SELFINDUCTANCE AND INDUCTORS


 Elwin Stevenson
 2 years ago
 Views:
Transcription
1 MISN SELFINDUCTANCE AND INDUCTORS SELFINDUCTANCE AND INDUCTORS by Pete Signell Michigan State Univesity 1. Intoduction A 2. SelfInductance L a. The Definition of L b. L When the Flux is Known c. L fo a Tooidal Solenoid L/l fo a Coaxial Cable a. Physical Desciption of a Coaxial Cable b. Getting B c. Getting L/l Inductive Enegy In a Cicuit a. Enegy Needed to Set up a Cuent b. The Enegy is Recoveable c. Location of the Enegy d. Enegy Flow When a Cuent is Stopped Acknowledgments Glossay Poject PHYSNET Physics Bldg. Michigan State Univesity East Lansing, MI 1
2 ID Sheet: MISN Title: SelfInductance and Inductos Autho: Pete Signell, Michigan State Univesity Vesion: 1/25/2001 Evaluation: Stage 0 Length: 1 h; 12 pages Input Skills: 1. Vocabulay: solenoid, tooid, inductance, induced voltage, induced cuent, induced magnetic field, FaadayHeny law, Lenz s law (MISN0142). 2. Use Ampee s law to detemine the magnetic field due to a long staight cuent (MISN0138). Output Skills (Knowledge): K1. Vocabulay: coaxial cable, heny, inducto, selfinductance. K2. Wite down Ampee s Law and fom it deive the selfinductance of a tooidal solenoid, explicitly justifying each step. K3. Wite down Ampee s Law and fom it deive the selfinductance pe unit length of a coaxial cable, explicitly justifying each step. K4. Stating fom the elation between powe, voltage and cuent in a steady state cicuit, deive the enegy stoed in the electic field of an inducto. K5. Descibe the flow of enegy: (a) when the cuent though an inducto is inceased; (b) when the cuent though an inducto is vey gadually deceased; and (c) when the cuent though an inducto is quickly deceased. PostOptions: 1. Two Element DCDiven LRC Cicuits (MISN0151). 2. Velocity of a Signal in a Coaxial Cable (MISN0150). THIS IS A DEVELOPMENTALSTAGE PUBLICATION OF PROJECT PHYSNET The goal of ou poject is to assist a netwok of educatos and scientists in tansfeing physics fom one peson to anothe. We suppot manuscipt pocessing and distibution, along with communication and infomation systems. We also wok with employes to identify basic scientific skills as well as physics topics that ae needed in science and technology. A numbe of ou publications ae aimed at assisting uses in acquiing such skills. Ou publications ae designed: (i) to be updated quickly in esponse to field tests and new scientific developments; (ii) to be used in both classoom and pofessional settings; (iii) to show the peequisite dependencies existing among the vaious chunks of physics knowledge and skill, as a guide both to mental oganization and to use of the mateials; and (iv) to be adapted quickly to specific use needs anging fom singleskill instuction to complete custom textbooks. New authos, eviewes and field testes ae welcome. PROJECT STAFF Andew Schnepp Eugene Kales Pete Signell Webmaste Gaphics Poject Diecto ADVISORY COMMITTEE D. Alan Bomley Yale Univesity E. Leonad Jossem The Ohio State Univesity A. A. Stassenbug S. U. N. Y., Stony Book Views expessed in a module ae those of the module autho(s) and ae not necessaily those of othe poject paticipants. c 2001, Pete Signell fo Poject PHYSNET, PhysicsAstonomy Bldg., Mich. State Univ., E. Lansing, MI 48824; (517) Fo ou libeal use policies see: 3 4
3 MISN SELFINDUCTANCE AND INDUCTORS by Pete Signell Michigan State Univesity 1. Intoduction A solenoid o a tooid, sometimes of miniatue size, is used in electonic cicuits to: (1) slow the ate of change of electic cuent; (2) tune a cicuit to a paticula oscillational fequency; o (3) contol the speed of tansmission of signals. In these applications one is making use of the fact that a change in the cuent going though the device poduces a change in the associated magnetic field and that in tun induces a cuent that opposes the change in the oiginal cuent. A device used in that manne is called an inducto and the stength of its changeopposing chaacte is called its selfinductance o simply its inductance. This (self) inductance is measued in the S.I. unit called the heny, so a cicuit designe may specify an inducto of, say, 35 millihenies to achieve one of the thee abovementioned aims. 2. SelfInductance L 2a. The Definition of L. Inductance in geneal includes potentials and cuents induced in one conducto by a timechanging cuent in anothe conducto, but selfinductance efes to potentials and cuents that ae induced in a single conducto by its own timechanging cuent. A device used fo this pupose usually has the shape of a solenoid o a tooid. A cuent flowing though an inducto of couse sets up a magnetic field so changes in the cuent poduce changes in the magnetic field. Such changes poduce an induced voltage dop in the inducto, a voltage dop that opposes the change in the cuent. The magnitude of the induced voltage is popotional to the timeateofchange of the cuent, as we have seen, 1 so we wite: V ind = L di dt. (1) 1 See Magnetic Inductance, MISN MISN A Figue 1. A tooidal solenoid. Hee L is a popotionality constant that depends on the geomety of the inducto and the inducto s mateial: it is called the inducto s inductance. An inducto s inductance can be much enhanced by placing its loops of wie aound a magnetic mateial such as ion. 2b. L When the Flux is Known. If one knows the flux though the suface bounded by a cicuit, then the selfinductance can be detemined by integating Eq. (1) to get: Φ = L I. (2) Fo example, if the flux enclosed by a loop is T m 2 when the cuent in the loop is 1.3 A, then the inductance of the loop is L = 20 mh. If the inducto is alteed to contain 15 successive loops glued togethe, then the flux met by the cuent will be inceased by a facto of 15 and so will L. 2c. L fo a Tooidal Solenoid. To find the selfinductance of a tooidal solenoid, we use Ampee s law and we daw the integation loop as a cicle of adius fom the cente of the tooid (see Fig. 1). If is less than the inne adius of the tooid, thee is zeo cuent going though any suface bounded by the integation loop. If is between the inne and oute adii of the tooid the the cuent going though the enclosed suface is NI whee N is the numbe of tuns of wie caying the cuent I. If is geate than the oute adius of the tooid, thee is again zeo net cuent going though any suface bounded by the integation loop (the cuent going one diection though the suface is exactly canceled by the cuent going the othe way though the suface). Then by Ampee s law, the magnetic field inside the solenoidal loops is: B() = 2 k m N I. (3) 5 6
4 MISN MISN dielectic (usually white, flexible) baided wie sheath V 0 cable R Figue 3. Longitudinal view of a coaxial cable in a hypothetical cicuit. cente wie (solid) plastic skin Figue 2. Cosssection of a coaxial cable. If the adius of each loop is much smalle than the tooidal adius, then will vay little ove the cosssectional aea A of each loop and we can take as a constant. To get the flux in a single loop of the winding, we need meely multiply the B by the loop aea A. Then fo the N windings in the inducto: Φ = N A B = 2 k m N 2 I A. (4) Then fom Eq. (2): L = 2 k m N 2 A. (5) Typical numbes fo a cicuit tooid will give an extemely small inductance unless the flux is enhanced though the use of an ion coe inside the solenoidal loops. This enhancement may be by a facto of five thousand o moe. 3. L/l fo a Coaxial Cable 3a. Physical Desciption of a Coaxial Cable. Physically, a coaxial cable looks like a fat ound wie: a good example is the cable that feeds television pogams to TV sets fom a cable company. If you wee to cut though such a cable you would see a cental conducting wie suounded by a dielectic mateial (usually flexible white plastic). This insulating mateial is suounded by a cylindical sheath woven fom conducting wies (woven to make it flexible). That oute conducto is, in tun, coveed by a thin skin of flexible insulating plastic (see Fig. 2). Cuent flows down one conducto and exactly the same amount of cuent flows in the opposite diection in the othe conducto. One can imagine a voltage souce connected between the inne and oute conductos at one end of the cable and a esisto connected between them at the othe end, as in Fig. 3. We will assume that the cuent taveling the inne wie is along its suface, at a adius i (a good assumption fo highfequency waves). We will call the adius of the oute conducto o. 3b. Getting B. Fo selfinductance we need flux pe unit cuent, and fo flux we need the magnetic field. Fo a coaxial cable the entie magnetic field is between the two conductos. That is easily seen because by Ampee s law thee is zeo net cuent cossing a cosssectional aea lage than the outside conducto (emembe that the cuents in the two conductos ae equal but opposite in diection). Fo the egion between the two conductos, Ampee s law immediately shows that the magnetic field is simple that of the inne conducto. Fo a long staight wie it is: 2 B = 2 k m I and of couse the diection of the field is eveywhee pependicula to the outwad cylindical adius fom the inne wie. 3c. Getting L/l. To get the flux in ou coaxial cable we must integate the component of the magnetic field nomal to a suface bounded by the loop of electical cuent. To make it easy we choose a suface that is Apply Ampee s law in you head o see Magnetic Fields fom Cuents, MISN A B D C, Figue 4. A longitudinal view of the coaxial cable showing the integation suface ABCD fo obtaining the flux. 7 8
5 MISN adial, unning between the two conductos and along the length of the cable (see Fig. 4). Now the element of flux at a paticula adius is just the value of B thee times the element of aea at that adius: ( ) 2 km I dφ = (B) da = l d. whee l is the length of the cable. Integating, o ( ) 2 km I Φ = dφ = l d = 2 k m I l ln ( o / i ). i Finally, then, the selfinductance pe unit length of cable is: L/l = Φ/l I = 2 k m ln ( o / i ). (6) The total inductance fo any paticula piece of cable can be obtained by multiplying L/l by the piece s length. 4. Inductive Enegy In a Cicuit 4a. Enegy Needed to Set up a Cuent. When a cicuit switch is closed, stating a flow of cuent though a cicuit, an inducto in the path of the cuent esists the ise of the cuent fom zeo by developing a countevoltage dop (a voltage ise). The souce of voltage in the cicuit must push the cuent past this voltage ise, doing wok equal to the voltage ise times the amount of chage pushed though it. The powe expended (the enegy pe unit time) is just the cuent (the chage pe unit time) times the selfinduced voltage: P = I L di dt. Using P = de/dt we can easily integate both sides of: to get: de = I L di E = 1 2 L I2. (7) MISN b. The Enegy is Recoveable. The enegy expended in setting up a cuent in an inducto is ecoveable if the cicuit s voltage souce is emoved. Assuming thee is still a complete cicuit without the voltage souce, the inducto will keep the cuent flowing until the enegy of Eq. (7) has been completely dissipated in the cicuit s esistances o pehaps tansfeed to a capacitance fo stoage thee. 4c. Location of the Enegy. We descibe the enegy of Eq. (7) as being stoed in the magnetic field of the inducto. That stoed enegy was zeo in the beginning of ou example, when the cicuit cuent was zeo so the inducto s magnetic field was zeo. As the cuent, hence the inducto s magnetic field, inceased the enegy in the magnetic field inceased as the cicuit s voltage souce supplied enegy to the field. If the cuent became steady, the inducto s magnetic field became steady along with it and thee was no longe a tansfe of enegy fom the voltage souce to the inducto s field. Howeve, the enegy stoed in the inducto s magnetic field stays stoed thee. 4d. Enegy Flow When a Cuent is Stopped. Suppose a steady cuent is flowing in a cicuit containing an inducto, and then one opens a switch so cuent can pesumably no longe flow: what happens to the enegy stoed in the inducto s magnetic field? The answe is that as the switch is opened the cuent will dop quickly, ceating a lage induced voltage that ionizes the ai acoss the switch gap and thus causes an electic ac. The enegy stoed in the inducto s magnetic field is thus dissipated in chemical and heat enegy in beaking down the ai in the switch gap and in buning the contact points in the switch and in buning anything else in the vicinity. In fact, the stoed enegy in a lage inducto can be extemely dangeous to anyone attempting to stop the cuent in a huy. Acknowledgments Pepaation of this module was suppoted in pat by the National Science Foundation, Division of Science Education Development and Reseach, though Gant #SED to Michigan State Univesity. This is the enegy that must be expended by the cicuit s enegy souce in ode to aise the cicuit cuent fom zeo to the value I. 9 10
6 MISN MISN ME1 Glossay coaxial cable: a cicuit element that looks like a long fat plasticcoveed wie, containing in successive cylindical layes: an inne solidwie conducto, a suounding flexible dielectic, and an oute baidedwie sheath. The cicuit s cuent goes down one conducto (solid o sheath) and back the othe. heny: the SI unit of selfinductance, abbeviated H and defined to be an ohmsecond. Thus: H Ω s = V s A 1. inducto: a cicuit element whose pupose is to povide selfinductance, an electical cicuit analog of mechanical inetia (mass). An inducto is usually in the shape of a solenoid o a tooid. The inductance of an inducto depends on the geomety of the inducto and the magnetic susceptibility of the mateials of which the inducto is constucted. Inductos in electonic cicuits typically ae in the mh ange. MODEL EXAM 1. See Output Skills K1K5 in this module s ID Sheet. As usual, Vocabulay means defining the wods as well as being able to use them popely. Bief Answes: 1. See this module s text. selfinductance: the negative of the induced voltage aound a loop divided by the timeateofchange of magnetic flux though any suface bounded by that loop: L = V induced / Φ. The minus sign shows that the induced voltage opposes the change in the flux. The SI unit of inductance is the heny (which see)
GAUSS S LAW APPLIED TO CYLINDRICAL AND PLANAR CHARGE DISTRIBUTIONS ` E MISN0133. CHARGE DISTRIBUTIONS by Peter Signell, Michigan State University
MISN0133 GAUSS S LAW APPLIED TO CYLINDRICAL AND PLANAR CHARGE DISTRIBUTIONS GAUSS S LAW APPLIED TO CYLINDRICAL AND PLANAR CHARGE DISTRIBUTIONS by Pete Signell, Michigan State Univesity 1. Intoduction..............................................
More informationTORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION
MISN034 TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION shaft TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION by Kiby Mogan, Chalotte, Michigan 1. Intoduction..............................................
More informationVoltage ( = Electric Potential )
V1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage
More informationSamples of conceptual and analytical/numerical questions from chap 21, C&J, 7E
CHAPTER 1 Magnetism CONCEPTUAL QUESTIONS Cutnell & Johnson 7E 3. ssm A chaged paticle, passing though a cetain egion of space, has a velocity whose magnitude and diection emain constant, (a) If it is known
More informationThe Role of Gravity in Orbital Motion
! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State
More informationGauss Law. Physics 231 Lecture 21
Gauss Law Physics 31 Lectue 1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing
More informationA r. (Can you see that this just gives the formula we had above?)
241 (SJP, Phys 1120) lectic flux, and Gauss' law Finding the lectic field due to a bunch of chages is KY! Once you know, you know the foce on any chage you put down  you can pedict (o contol) motion
More informationDeflection of Electrons by Electric and Magnetic Fields
Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An
More information2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,
3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects
More informationMagnetic Field and Magnetic Forces. Young and Freedman Chapter 27
Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew  electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field
More informationThe Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = W/q 0 1V [Volt] =1 Nm/C
Geneal Physics  PH Winte 6 Bjoen Seipel The Electic Potential, Electic Potential Enegy and Enegy Consevation Electic Potential Enegy U is the enegy of a chaged object in an extenal electic field (Unit
More informationProblem Set 6: Solutions
UNIVESITY OF ALABAMA Depatment of Physics and Astonomy PH 164 / LeClai Fall 28 Poblem Set 6: Solutions 1. Seway 29.55 Potons having a kinetic enegy of 5. MeV ae moving in the positive x diection and ente
More informationConcept and Experiences on using a Wikibased System for Softwarerelated Seminar Papers
Concept and Expeiences on using a Wikibased System fo Softwaeelated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wthaachen.de,
More informationYour Comments. did not understand any of this wtf. so over my head I have no idea what I am doing :(
You Comments Llamas, and why it is so difficult to get someone to buy me one as a pet. All ask is that it beathes fie. have simple needs people. Seiously. p.s. this lectue was had. did not undestand any
More informationRevision Guide for Chapter 11
Revision Guide fo Chapte 11 Contents Student s Checklist Revision Notes Momentum... 4 Newton's laws of motion... 4 Gavitational field... 5 Gavitational potential... 6 Motion in a cicle... 7 Summay Diagams
More informationPoynting Vector and Energy Flow in a Capacitor Challenge Problem Solutions
Poynting Vecto an Enegy Flow in a Capacito Challenge Poblem Solutions Poblem 1: A paallelplate capacito consists of two cicula plates, each with aius R, sepaate by a istance. A steay cuent I is flowing
More informationProblems on Force Exerted by a Magnetic Fields from Ch 26 T&M
Poblems on oce Exeted by a Magnetic ields fom Ch 6 TM Poblem 6.7 A cuentcaying wie is bent into a semicicula loop of adius that lies in the xy plane. Thee is a unifom magnetic field B Bk pependicula to
More informationForces & Magnetic Dipoles. r r τ = μ B r
Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent
More informationYARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH
nd INTERNATIONAL TEXTILE, CLOTHING & ESIGN CONFERENCE Magic Wold of Textiles Octobe 03 d to 06 th 004, UBROVNIK, CROATIA YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH Jana VOBOROVA; Ashish GARG; Bohuslav
More informationVoltage ( = Electric Potential )
V1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is
More information2  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 1
 ELECTROSTATIC POTENTIAL AND CAPACITANCE Page. Line Integal of Electic Field If a unit positive chage is displaced by `given by dw E. dl dl in an electic field of intensity E, wok done is Line integation
More informationPhysics 235 Chapter 5. Chapter 5 Gravitation
Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus
More informationLesson 7 Gauss s Law and Electric Fields
Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual
More informationVector Calculus: Are you ready? Vectors in 2D and 3D Space: Review
Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.7. find the vecto defined
More informationPAN STABILITY TESTING OF DC CIRCUITS USING VARIATIONAL METHODS XVIII  SPETO  1995. pod patronatem. Summary
PCE SEMINIUM Z PODSTW ELEKTOTECHNIKI I TEOII OBWODÓW 8  TH SEMIN ON FUNDMENTLS OF ELECTOTECHNICS ND CICUIT THEOY ZDENĚK BIOLEK SPŠE OŽNO P.., CZECH EPUBLIC DLIBO BIOLEK MILITY CDEMY, BNO, CZECH EPUBLIC
More informationIntroduction to Electric Potential
Univesiti Teknologi MARA Fakulti Sains Gunaan Intoduction to Electic Potential : A Physical Science Activity Name: HP: Lab # 3: The goal of today s activity is fo you to exploe and descibe the electic
More informationFXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.
Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing
More informationExperiment 6: Centripetal Force
Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee
More informationGauss s law relates to total electric flux through a closed surface to the total enclosed charge.
Chapte : Gauss s Law Gauss s Law is an altenative fomulation of the elation between an electic field and the souces of that field in tems of electic flu. lectic Flu Φ though an aea ~ Numbe of Field Lines
More informationChapter 30: Magnetic Fields Due to Currents
d Chapte 3: Magnetic Field Due to Cuent A moving electic chage ceate a magnetic field. One of the moe pactical way of geneating a lage magnetic field (.11 T) i to ue a lage cuent flowing though a wie.
More informationUNIT CIRCLE TRIGONOMETRY
UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + =   
More informationHour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and
Hou Exam No. Please attempt all of the following poblems befoe the due date. All poblems count the same even though some ae moe complex than othes. Assume that c units ae used thoughout. Poblem A photon
More informationContinuous Compounding and Annualization
Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem
More informationChapter 3 Savings, Present Value and Ricardian Equivalence
Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,
More informationDERIVATION OF ORBITS IN INVERSE SQUARE LAW FORCE FIELDS
MISN0106 DERIVATION OF ORBITS IN INVERSE SQUARE LAW FORCE FIELDS Force Center (also the coordinate center) satellite DERIVATION OF ORBITS IN INVERSE SQUARE LAW FORCE FIELDS by Peter Signell 1. Introduction..............................................
More informationSpirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project
Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.
More information12. Rolling, Torque, and Angular Momentum
12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.
More information4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first nonzero digit to
. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate
More informationExperiment MF Magnetic Force
Expeiment MF Magnetic Foce Intoduction The magnetic foce on a cuentcaying conducto is basic to evey electic moto  tuning the hands of electic watches and clocks, tanspoting tape in Walkmans, stating
More informationEpisode 401: Newton s law of universal gravitation
Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce
More informationDetermining solar characteristics using planetary data
Detemining sola chaacteistics using planetay data Intoduction The Sun is a G type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this inestigation
More informationAn Epidemic Model of Mobile Phone Virus
An Epidemic Model of Mobile Phone Vius Hui Zheng, Dong Li, Zhuo Gao 3 Netwok Reseach Cente, Tsinghua Univesity, P. R. China zh@tsinghua.edu.cn School of Compute Science and Technology, Huazhong Univesity
More informationComparing Availability of Various Rack Power Redundancy Configurations
Compaing Availability of Vaious Rack Powe Redundancy Configuations White Pape 48 Revision by Victo Avela > Executive summay Tansfe switches and dualpath powe distibution to IT equipment ae used to enhance
More informationAP Physics Electromagnetic Wrap Up
AP Physics Electomagnetic Wap Up Hee ae the gloious equations fo this wondeful section. F qsin This is the equation fo the magnetic foce acting on a moing chaged paticle in a magnetic field. The angle
More informationChapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.
Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming
More informationChapter 2. Electrostatics
Chapte. Electostatics.. The Electostatic Field To calculate the foce exeted by some electic chages,,, 3,... (the souce chages) on anothe chage Q (the test chage) we can use the pinciple of supeposition.
More informationCollege of Engineering Bachelor of Computer Science
2 0 0 7 w w w. c n u a s. e d u College of Engineeing Bachelo of Compute Science This bochue Details the BACHELOR OF COMPUTER SCIENCE PROGRAM available though CNU s College of Engineeing. Fo ou most uptodate
More informationPhysics HSC Course Stage 6. Space. Part 1: Earth s gravitational field
Physics HSC Couse Stage 6 Space Pat 1: Eath s gavitational field Contents Intoduction... Weight... 4 The value of g... 7 Measuing g...8 Vaiations in g...11 Calculating g and W...13 You weight on othe
More informationAlgebra and Trig. I. A point is a location or position that has no size or dimension.
Algeba and Tig. I 4.1 Angles and Radian Measues A Point A A B Line AB AB A point is a location o position that has no size o dimension. A line extends indefinitely in both diections and contains an infinite
More information867 Product Transfer and Resale Report
867 Poduct Tansfe and Resale Repot Functional Goup ID=PT Intoduction: This X12 Tansaction Set contains the fomat and establishes the data contents of the Poduct Tansfe and Resale Repot Tansaction Set (867)
More informationVISCOSITY OF BIODIESEL FUELS
VISCOSITY OF BIODIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use
More informationProblems of the 2 nd International Physics Olympiads (Budapest, Hungary, 1968)
Poblems of the nd ntenational Physics Olympiads (Budapest Hungay 968) Péte Vankó nstitute of Physics Budapest Univesity of Technical Engineeing Budapest Hungay Abstact Afte a shot intoduction the poblems
More informationFigure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!
1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the
More informationComparing Availability of Various Rack Power Redundancy Configurations
Compaing Availability of Vaious Rack Powe Redundancy Configuations By Victo Avela White Pape #48 Executive Summay Tansfe switches and dualpath powe distibution to IT equipment ae used to enhance the availability
More informationChapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43
Chapte 13 Gavitation Poblems: 1, 4, 5, 7, 18, 19, 5, 9, 31, 33, 43 Evey object in the univese attacts evey othe object. This is called gavitation. We e use to dealing with falling bodies nea the Eath.
More informationLecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3
Lectue 16: Colo and Intensity and he made him a coat of many colous. Genesis 37:3 1. Intoduction To display a pictue using Compute Gaphics, we need to compute the colo and intensity of the light at each
More informationThe LCOE is defined as the energy price ($ per unit of energy output) for which the Net Present Value of the investment is zero.
Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the
More informationSTUDENT RESPONSE TO ANNUITY FORMULA DERIVATION
Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts
More informationSemipartial (Part) and Partial Correlation
Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated
More informationDo Vibrations Make Sound?
Do Vibations Make Sound? Gade 1: Sound Pobe Aligned with National Standads oveview Students will lean about sound and vibations. This activity will allow students to see and hea how vibations do in fact
More informationCHAPTER 5 GRAVITATIONAL FIELD AND POTENTIAL
CHATER 5 GRAVITATIONAL FIELD AND OTENTIAL 5. Intoduction. This chapte deals with the calculation of gavitational fields and potentials in the vicinity of vaious shapes and sizes of massive bodies. The
More informationLearning Objectives. Decreasing size. ~10 3 m. ~10 6 m. ~10 10 m 1/22/2013. Describe ionic, covalent, and metallic, hydrogen, and van der Waals bonds.
Lectue #0 Chapte Atomic Bonding Leaning Objectives Descibe ionic, covalent, and metallic, hydogen, and van de Waals bonds. Which mateials exhibit each of these bonding types? What is coulombic foce of
More informationCarterPenrose diagrams and black holes
CatePenose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example
More informationINITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS
INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in
More informationQuestions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing
M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow
More informationFirstmark Credit Union Commercial Loan Department
Fistmak Cedit Union Commecial Loan Depatment Thank you fo consideing Fistmak Cedit Union as a tusted souce to meet the needs of you business. Fistmak Cedit Union offes a wide aay of business loans and
More informationAntiLock Braking System Training Program
COVERST.EPS ac T to $2.50 BS A AntiLock Baking System Taining Pogam Student Manual TP9738 Revised 399 Module 1 ABS Components and System Opeation Module 2 ABS Diagnosis and Repai Module 3 ATC Opeation,
More informationStructure and evolution of circumstellar disks during the early phase of accretion from a parent cloud
Cente fo Tubulence Reseach Annual Reseach Biefs 2001 209 Stuctue and evolution of cicumstella disks duing the ealy phase of accetion fom a paent cloud By Olusola C. Idowu 1. Motivation and Backgound The
More informationLab M4: The Torsional Pendulum and Moment of Inertia
M4.1 Lab M4: The Tosional Pendulum and Moment of netia ntoduction A tosional pendulum, o tosional oscillato, consists of a disklike mass suspended fom a thin od o wie. When the mass is twisted about the
More informationGravitation. AP Physics C
Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What
More informationExam 3: Equation Summary
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P
More informationPHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013
PHYSICS 111 HOMEWORK SOLUTION #13 May 1, 2013 0.1 In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 2.10 kg and 21.0
More informationCONDUCTION OBJECTIVES
CONDUCTION INTRODUCTION This package deals pimaily with the mechanisms by which electical excitation speads ove the membanes of excitable cells. Many physiological phenomena depend on the conduction of
More informationDisplacement, Velocity And Acceleration
Displacement, Velocity And Acceleation Vectos and Scalas Position Vectos Displacement Speed and Velocity Acceleation Complete Motion Diagams Outline Scala vs. Vecto Scalas vs. vectos Scala : a eal numbe,
More informationProblem Set # 9 Solutions
Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new highspeed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease
More informationThe transport performance evaluation system building of logistics enterprises
Jounal of Industial Engineeing and Management JIEM, 213 6(4): 194114 Online ISSN: 213953 Pint ISSN: 2138423 http://dx.doi.og/1.3926/jiem.784 The tanspot pefomance evaluation system building of logistics
More informationStrength Analysis and Optimization Design about the key parts of the Robot
Intenational Jounal of Reseach in Engineeing and Science (IJRES) ISSN (Online): 23209364, ISSN (Pint): 23209356 www.ijes.og Volume 3 Issue 3 ǁ Mach 2015 ǁ PP.2529 Stength Analysis and Optimization Design
More information92.131 Calculus 1 Optimization Problems
9 Calculus Optimization Poblems ) A Noman window has the outline of a semicicle on top of a ectangle as shown in the figue Suppose thee is 8 + π feet of wood tim available fo all 4 sides of the ectangle
More informationLab #7: Energy Conservation
Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 14 Intoduction: Pehaps one of the most unusual
More informationAN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM
AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,
More informationLesson 13 Applications of Timevarying Circuits
Lesson 13 Applications of Timevaying Cicuits Lawence. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 13. Intoduction In this lesson we ll look at a numbe
More informationFinancing Terms in the EOQ Model
Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad
More informationPublic Health and Transportation Coalition (PHiT) Vision, Mission, Goals, Objectives, and Work Plan August 2, 2012
Public Health and Tanspotation Coalition (PHiT) Vision, Mission, Goals, Objectives, and Wok Plan 2, 2012 Vision We envision Maine as a place whee people of all ages and abilities can move about in ways
More informationUNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Approximate time two 100minute sessions
Name St.No.  Date(YY/MM/DD) / / Section Goup# UNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Appoximate time two 100minute sessions OBJECTIVES I began to think of gavity extending to the ob of the moon,
More informationCharges, Coulomb s Law, and Electric Fields
Q&E 1 Chages, Coulomb s Law, and Electic ields Some expeimental facts: Expeimental fact 1: Electic chage comes in two types, which we call (+) and ( ). An atom consists of a heavy (+) chaged nucleus suounded
More information883 Brochure A5 GENE ss vernis.indd 12
ess x a eu / u e a. p o.eu c e / :/ http EURAXESS Reseaches in Motion is the gateway to attactive eseach caees in Euope and to a pool of woldclass eseach talent. By suppoting the mobility of eseaches,
More informationPhys 2101 Gabriela González. cos. sin. sin
1 Phys 101 Gabiela González a m t t ma ma m m T α φ ω φ sin cos α τ α φ τ sin m m α τ I We know all of that aleady!! 3 The figue shows the massive shield doo at a neuton test facility at Lawence Livemoe
More informationDefinitions and terminology
I love the Case & Fai textbook but it is out of date with how monetay policy woks today. Please use this handout to supplement the chapte on monetay policy. The textbook assumes that the Fedeal Reseve
More information2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES
. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an
More informationIBM Research Smarter Transportation Analytics
IBM Reseach Smate Tanspotation Analytics Laua Wynte PhD, Senio Reseach Scientist, IBM Watson Reseach Cente lwynte@us.ibm.com INSTRUMENTED We now have the ability to measue, sense and see the exact condition
More informationDesign of Wind Energy System on the Building Tower Applications
ISSN(Online): 398753 ISSN (Pint) :34767 (An ISO 397: 7 Cetified Oganization) Vol. 4, Issue, Febuay 5 Design of Wind Enegy System on the Building owe Applications D.Anusha, L V Suesh Kuma, G.V. Nagesh
More information12.1. FÖRSTER RESONANCE ENERGY TRANSFER
ndei Tokmakoff, MIT epatment of Chemisty, 3/5/8 11 1.1. FÖRSTER RESONNCE ENERGY TRNSFER Föste esonance enegy tansfe (FR) efes to the nonadiative tansfe of an electonic excitation fom a dono molecule to
More informationAdvanced Control of Active Filters. in a Battery Charger Application. Martin Bojrup
Advanced Contol of Active Filtes in a Battey Chage Application Matin Bojup Lund 999 ii Cove pictue Measuement on the dynamic esponse of the MRI hamonic filte contolle: load cuent (top), esulting line cuent
More informationLesson 8 Ampère s Law and Differential Operators
Lesson 8 Ampèe s Law and Diffeential Opeatos Lawence Rees 7 You ma make a single cop of this document fo pesonal use without witten pemission 8 Intoduction Thee ae significant diffeences between the electic
More informationSoftware Engineering and Development
I T H E A 67 Softwae Engineeing and Development SOFTWARE DEVELOPMENT PROCESS DYNAMICS MODELING AS STATE MACHINE Leonid Lyubchyk, Vasyl Soloshchuk Abstact: Softwae development pocess modeling is gaining
More informationModel Question Paper Mathematics Class XII
Model Question Pape Mathematics Class XII Time Allowed : 3 hous Maks: 100 Ma: Geneal Instuctions (i) The question pape consists of thee pats A, B and C. Each question of each pat is compulsoy. (ii) Pat
More informationCoordinate Systems L. M. Kalnins, March 2009
Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean
More informationConverting knowledge Into Practice
Conveting knowledge Into Pactice Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 2 0 1 0 C o p y i g h t s V l a d i m i R i b a k o v 1 Disclaime and Risk Wanings Tading
More informationest using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.
9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,
More informationDYNAMICS AND STRUCTURAL LOADING IN WIND TURBINES
DYNAMIS AND STRUTURAL LOADING IN WIND TURBINES M. Ragheb 12/30/2008 INTRODUTION The loading egimes to which wind tubines ae subject to ae extemely complex equiing special attention in thei design, opeation
More information