner OR.nocustombnner.IR OR.noIR #IDENTIFIER_FOR_THIS_SITE div#pgecontiner.depends_on_page_ty PE llowsidebr mens tht there
More informationCapacitance and Dielectrics
2.2 This is the Nerest One He 803 P U Z Z L E R Mny electronic components crry wrning lel like this one. Wht is there insie these evices tht mkes them so ngerous? Why wouln t you e sfe if you unplugge
More informationData replication in mobile computing
Technicl Report, My 2010 Dt repliction in mobile computing Bchelor s Thesis in Electricl Engineering Rodrigo Christovm Pmplon HALMSTAD UNIVERSITY, IDE SCHOOL OF INFORMATION SCIENCE, COMPUTER AND ELECTRICAL
More information1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator
AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.
More informationORBITAL MANEUVERS USING LOWTHRUST
Proceedings of the 8th WSEAS Interntionl Conference on SIGNAL PROCESSING, ROBOICS nd AUOMAION ORBIAL MANEUVERS USING LOWHRUS VIVIAN MARINS GOMES, ANONIO F. B. A. PRADO, HÉLIO KOII KUGA Ntionl Institute
More informationhrb==. trn't Pltysics 48 (Fail 201I) Chupter 30: Induction and fnductunce Reading: pages 791813 O utline:
Pltysics 48 (Fil 201I) Chupter 30: Induction nd fnductunce hrb==. trn't 7 "For every minute you re ngry, you lose 60 seconds ofhppiness. "  Rlph Wldo Emerson "Consider how much more often you suffer
More informationDerivatives and Rates of Change
Section 2.1 Derivtives nd Rtes of Cnge 2010 Kiryl Tsiscnk Derivtives nd Rtes of Cnge Te Tngent Problem EXAMPLE: Grp te prbol y = x 2 nd te tngent line t te point P(1,1). Solution: We ve: DEFINITION: Te
More information