for the magnetic induction at the point P with coordinate x produced by an increment of current

Size: px
Start display at page:

Download "for the magnetic induction at the point P with coordinate x produced by an increment of current"

Transcription

1 5. tatng wth th ffnta psson B fo th magntc nucton at th pont P wth coonat pouc by an ncmnt of cunt at, show pcty that fo a oop cayng a cunt th magntc nucton at P s B Ω wh Ω s th so ang subtn by th oop at th pont P. Ths cospons to a magntc scaa potnta,. Th sgn connton fo th so ang s that Ω s post f th pont P ws th nn s of th sufac spannng th oop, that s, f a unt noma n to th sufac s fn by th cton of cunt fow a th ght han, Ω s post f n ponts away fom th pont P, an ngat othws. Ths s th sam connton as n cton.6 fo th ctc po ay. ( B( c c a c a ( ( a ( a a a Ω Ω( B Ω( B( Ω( 5. ght ccua sono of fnt ngth L an aus a has N tuns p unt ngth an cas a cunt. how that th magntc nucton on th cyn as n th mt s N B ( θ θ wh th angs a fn n th fgu at pag 5.

2 B fo on oop B NL N N asn θ sn θ a θ sn θ θa N N sn θ N N B BN N θ sn θ θ θ θ θ θ θ [ ] 5.6 cynca conucto of aus a has a ho of aus b bo paa to, an cnt a stanc fom, th cyn as ba. Th cunt nsty s unfom thoughout th manng mta of th cyn an s paa to th as. Us mp s aw an pncp of na supposton to fn th magntu an th cton of th magntc fu nsty n th ho. Bcaty Bnoho Bho ϕ ϕ J J J J Bcaty ϕ ϕ ( ϕ ϕ Bcaty J J B caty ( a b ( a b 5. ccua oop of w cayng a cunt s ocat wth ts cnt at th ogn of coonats an th noma to ts pan hang sphca angs θ,ψ. Th s an app magntc f, B X B (βy an B y B (β. (a Cacuat th foc actng on th oop wthout mang any appomaton. Compa you sut wth th appomat sut (5.69. Commnt. (b Cacuat th toqu n owst o. Can you uc anythng about th hgh o contbutons? Do th ansh fo th ccua oop? What about fo oth shaps? (a

3 B B y B y ( β ( β n sn θ ϕ sn θsnϕy θ F B F B F F B C C C ( B ( B ( By C C B β ( β y y β ( β βy s F ( ( B βy n Bβ sn sn s y n B β θ ϕ s F Bβ sn θ sn ϕa F ( y B β n Bβ sn θ ϕa s F ( ( { F B By By By By } C C y B y By y ( β ( β B B y F m m F Bβ snθ sna Bβmy Qsnθ sna Fy Bβ snθ a Bβm fom 5.69 F ( m B F B βm B βm y y N m B m B m B y B m m ( y Eact toqu N J B J B B J y pojcton of oop aa { ( β ( β ( β ( β } J B y B y B y B y J f w can soat th owst o tm, a oths a hgh o contbutons. Not that 5.5 m J ( yj J y J J y J yyj m my m y

4 5. sph of aus a cas a unfom sufac chag stbuton σ. Th sph s otat about a amt wth constant angua octy ω. Fn th cto potnta an magntc fu nsty both ns an outs th sph. Choos th aong th as K σ σ ω ω ω & asn θ ϕ asn θ sn ϕ j aθ j {( a ( a j} σ ω σ ω snθ snϕ ω snθ ϕ asnθ ϕ asnθ snϕ a θ K ( a * a K Y (, (, m θ ϕ Ym θ ϕ m a * Y m( θϕ, K( Ym( θ, ϕ Ω m σωa * Y (, {( sn sn ( sn m θϕ θ ϕ θ ϕ j} Ym ( θ, ϕ Ω m σω a 8 8 (, ( Y ( m θϕ Y Y Y Y j * Ym( θ, ϕ Ω m σω a 8 8 ( (, (, Y θϕ Y θϕ ( Y( θϕ, Y ( θϕ, j σω a sn { ( sn sn ( sn σωa θ θ ϕ θ ϕ j} Bcaus othogona, th ntgaton bcoms σ a ω σ a ω sn θ [ snϕ ϕy] sn θϕ σ aω n sn θϕ Bn n σaω ( mnn m m a, hnt : out B out m a σω

5 5.8 ccua oop of w hang a aus a an cayng a cunt s ocat n acuum wth ts cnt a stanc away fom a sm nfnt sab of pmabty. Fn th foc actng on th oop whn (a Th pan of th oop s paa to th fac of th sab, (b Th pan of th oop s ppncua to th fac of th sab. (c Dtmn th mtng fom of you answ to pats a an b whn a. Can you obtan ths mtng aus on som smp an ct way? (a & J δ a δ θ a a Fom (5.8 & (5.9 a!! B P ( θ!!! a a a Bθ P (! a F J( B( ( θ!! ê δ a δ θ P ( θ θ ê a a a!!! a a δ a δ θ P ( θ a (! a a a a ( a (!! a a a a P P! ( a a a a a a ( a upp : a ow : a ( b ( ( J δ ± a δ a Fom pobm ( b J a J B, B ( F J( B( J B J B 5 (, (

6 a a B J( a J B J a J a a J( a J J( a J ( F J B J B J B ( J B a ( ( ( J a J a ( ± ( a J ( ( ( a ± ± ( J B ( ( s ( a a ± ± ( J ( a J a co 5.9 magntcay ha mata s n th shap of a ght ccua cyn of ngth L an aus a. Th cyn has a pmannt magntaton, unfom thoughout ts oum an paa to ts as. (a Dtmn th magntc f H an magntc nucton B at a ponts on th as of th cyn, both ns an outs. (b Pot th atos on th as as functons of fo. (a Magntc ha mata, s 5.9(c, scaa potnta H( ΦM ( M( n M( Φ M ( a s M, on th top M( M M( & n M( M, on th bottom, oth n M( M a M a Φ M ( a s L L a a M L L L L

7 L ( ns M L L Φ M ( a a L L M H( Φ M L L a a L L M B ( H M L L a a L ( outs M L L Φ M ( a a L L L M H( Φ M ( L L a a L L M B ( H M H L L a a 5. (a tatng fom th foc quaton (5. an th fact that a magntaton M ns a oum boun by a sufac s quant to a oum cunt nsty an a sufac cunt nsty, show that n th absnc of macoscopc conucton cunts th tota magntc foc on th boy can b wttn F ( M B ( M n B a wh s th app magntc nucton (not ncung that of th boy n quston. Th foc s now pss n tms of th ffct chag nsts an. f th stbuton of magntaton s not scontnuous, th sufac can b at nfnty an th foc gn by just th oum ntga. (b sph of aus wth unfom magntaton ha ts cnt at th ogn of coonats an ts cton of magntaton mang sphca angs,. f th tna magntc f s th sam as n Pobm 5., us th psson of pat a to auat th componnts of th foc actng on th sph.

8 (b (a F J B K Ba Jm M K m M n F M B M n Ba B M M n Ba ( M B ( M B ( B M M ( B B ( M ( M B ( B M B ( M Q B ( M n Ba B ( M n a [ ( B n M ( B M n ] a F [( M B ( B M ( M B ] [ ( B n M ( B M ] us ( C D ( C D ( n C Da ( B M ( B M ( n B Ma ( n B ( M B ( M B ( n M Ba F [ ( M B ( M B ] [( n M B ( B ] ( M B ( n M Ba M M( sn θ ϕ,sn θsn ϕ,θ B B y B M n a Ma ( β, β, ( β snθsn ϕ, β snθ θ, ( θ ϕ θ ϕ θ n sn,sn sn, Thn F M n B a n a ( θθ sn θsn θ ( ϕ ϕ ( β sn θsn ϕ, β sn θ θ, MB Ω MB β sn θsn ϕ,sn θ ϕ, 5. how that n gna a ong, staght ba of unfom coss sctona aa wth unfom ngthws magntaton M, whn pac wth ts fat n aganst an nfnty pmab fat sufac, ahs wth a foc gn appomaty by F M ctostatc consatons n cton... at you scusson to th

9 Ths pobm s bst so by consng an mag magnt. Th nfnt pmabty of th at sufac nsus that th magntc f must b ppncua to th sufac. s a sut, ths s sma to th ctostatc cas of ctc f ns bng ppncua to th sufac of a pfct conucto. Fo magntostatcs, ths mans that w may us a magntc scaa potnta _M (snc th a no f cunts subjct to th conton _M at (tang th sufac to n th -y pan at. Th mag pobm s thn st up as foows W M B H t s mpotant to not that, wh w so ths pobm usng an mag magnt, th ony quantts that show up n ths ngy ntga a th actua soucs of magntaton ~M an th actua magntc nucton ~B. W pac th magnt at a stanc fom th sufac so that M a M, L, othws s a sut Z L L W M a B ( M B Z wh w ha appomat that th magntc nucton s oughy unfom acoss th fac of th magnt. Usng th mag magnt stup, th a two soucs of magntc nucton B B B a mag Usng ( w s that L Ba um a ( L a L L Bmag um a ( L a ( L H w ha shft th coonats such that th a magnt s btwn an L an th mag magnt s btwn --L an -. n pncp, w may nst ths pssons nto (5 to comput th magntostatc ngy. How, as a smpfcaton, w not that th ntga of ~M.Ba gs a poston npnnt ( npnnt sf ngy. Hnc ths w not contbut to th

10 foc. s a sut, w ony n to nst Bmag nto (5. Ths gs us Z L L W um Z ( a a ( L Z L um a ( a ( L Z um a ( L a a ( L Th foc s thn W L L F um a ( ( ( L a a L L L -u M a L a L - um wh n th ast n w us L a (a conton that w n anyway to nsu that B s nay unfom on th n caps. Not that w cou ha atnaty us th sut of Pobm 5. F ( M B ( M n B a s wh th app magntc nucton ~B s gn by Bmag n (6 wth. nc th magntaton s unfom, th foc ass nty fom th sufac tm F ( M n B a M B ( B ( L a s [ ] L L L [ (] um M B L B um a L a L a L What w ha on h s to cacuat th foc though th magntostatc ngy F W( wh ~ nots th poston of th ba magnt. Ths s th magntostatc quant of th foc scusson n cton., whch stats that \Focs actng btwn chag bos can b obtan by cacuatng th chang n th tota ctostatc ngy of th systm un sma

11 tua spacmnts." n fact, ths statmnt s tu n gna, po w us th compt (ctostatc pus magntostatc ngy of th systm. Cuousy, a conucto wth sufac-chag nsty _ fs an outwa foc of th fom σ F ε whch s oughy th ctostatc quant of F um 5. (a how that a sufac cunt nsty K / fowng n th aa cton on a ght ccua cynca sufac of aus poucs ns th cyn a unfom magntc nucton n a cton ppncua to th cyn as. how that th f outs s that of a two mnsona po. (b Cacuat th tota magnt statc f ngy p unt ngth. How s t ns an outs th cyn? (c What s th nuctanc p unt ngth of th systm, w as a ng ccut wth cunt fowng up on s of th cyn an bac th oth? (a J ( K( δ ( K( ( J sn y by.9 m m ( [ ( ] ( K ( m( ( [ ( ] ( K ( th ntga pc up m by othogonaty of ns. Usng sn sn an ( m ( m m m m

12 [ ] [ ] [ ] K K δ δ Q th asymptotc fom. appyng Γ Γ δ δ δ y B, sn, sn,, sn y y (b sn B B W υ

13 (c nc w ha ony on ccut. Compang to th abo w a off.

Load Equations. So let s look at a single machine connected to an infinite bus, as illustrated in Fig. 1 below.

Load Equations. So let s look at a single machine connected to an infinite bus, as illustrated in Fig. 1 below. oa Euatons Thoughout all of chapt 4, ou focus s on th machn tslf, thfo w wll only pfom a y smpl tatmnt of th ntwok n o to s a complt mol. W o that h, but alz that w wll tun to ths ssu n Chapt 9. So lt

More information

Rectification and Depth Computation

Rectification and Depth Computation Dpatmnt of Comput Engnng Unvst of Cafona at Santa Cuz Rctfcaton an Dpth Computaton CMPE 64: mag Anass an Comput Vson Spng 0 Ha ao 4/6/0 mag cosponncs Dpatmnt of Comput Engnng Unvst of Cafona at Santa Cuz

More information

Massachusetts Institute of Technology Introduction to Plasma Physics

Massachusetts Institute of Technology Introduction to Plasma Physics Massachustts Insttut of Tchnology Intoducton to Plasma Physcs NAME 6.65J,8.63J,.6J R. Pak Dcmb 5 Fnal Eam :3-4:3 PM NOTES: Th a 8 pags to th am, plus on fomula sht. Mak su that you copy s complt. Each

More information

Applications of Lagrange Equations

Applications of Lagrange Equations Applcaton of agang Euaton Ca Stuy : Elctc Ccut ng th agang uaton of oton, vlop th athatcal ol fo th ccut hown n Fgu.Sulat th ult by SIMI. Th ccuty paat a: 0.0 H, 0.00 H, 0.00 H, C 0.0 F, C 0. F, 0 Ω, Ω

More information

APPENDIX H CONSTANT VOLTAGE BEHIND TRANSIENT REACTANCE GENERATOR MODEL

APPENDIX H CONSTANT VOLTAGE BEHIND TRANSIENT REACTANCE GENERATOR MODEL APPNDIX H CONSAN VOAG BHIND RANSIN RACANC GNRAOR MOD h mprov two gnrator mo uss th constant votag bhn transnt ractanc gnrator mo. hs mo gnors magntc sancy; assums th opratng ractanc of th gnrator s th

More information

5- Scattering Stationary States

5- Scattering Stationary States Lctu 19 Pyscs Dpatmnt Yamou Unvsty 1163 Ibd Jodan Pys. 441: Nucla Pyscs 1 Pobablty Cunts D. Ndal Esadat ttp://ctaps.yu.du.jo/pyscs/couss/pys641/lc5-3 5- Scattng Statonay Stats Rfnc: Paagaps B and C Quantum

More information

Period vs. Length of a Pendulum

Period vs. Length of a Pendulum Gaphcal Mtho n Phc Gaph Intptaton an Lnazaton Pat 1: Gaphng Tchnqu In Phc w u a vat of tool nclung wo, quaton, an gaph to mak mol of th moton of objct an th ntacton btwn objct n a tm. Gaph a on of th bt

More information

MULTIPOLE FIELDS. Multipoles, 2 l poles. Monopoles, dipoles, quadrupoles, octupoles... Electric Dipole R 1 R 2. P(r,θ,φ) e r

MULTIPOLE FIELDS. Multipoles, 2 l poles. Monopoles, dipoles, quadrupoles, octupoles... Electric Dipole R 1 R 2. P(r,θ,φ) e r MULTIPOLE FIELDS Mutpoes poes. Monopoes dpoes quadupoes octupoes... 4 8 6 Eectc Dpoe +q O θ e R R P(θφ) -q e The potenta at the fed pont P(θφ) s ( θϕ )= q R R Bo E. Seneus : Now R = ( e) = + cosθ R = (

More information

1.050 Engineering Mechanics I. Summary of variables/concepts. Lecture 27-37

1.050 Engineering Mechanics I. Summary of variables/concepts. Lecture 27-37 .5 Engneeng Mechancs I Summa of vaabes/concepts Lectue 7-37 Vaabe Defnton Notes & ments f secant f tangent f a b a f b f a Convet of a functon a b W v W F v R Etena wok N N δ δ N Fee eneg an pementa fee

More information

FI 3103 Quantum Physics

FI 3103 Quantum Physics 7//7 FI 33 Quantum Physics Axan A. Iskana Physics of Magntism an Photonics sach oup Institut Tknoogi Banung Schoing Equation in 3D Th Cnta Potntia Hyognic Atom 7//7 Schöing quation in 3D Fo a 3D pobm,

More information

Homework: Due

Homework: Due hw-.nb: //::9:5: omwok: Du -- Ths st (#7) s du on Wdnsday, //. Th soluton fom Poblm fom th xam s found n th mdtm solutons. ü Sakua Chap : 7,,,, 5. Mbach.. BJ 6. ü Mbach. Th bass stats of angula momntum

More information

Differential Kinematics

Differential Kinematics Lctu Diffntia Kinmatic Acknowgmnt : Pof. Ouama Khatib, Robotic Laboato, tanfo Univit, UA Pof. Ha Aaa, AI Laboato, MIT, UA Guiing Qution In obotic appication, not on th poition an ointation, but th vocit

More information

( V ) 0 in the above equation, but retained to keep the complete vector identity for V in equation.

( V ) 0 in the above equation, but retained to keep the complete vector identity for V in equation. Cuvlna Coodnats Outln:. Otogonal cuvlna coodnat systms. Dffntal opatos n otogonal cuvlna coodnat systms. Dvatvs of t unt vctos n otogonal cuvlna coodnat systms 4. Incompssbl N-S quatons n otogonal cuvlna

More information

First looking at the scalar potential term, suppose that the displacement is given by u = φ. If one can find a scalar φ such that u = φ. u x.

First looking at the scalar potential term, suppose that the displacement is given by u = φ. If one can find a scalar φ such that u = φ. u x. 7.4 Eastodynams 7.4. Propagaton of Wavs n East Sods Whn a strss wav travs throgh a matra, t ass matra parts to dspa by. It an b shown that any vtor an b wrttn n th form φ + ra (7.4. whr φ s a saar potnta

More information

EE243 Advanced Electromagnetic Theory Lec # 22 Scattering and Diffraction. Reading: Jackson Chapter 10.1, 10.3, lite on both 10.2 and 10.

EE243 Advanced Electromagnetic Theory Lec # 22 Scattering and Diffraction. Reading: Jackson Chapter 10.1, 10.3, lite on both 10.2 and 10. Appid M Fa 6, Nuuth Lctu # V //6 43 Advancd ctomagntic Thoy Lc # Scatting and Diffaction Scatting Fom Sma Obcts Scatting by Sma Dictic and Mtaic Sphs Coction of Scatts Sphica Wav xpansions Scaa Vcto Rading:

More information

Physics 505 Electricity and Magnetism Fall 2003 Prof. G. Raithel. Problem Set 7 Maximal score: 25 Points. 1. Jackson, Problem Points.

Physics 505 Electricity and Magnetism Fall 2003 Prof. G. Raithel. Problem Set 7 Maximal score: 25 Points. 1. Jackson, Problem Points. Physics 505 Eecticity and Magnetism Fa 00 Pof. G. Raithe Pobem et 7 Maxima scoe: 5 Points. Jackson, Pobem 5. 6 Points Conside the i-th catesian component of the B-Fied, µ 0 I B(x) ˆx i ˆx i d (x x ) x

More information

( ) + is the distance from the point of interest to the location of the charge q i

( ) + is the distance from the point of interest to the location of the charge q i Elctcal Engy and apactanc 57. Bcaus lctc ocs a consvatv, th kntc ngy gand s qual to th dcas n lctcal potntal ngy, o + + 4 4 KE PE q( ).. so th coct choc s (a).. Fom consvaton o ngy, KE + PE KE + PE, o

More information

COLLEGE OF FOUNDATION AND GENERAL STUDIES PUTRAJAYA CAMPUS FINAL EXAMINATION TRIMESTER /2017

COLLEGE OF FOUNDATION AND GENERAL STUDIES PUTRAJAYA CAMPUS FINAL EXAMINATION TRIMESTER /2017 COLLEGE OF FOUNDATION AND GENERAL STUDIES PUTRAJAYA CAMPUS FINAL EXAMINATION TRIMESTER 1 016/017 PROGRAMME SUBJECT CODE : Foundaton n Engneeng : PHYF115 SUBJECT : Phscs 1 DATE : Septembe 016 DURATION :

More information

Grid Transformations for CFD Calculations

Grid Transformations for CFD Calculations Coll of Ennn an Comput Scnc Mchancal Ennn Dpatmnt ME 69 Computatonal lu Dnamcs Spn Tct: 5754 Instuct: La Catto Intoucton G Tansfmatons f CD Calculatons W want to ca out ou CD analss n altnatv conat sstms.

More information

GMm. 10a-0. Satellite Motion. GMm U (r) - U (r ) how high does it go? Escape velocity. Kepler s 2nd Law ::= Areas Angular Mom. Conservation!!!!

GMm. 10a-0. Satellite Motion. GMm U (r) - U (r ) how high does it go? Escape velocity. Kepler s 2nd Law ::= Areas Angular Mom. Conservation!!!! F Satllt Moton 10a-0 U () - U ( ) 0 f ow g dos t go? scap locty Kpl s nd Law ::= Aas Angula Mo. Consaton!!!! Nwton s Unsal Law of Gaty 10a-1 M F F 1) F acts along t ln connctng t cnts of objcts Cntal Foc

More information

Physics 11b Lecture #2. Electric Field Electric Flux Gauss s Law

Physics 11b Lecture #2. Electric Field Electric Flux Gauss s Law Physcs 11b Lectue # Electc Feld Electc Flux Gauss s Law What We Dd Last Tme Electc chage = How object esponds to electc foce Comes n postve and negatve flavos Conseved Electc foce Coulomb s Law F Same

More information

Review of Vector Algebra and Vector Calculus Operations

Review of Vector Algebra and Vector Calculus Operations Revew of Vecto Algeba and Vecto Calculus Opeatons Tpes of vaables n Flud Mechancs Repesentaton of vectos Dffeent coodnate sstems Base vecto elatons Scala and vecto poducts Stess Newton s law of vscost

More information

Classical Electrodynamics

Classical Electrodynamics A Fst Look at Quantum Physcs Cassca Eectodynamcs Chapte 4 Mutpoes, Eectostatcs of Macoscopc Meda, Deectcs Cassca Eectodynamcs Pof. Y. F. Chen Contents A Fst Look at Quantum Physcs 4. Mutpoe Expanson 4.

More information

Remark: Positive work is done on an object when the point of application of the force moves in the direction of the force.

Remark: Positive work is done on an object when the point of application of the force moves in the direction of the force. Unt 5 Work and Energy 5. Work and knetc energy 5. Work - energy theore 5.3 Potenta energy 5.4 Tota energy 5.5 Energy dagra o a ass-sprng syste 5.6 A genera study o the potenta energy curve 5. Work and

More information

1. A body will remain in a state of rest, or of uniform motion in a straight line unless it

1. A body will remain in a state of rest, or of uniform motion in a straight line unless it Pncples of Dnamcs: Newton's Laws of moton. : Foce Analss 1. A bod wll eman n a state of est, o of unfom moton n a staght lne unless t s acted b etenal foces to change ts state.. The ate of change of momentum

More information

A L A BA M A L A W R E V IE W

A L A BA M A L A W R E V IE W A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N

More information

Diffraction. Diffraction: general Fresnel vs. Fraunhofer diffraction Several coherent oscillators Single-slit diffraction. Phys 322 Lecture 28

Diffraction. Diffraction: general Fresnel vs. Fraunhofer diffraction Several coherent oscillators Single-slit diffraction. Phys 322 Lecture 28 Chapt 10 Phys 3 Lctu 8 Dffacton Dffacton: gnal Fsnl vs. Faunhof dffacton Sval cohnt oscllatos Sngl-slt dffacton Dffacton Gmald, 1600s: dffacto, dvaton of lght fom lna popagaton Dffacton s a consqunc of

More information

Structure and Features

Structure and Features Thust l Roll ans Thust Roll ans Stutu an atus Thust ans onsst of a psly ma a an olls. Thy hav hh ty an hh loa apats an an b us n small spas. Thust l Roll ans nopoat nl olls, whl Thust Roll ans nopoat ylnal

More information

Anouncements. Conjugate Gradients. Steepest Descent. Outline. Steepest Descent. Steepest Descent

Anouncements. Conjugate Gradients. Steepest Descent. Outline. Steepest Descent. Steepest Descent oucms Couga Gas Mchal Kazha (6.657) Ifomao abou h Sma (6.757) hav b pos ol: hp://www.cs.hu.u/~msha Tch Spcs: o M o Tusay afoo. o Two paps scuss ach w. o Vos fo w s caa paps u by Thusay vg. Oul Rvw of Sps

More information

ALL QUESTIONS ARE WORTH 20 POINTS. WORK OUT FIVE PROBLEMS.

ALL QUESTIONS ARE WORTH 20 POINTS. WORK OUT FIVE PROBLEMS. GNRAL PHYSICS PH -3A (D. S. Mov) Test (/3/) key STUDNT NAM: STUDNT d #: -------------------------------------------------------------------------------------------------------------------------------------------

More information

Diffraction. Diffraction: general Fresnel vs. Fraunhofer diffraction Several coherent oscillators Single-slit diffraction. Phys 322 Lecture 28

Diffraction. Diffraction: general Fresnel vs. Fraunhofer diffraction Several coherent oscillators Single-slit diffraction. Phys 322 Lecture 28 Chapt 10 Phys 3 Lctu 8 Dffacton Dffacton: gnal Fsnl vs. Faunhof dffacton Sval cohnt oscllatos Sngl-slt dffacton Dffacton Gmald, 1600s: dffacto, dvaton of lght fom lna popagaton Dffacton s a consqunc of

More information

P a g e 5 1 of R e p o r t P B 4 / 0 9

P a g e 5 1 of R e p o r t P B 4 / 0 9 P a g e 5 1 of R e p o r t P B 4 / 0 9 J A R T a l s o c o n c l u d e d t h a t a l t h o u g h t h e i n t e n t o f N e l s o n s r e h a b i l i t a t i o n p l a n i s t o e n h a n c e c o n n e

More information

Chapter 3 Vector Integral Calculus

Chapter 3 Vector Integral Calculus hapte Vecto Integal alculus I. Lne ntegals. Defnton A lne ntegal of a vecto functon F ove a cuve s F In tems of components F F F F If,, an ae functon of t, we have F F F F t t t t E.. Fn the value of the

More information

The Random Phase Approximation:

The Random Phase Approximation: Th Random Phas Appoxmaton: Elctolyts, Polym Solutons and Polylctolyts I. Why chagd systms a so mpotant: thy a wat solubl. A. bology B. nvonmntally-fndly polym pocssng II. Elctolyt solutons standad dvaton

More information

GRAVITATION. (d) If a spring balance having frequency f is taken on moon (having g = g / 6) it will have a frequency of (a) 6f (b) f / 6

GRAVITATION. (d) If a spring balance having frequency f is taken on moon (having g = g / 6) it will have a frequency of (a) 6f (b) f / 6 GVITTION 1. Two satllits and o ound a plant P in cicula obits havin adii 4 and spctivly. If th spd of th satllit is V, th spd of th satllit will b 1 V 6 V 4V V. Th scap vlocity on th sufac of th ath is

More information

Scalars and Vectors Scalar

Scalars and Vectors Scalar Scalas and ectos Scala A phscal quantt that s completel chaacteed b a eal numbe (o b ts numecal value) s called a scala. In othe wods a scala possesses onl a magntude. Mass denst volume tempeatue tme eneg

More information

PHYS Week 5. Reading Journals today from tables. WebAssign due Wed nite

PHYS Week 5. Reading Journals today from tables. WebAssign due Wed nite PHYS 015 -- Week 5 Readng Jounals today fom tables WebAssgn due Wed nte Fo exclusve use n PHYS 015. Not fo e-dstbuton. Some mateals Copyght Unvesty of Coloado, Cengage,, Peason J. Maps. Fundamental Tools

More information

Dynamics of Rigid Bodies

Dynamics of Rigid Bodies Dynamcs of Rgd Bodes A gd body s one n whch the dstances between consttuent patcles s constant thoughout the moton of the body,.e. t keeps ts shape. Thee ae two knds of gd body moton: 1. Tanslatonal Rectlnea

More information

4/12/2018. PHY 712 Electrodynamics 9-9:50 AM MWF Olin 105. Plan for Lecture 34: Review radiating systems

4/12/2018. PHY 712 Electrodynamics 9-9:50 AM MWF Olin 105. Plan for Lecture 34: Review radiating systems PHY 7 Eodynams 9-9:5 M MWF On 5 Pan o u : Rvw adang sysms Souon o Maxw s quaons wh sous Tm pod sous Examps //8 PHY 7 Spng 8 -- u //8 PHY 7 Spng 8 -- u Gna vw -- SI uns mosop o vauum om ( P M ): Couombs

More information

2 dependence in the electrostatic force means that it is also

2 dependence in the electrostatic force means that it is also lectc Potental negy an lectc Potental A scala el, nvolvng magntues only, s oten ease to wo wth when compae to a vecto el. Fo electc els not havng to begn wth vecto ssues woul be nce. To aange ths a scala

More information

Engineering Mechanics. Force resultants, Torques, Scalar Products, Equivalent Force systems

Engineering Mechanics. Force resultants, Torques, Scalar Products, Equivalent Force systems Engneeng echancs oce esultants, Toques, Scala oducts, Equvalent oce sstems Tata cgaw-hll Companes, 008 Resultant of Two oces foce: acton of one bod on anothe; chaacteed b ts pont of applcaton, magntude,

More information

September 27, Introduction to Ordinary Differential Equations. ME 501A Seminar in Engineering Analysis Page 1. Outline

September 27, Introduction to Ordinary Differential Equations. ME 501A Seminar in Engineering Analysis Page 1. Outline Introucton to Ornar Dffrntal Equatons Sptmbr 7, 7 Introucton to Ornar Dffrntal Equatons Larr artto Mchancal Engnrng AB Smnar n Engnrng Analss Sptmbr 7, 7 Outln Rvw numrcal solutons Bascs of ffrntal quatons

More information

2.8 Variational Approach in Finite Element Formulation [Bathe P ] Principle of Minimum Total Potential Energy

2.8 Variational Approach in Finite Element Formulation [Bathe P ] Principle of Minimum Total Potential Energy .8 Varatona Approach n Fnt Emnt Formuaton [Bath P.-6] In unrstanng th phnomna occurrng n natur, w ar qut us to ffrnta quatons to scrb th phnomna mathmatca bas on basc phsca prncps, nam consrvaton aws n

More information

An action with positive kinetic energy term for general relativity. T. Mei

An action with positive kinetic energy term for general relativity. T. Mei An ton wt post nt ny t fo n tty T (Dptnt of Jon Cnt Cn o Unsty Wn H PRO Pop s Rp of Cn E-: to@nn tow@pwn ) Astt: At fst w stt so sts n X: 7769 n tn sn post nt ny oont onton n y X: 7769 w psnt n ton wt

More information

From Structural Analysis to FEM. Dhiman Basu

From Structural Analysis to FEM. Dhiman Basu From Structural Analyss to FEM Dhman Basu Acknowldgmnt Followng txt books wr consultd whl prparng ths lctur nots: Znkwcz, OC O.C. andtaylor Taylor, R.L. (000). Th FntElmnt Mthod, Vol. : Th Bass, Ffth dton,

More information

CIVL 7/ D Boundary Value Problems - Axisymmetric Elements 1/8

CIVL 7/ D Boundary Value Problems - Axisymmetric Elements 1/8 CIVL 7/8 -D Bounday Valu Poblms - xsymmtc Elmnts /8 xsymmtc poblms a somtms fd to as adally symmtc poblms. hy a gomtcally th-dmnsonal but mathmatcally only two-dmnsonal n th physcs of th poblm. In oth

More information

Plasma Sheaths and Langmuir probes

Plasma Sheaths and Langmuir probes Cass nots fo EE5383/Phys 5383 Spng Ths documnt s fo nstuctona us ony and may not b copd o dstbutd outsd of EE5383/Phys 5383 Last wk w ookd at msson of ght fom a pasma. In ffct ght msson s th smpst dagnostc

More information

Gauss Law. 2. Gauss s Law: connects charge and field 3. Applications of Gauss s Law

Gauss Law. 2. Gauss s Law: connects charge and field 3. Applications of Gauss s Law Gauss Law 1. Review on 1) Couomb s Law (charge and force) 2) Eectric Fied (fied and force) 2. Gauss s Law: connects charge and fied 3. Appications of Gauss s Law Couomb s Law and Eectric Fied Couomb s

More information

Clausius-Clapeyron Equation

Clausius-Clapeyron Equation ausius-apyron Equation 22000 p (mb) Liquid Soid 03 6. Vapor 0 00 374 (º) oud drops first form whn th aporization quiibrium point is rachd (i.., th air parc bcoms saturatd) Hr w dop an quation that dscribs

More information

176 5 t h Fl oo r. 337 P o ly me r Ma te ri al s

176 5 t h Fl oo r. 337 P o ly me r Ma te ri al s A g la di ou s F. L. 462 E l ec tr on ic D ev el op me nt A i ng er A.W.S. 371 C. A. M. A l ex an de r 236 A d mi ni st ra ti on R. H. (M rs ) A n dr ew s P. V. 326 O p ti ca l Tr an sm is si on A p ps

More information

The angle between L and the z-axis is found from

The angle between L and the z-axis is found from Poblm 6 This is not a ifficult poblm but it is a al pain to tansf it fom pap into Mathca I won't giv it to you on th quiz, but know how to o it fo th xam Poblm 6 S Figu 6 Th magnitu of L is L an th z-componnt

More information

University of Bahrain College of Science Dept. of Physics PHYCS 102 FINAL EXAM

University of Bahrain College of Science Dept. of Physics PHYCS 102 FINAL EXAM Unversty o Bahran College o Scence Dept. o Physcs PHYCS 10 FINAL XAM Date: 15/1/001 Tme:Two Hours Name:-------------------------------------------------ID#---------------------- Secton:----------------

More information

Ερωτήσεις και ασκησεις Κεφ. 10 (για μόρια) ΠΑΡΑΔΟΣΗ 29/11/2016. (d)

Ερωτήσεις και ασκησεις Κεφ. 10 (για μόρια) ΠΑΡΑΔΟΣΗ 29/11/2016. (d) Ερωτήσεις και ασκησεις Κεφ 0 (για μόρια ΠΑΡΑΔΟΣΗ 9//06 Th coffcnt A of th van r Waals ntracton s: (a A r r / ( r r ( (c a a a a A r r / ( r r ( a a a a A r r / ( r r a a a a A r r / ( r r 4 a a a a 0 Th

More information

Math 656 March 10, 2011 Midterm Examination Solutions

Math 656 March 10, 2011 Midterm Examination Solutions Math 656 March 0, 0 Mdtrm Eamnaton Soltons (4pts Dr th prsson for snh (arcsnh sng th dfnton of snh w n trms of ponntals, and s t to fnd all als of snh (. Plot ths als as ponts n th compl plan. Mak sr or

More information

Molecules and electronic, vibrational and rotational structure

Molecules and electronic, vibrational and rotational structure Molculs and ctonic, ational and otational stuctu Max on ob 954 obt Oppnhim Ghad Hzbg ob 97 Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs Hamiltonian fo a molcul h h H i m M i V i fs to ctons, to

More information

Physics Exam 3

Physics Exam 3 Physcs 114 1 Exam 3 The numbe of ponts fo each secton s noted n backets, []. Choose a total of 35 ponts that wll be gaded that s you may dop (not answe) a total of 5 ponts. Clealy mak on the cove of you

More information

(A) the function is an eigenfunction with eigenvalue Physical Chemistry (I) First Quiz

(A) the function is an eigenfunction with eigenvalue Physical Chemistry (I) First Quiz 96- Physcl Chmstry (I) Frst Quz lctron rst mss m 9.9 - klogrm, Plnck constnt h 6.66-4 oul scon Sp of lght c. 8 m/s, lctron volt V.6-9 oul. Th functon F() C[cos()+sn()] s n gnfuncton of /. Th gnvlu s (A)

More information

TEST-03 TOPIC: MAGNETISM AND MAGNETIC EFFECT OF CURRENT Q.1 Find the magnetic field intensity due to a thin wire carrying current I in the Fig.

TEST-03 TOPIC: MAGNETISM AND MAGNETIC EFFECT OF CURRENT Q.1 Find the magnetic field intensity due to a thin wire carrying current I in the Fig. TEST-03 TPC: MAGNETSM AND MAGNETC EFFECT F CURRENT Q. Fnd the magnetc feld ntensty due to a thn we cayng cuent n the Fg. - R 0 ( + tan) R () 0 ( ) R 0 ( + ) R 0 ( + tan ) R Q. Electons emtted wth neglgble

More information

Chapter 23: Magnetic Field Shielding

Chapter 23: Magnetic Field Shielding ELECTROMAGNETIC COMPATIBILITY ANDBOOK 1 Chapt : Magntc Fld Shldng.1 Usng th Bt-Savat law, vfy th magntc fld xpssn X (pvdd by yu nstuct) gvn n th cunt dstbutns and th magntc flds tabl n ths chapt.. Usng

More information

T h e C S E T I P r o j e c t

T h e C S E T I P r o j e c t T h e P r o j e c t T H E P R O J E C T T A B L E O F C O N T E N T S A r t i c l e P a g e C o m p r e h e n s i v e A s s es s m e n t o f t h e U F O / E T I P h e n o m e n o n M a y 1 9 9 1 1 E T

More information

Signal Circuit and Transistor Small-Signal Model

Signal Circuit and Transistor Small-Signal Model Snal cut an anto Sall-Snal Mol Lctu not: Sc. 5 Sa & Sth 6 th E: Sc. 5.5 & 6.7 Sa & Sth 5 th E: Sc. 4.6 & 5.6 F. Najaba EE65 Wnt 0 anto pl lopnt Ba & Snal Ba Snal only Ba Snal - Ba? MOS... : : S... MOS...

More information

ELECTROMAGNETISM. at a point whose position vector with respect to a current element i d l is r. According to this law :

ELECTROMAGNETISM. at a point whose position vector with respect to a current element i d l is r. According to this law : ELECTROMAGNETISM ot-svt Lw: Ths w s used to fnd the gnetc fed d t pont whose poston vecto wth espect to cuent eeent d s. Accodng to ths w : µ d ˆ d = 4π d d The tot fed = d θ P whee ˆ s unt vecto n the

More information

Chapter Fifiteen. Surfaces Revisited

Chapter Fifiteen. Surfaces Revisited Chapte Ffteen ufaces Revsted 15.1 Vecto Descpton of ufaces We look now at the vey specal case of functons : D R 3, whee D R s a nce subset of the plane. We suppose s a nce functon. As the pont ( s, t)

More information

English Made Easy: Foundation Book 1 Notes for parents

English Made Easy: Foundation Book 1 Notes for parents a nh Ma ay: Fnan 1 pan h b n hp y ch an ay an by cn n h n n ach h n h aphab. h h achn an ca phnc. h nan, achn an wn ac w nca y ch an h na ach, a w a h n n ach a an hw wn n h pa. y cpn h pa h b, y ch w

More information

Multi-linear Systems and Invariant Theory. in the Context of Computer Vision and Graphics. Class 4: Mutli-View 3D-from-2D. CS329 Stanford University

Multi-linear Systems and Invariant Theory. in the Context of Computer Vision and Graphics. Class 4: Mutli-View 3D-from-2D. CS329 Stanford University Mult-lna Sytm and Invaant hoy n th Contxt of Comut Von and Gahc Cla 4: Mutl-Vw 3D-fom-D CS39 Stanfod Unvty Amnon Shahua Cla 4 Matal W Wll Cov oday Eola Gomty and Fundamntal Matx h lan+aallax modl and latv

More information

19 The Born-Oppenheimer Approximation

19 The Born-Oppenheimer Approximation 9 The Bon-Oppenheme Appoxmaton The full nonelatvstc Hamltonan fo a molecule s gven by (n a.u.) Ĥ = A M A A A, Z A + A + >j j (883) Lets ewte the Hamltonan to emphasze the goal as Ĥ = + A A A, >j j M A

More information

2/4/2012. τ = Reasoning Strategy 1. Select the object to which the equations for equilibrium are to be applied. Ch 9. Rotational Dynamics

2/4/2012. τ = Reasoning Strategy 1. Select the object to which the equations for equilibrium are to be applied. Ch 9. Rotational Dynamics /4/ Ch 9. Rtatna Dynamcs In pue tansatna mtn, a pnts n an bject tae n paae paths. ces an Tques Net ce acceeatn. What causes an bject t hae an angua acceeatn? TORQUE 9. The ctn ces an Tques n Rg Objects

More information

8 - GRAVITATION Page 1

8 - GRAVITATION Page 1 8 GAVITATION Pag 1 Intoduction Ptolmy, in scond cntuy, gav gocntic thoy of plantay motion in which th Eath is considd stationay at th cnt of th univs and all th stas and th plants including th Sun volving

More information

The Forming Theory and the NC Machining for The Rotary Burs with the Spectral Edge Distribution

The Forming Theory and the NC Machining for The Rotary Burs with the Spectral Edge Distribution oden Appled Scence The Fomn Theoy and the NC achnn fo The Rotay us wth the Spectal Ede Dstbuton Huan Lu Depatment of echancal Enneen, Zhejan Unvesty of Scence and Technoloy Hanzhou, c.y. chan, 310023,

More information

Rotary motion

Rotary motion ectue 8 RTARY TN F THE RGD BDY Notes: ectue 8 - Rgd bod Rgd bod: j const numbe of degees of feedom 6 3 tanslatonal + 3 ota motons m j m j Constants educe numbe of degees of feedom non-fee object: 6-p

More information

Midterm Exam. CS/ECE 181B Intro to Computer Vision. February 13, :30-4:45pm

Midterm Exam. CS/ECE 181B Intro to Computer Vision. February 13, :30-4:45pm Nam: Midtm am CS/C 8B Into to Comput Vision Fbua, 7 :-4:45pm las spa ouslvs to th dg possibl so that studnts a vnl distibutd thoughout th oom. his is a losd-boo tst. h a also a fw pags of quations, t.

More information

SIMPLIFICATIONS OF SYNCHRONOUS MACHINE PARAMETERS IN STABILITY STUDIES AND REACTIVE CAPABILITY LIMITS

SIMPLIFICATIONS OF SYNCHRONOUS MACHINE PARAMETERS IN STABILITY STUDIES AND REACTIVE CAPABILITY LIMITS Rgua pap SIMPIFICATIONS OF SYNCHRONOUS MACHINE PARAMETERS IN STABIITY STUDIES AND REACTIVE CAPABIITY IMITS Sjan MAZAICA, Mćo GAĆANOVIĆ 2 Abstact: In ths pap, bfy, th pobm of pow sytm stabty s cons. Aft

More information

Rigid Bodies: Equivalent Systems of Forces

Rigid Bodies: Equivalent Systems of Forces Engneeng Statcs, ENGR 2301 Chapte 3 Rgd Bodes: Equvalent Sstems of oces Intoducton Teatment of a bod as a sngle patcle s not alwas possble. In geneal, the se of the bod and the specfc ponts of applcaton

More information

Jackson 3.3 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jackson 3.3 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell Jackson 3.3 Homewok Pobem Soution D. Chistophe S. Baid Univesity of Massachusetts Lowe POBLEM: A thin, fat, conducting, cicua disc of adius is ocated in the x-y pane with its cente at the oigin, and is

More information

the output is Thus, the output lags in phase by θ( ωo) radians Rewriting the above equation we get

the output is Thus, the output lags in phase by θ( ωo) radians Rewriting the above equation we get Th output y[ of a frquncy-sctiv LTI iscrt-tim systm with a frquncy rspons H ( xhibits som ay rativ to th input caus by th nonro phas rspons θ( ω arg{ H ( } of th systm For an input A cos( ωo n + φ, < n

More information

dt d Chapter 30: 1-Faraday s Law of induction (induced EMF) Chapter 30: 1-Faraday s Law of induction (induced Electromotive Force)

dt d Chapter 30: 1-Faraday s Law of induction (induced EMF) Chapter 30: 1-Faraday s Law of induction (induced Electromotive Force) Chaptr 3: 1-Faraday s aw of induction (inducd ctromotiv Forc) Variab (incrasing) Constant Variab (dcrasing) whn a magnt is movd nar a wir oop of ara A, currnt fows through that wir without any battris!

More information

ES 330 Electronics II Homework # 5 (Fall 2016 Due Wednesday, October 4, 2017)

ES 330 Electronics II Homework # 5 (Fall 2016 Due Wednesday, October 4, 2017) Pag1 Na olutions E 33 Elctonics II Howok # 5 (Fall 216 Du Wdnsday, Octob 4, 217) Pobl 1 (25 pots) A coon-itt aplifi uss a BJT with cunt ga = 1 whn biasd at I =.5 A. It has a collcto sistanc of = 1 k. (a)

More information

PHY126 Summer Session I, 2008

PHY126 Summer Session I, 2008 PHY6 Summe Sesson I, 8 Most of nfomaton s avalable at: http://nngoup.phscs.sunsb.edu/~chak/phy6-8 ncludng the sllabus and lectue sldes. Read sllabus and watch fo mpotant announcements. Homewok assgnment

More information

Filter Design Techniques

Filter Design Techniques Fltr Dsgn chnqus Fltr Fltr s systm tht psss crtn frquncy componnts n totlly rcts ll othrs Stgs of th sgn fltr Spcfcton of th sr proprts of th systm ppromton of th spcfcton usng cusl scrt-tm systm Rlzton

More information

Hydrogen atom. Energy levels and wave functions Orbital momentum, electron spin and nuclear spin Fine and hyperfine interaction Hydrogen orbitals

Hydrogen atom. Energy levels and wave functions Orbital momentum, electron spin and nuclear spin Fine and hyperfine interaction Hydrogen orbitals Hydogn atom Engy lvls and wav functions Obital momntum, lcton spin and nucla spin Fin and hypfin intaction Hydogn obitals Hydogn atom A finmnt of th Rydbg constant: R ~ 109 737.3156841 cm -1 A hydogn mas

More information

2/24/2014. The point mass. Impulse for a single collision The impulse of a force is a vector. The Center of Mass. System of particles

2/24/2014. The point mass. Impulse for a single collision The impulse of a force is a vector. The Center of Mass. System of particles /4/04 Chapte 7 Lnea oentu Lnea oentu of a Sngle Patcle Lnea oentu: p υ It s a easue of the patcle s oton It s a vecto, sla to the veloct p υ p υ p υ z z p It also depends on the ass of the object, sla

More information

GRAVITATION 4) R. max. 2 ..(1) ...(2)

GRAVITATION 4) R. max. 2 ..(1) ...(2) GAVITATION PVIOUS AMCT QUSTIONS NGINING. A body is pojctd vtically upwads fom th sufac of th ath with a vlocity qual to half th scap vlocity. If is th adius of th ath, maximum hight attaind by th body

More information

Chapter 23: Electric Potential

Chapter 23: Electric Potential Chapte 23: Electc Potental Electc Potental Enegy It tuns out (won t show ths) that the tostatc foce, qq 1 2 F ˆ = k, s consevatve. 2 Recall, fo any consevatve foce, t s always possble to wte the wok done

More information

III. Electromechanical Energy Conversion

III. Electromechanical Energy Conversion . Electoancal Enegy Coneson Schematc epesentaton o an toancal enegy coneson ece coppe losses coe losses (el losses) ancal losses Deental enegy nput om tcal souce: W V t Rt e t t W net ancal enegy output

More information

Field due to a collection of N discrete point charges: r is in the direction from

Field due to a collection of N discrete point charges: r is in the direction from Physcs 46 Fomula Shee Exam Coulomb s Law qq Felec = k ˆ (Fo example, f F s he elecc foce ha q exes on q, hen ˆ s a un veco n he decon fom q o q.) Elecc Feld elaed o he elecc foce by: Felec = qe (elecc

More information

A Note on Estimability in Linear Models

A Note on Estimability in Linear Models Intrnatonal Journal of Statstcs and Applcatons 2014, 4(4): 212-216 DOI: 10.5923/j.statstcs.20140404.06 A Not on Estmablty n Lnar Modls S. O. Adymo 1,*, F. N. Nwob 2 1 Dpartmnt of Mathmatcs and Statstcs,

More information

c- : r - C ' ',. A a \ V

c- : r - C ' ',. A a \ V HS PAGE DECLASSFED AW EO 2958 c C \ V A A a HS PAGE DECLASSFED AW EO 2958 HS PAGE DECLASSFED AW EO 2958 = N! [! D!! * J!! [ c 9 c 6 j C v C! ( «! Y y Y ^ L! J ( ) J! J ~ n + ~ L a Y C + J " J 7 = [ " S!

More information

Winnie flies again. Winnie s Song. hat. A big tall hat Ten long toes A black magic wand A long red nose. nose. She s Winnie Winnie the Witch.

Winnie flies again. Winnie s Song. hat. A big tall hat Ten long toes A black magic wand A long red nose. nose. She s Winnie Winnie the Witch. Wnn f gn ht Wnn Song A g t ht Tn ong to A k g wnd A ong d no. no Sh Wnn Wnn th Wth. y t d to A ong k t Bg gn y H go wth Wnn Whn h f. wnd ootk H Wu Wu th t. Ptu Dtony oo hopt oon okt hng gd ho y ktod nh

More information

E F. and H v. or A r and F r are dual of each other.

E F. and H v. or A r and F r are dual of each other. A Duality Thom: Consid th following quations as an xampl = A = F μ ε H A E A = jωa j ωμε A + β A = μ J μ A x y, z = J, y, z 4π E F ( A = jω F j ( F j β H F ωμε F + β F = ε M jβ ε F x, y, z = M, y, z 4π

More information

Xa.Ktlyn@nam.u ERASMUS MUNDUS MASTER STEPS 0&0//0 Intoucton to Vcto Contol of Pmannt Magnt Synchonou Machn ung Engtc Macocopc Rpntaton Xa.Ktlyn@nam.u Aocat Pofo n Elctcal Engnng PhD - HR At t Mét PaTch

More information

Basic Electrical Engineering for Welding [ ] --- Introduction ---

Basic Electrical Engineering for Welding [ ] --- Introduction --- Basc Elctrcal Engnrng for Wldng [] --- Introducton --- akayosh OHJI Profssor Ertus, Osaka Unrsty Dr. of Engnrng VIUAL WELD CO.,LD t-ohj@alc.co.jp OK 15 Ex. Basc A.C. crcut h fgurs n A-group show thr typcal

More information

Fourier transforms (Chapter 15) Fourier integrals are generalizations of Fourier series. The series representation

Fourier transforms (Chapter 15) Fourier integrals are generalizations of Fourier series. The series representation Pof. D. I. Nass Phys57 (T-3) Sptmb 8, 03 Foui_Tansf_phys57_T3 Foui tansfoms (Chapt 5) Foui intgals a gnalizations of Foui sis. Th sis psntation a0 nπx nπx f ( x) = + [ an cos + bn sin ] n = of a function

More information

Kinetics. Central Force Motion & Space Mechanics

Kinetics. Central Force Motion & Space Mechanics Kintics Cntal Foc Motion & Spac Mcanics Outlin Cntal Foc Motion Obital Mcanics Exampls Cntal-Foc Motion If a paticl tavls un t influnc of a foc tat as a lin of action ict towas a fix point, tn t motion

More information

Advanced Quantum Mechanics

Advanced Quantum Mechanics Advanced Quantum Mechanics Rajdeep Sensama sensama@theoy.tif.es.in Scatteing Theoy Ref : Sakuai, Moden Quantum Mechanics Tayo, Quantum Theoy of Non-Reativistic Coisions Landau and Lifshitz, Quantum Mechanics

More information

Chapter 8. Linear Momentum, Impulse, and Collisions

Chapter 8. Linear Momentum, Impulse, and Collisions Chapte 8 Lnea oentu, Ipulse, and Collsons 8. Lnea oentu and Ipulse The lnea oentu p of a patcle of ass ovng wth velocty v s defned as: p " v ote that p s a vecto that ponts n the sae decton as the velocty

More information

Why CEHCH? Completion of this program provides participants direct access to sit for the NAB HCBS exam.

Why CEHCH? Completion of this program provides participants direct access to sit for the NAB HCBS exam. Wy? T fus f s pns n us s p ssn pnns f w-bn & uny bs ss pg. Ts us s p n n ff s s bs p wn & uny bs ss pfssn. W uny n s f O s n n ns qu f sp u D, sussfu pn f s n us w nb yu ns n knwg bs. T O un f & sp n O

More information

Physics 202, Lecture 5. Today s Topics. Announcements: Homework #3 on WebAssign by tonight Due (with Homework #2) on 9/24, 10 PM

Physics 202, Lecture 5. Today s Topics. Announcements: Homework #3 on WebAssign by tonight Due (with Homework #2) on 9/24, 10 PM Physics 0, Lctu 5 Today s Topics nnouncmnts: Homwok #3 on Wbssign by tonight Du (with Homwok #) on 9/4, 10 PM Rviw: (Ch. 5Pat I) Elctic Potntial Engy, Elctic Potntial Elctic Potntial (Ch. 5Pat II) Elctic

More information

Physics 240: Worksheet 15 Name

Physics 240: Worksheet 15 Name Physics 40: Woksht 15 Nam Each of ths poblms inol physics in an acclatd fam of fnc Althouh you mind wants to ty to foc you to wok ths poblms insid th acclatd fnc fam (i.. th so-calld "won way" by som popl),

More information

6. Introduction to Transistor Amplifiers: Concepts and Small-Signal Model

6. Introduction to Transistor Amplifiers: Concepts and Small-Signal Model 6. ntoucton to anssto mples: oncepts an Small-Sgnal Moel Lectue notes: Sec. 5 Sea & Smth 6 th E: Sec. 5.4, 5.6 & 6.3-6.4 Sea & Smth 5 th E: Sec. 4.4, 4.6 & 5.3-5.4 EE 65, Wnte203, F. Najmaba Founaton o

More information

The Hyperelastic material is examined in this section.

The Hyperelastic material is examined in this section. 4. Hyprlastcty h Hyprlastc matral s xad n ths scton. 4..1 Consttutv Equatons h rat of chang of ntrnal nrgy W pr unt rfrnc volum s gvn by th strss powr, whch can b xprssd n a numbr of dffrnt ways (s 3.7.6):

More information