Plasma Internal Inductance in Presence of Toroidal Field Ripple of Tokamak

Size: px
Start display at page:

Download "Plasma Internal Inductance in Presence of Toroidal Field Ripple of Tokamak"

Transcription

1 Journl of Nucler nd Prticle Physics 13, 3(4): O: 1.593/j.jn Plsm nternl nductnce in Presence of Toroidl Field Rile of Tokmk A. Slr Elhi *, M. Ghornneviss Plsm Physics Reserch Center, Science nd Reserch Brnch, slmic Azd University, Tehrn, rn Astrct n this reserch we investigted the effects of toroidl field rile of tokmk on the lsm internl inductnce. For this urose, rry of mgnetic roes nd lso dimgnetic loo with its comenstion coil were designed, constructed, nd instlled on outer surfce of the R-T1 tokmk. Amlitude of the TF rile is otined.1, nd lso the effect of the TF rile on the lsm internl inductnce ws discussed. n the high field side region of tokmk chmer, the TF rile effect is incresing of the lsm internl inductnce, wheres the low field side hs inverse sitution. Keywords Tokmk, Toroidl Field Rile, Pls m nternl nductnce 1. ntroduction Usully tokmks lsm equiliri re investigted s two-dimensionl (xisymmetric) systems. Although this symmetry offers mny dvntges for its nlysis, ut relistic tokmks consists of finite numer of Toroidl Field (TF) coils. Then, this discreteness yields the toroidl field riles ( eriodic vrition of the toroidl mgnetic field). n other words, relistic tokmks could not e xisymmetric configurtions. Most of the TF rile studies hve een done on effects of the TF rile on confinement of the high energy lh rticles, formtion of internl trnsort rriers, lsm rottion, nd H-mode erformnce. n R-T1 Tokmk, which is smll, low Bet nd lrge sect rtio tokmk with circulr cross section (see Tle 1), the numer N of TF coils is 16, nd then the eriod of the TF rile ws.5. n this er we resent the effects of the TF rile on the ls m internl inductnce in R-T1 tokmk. etermintion of the internl inductnce is essentil for tokmk exeriments nd otimized oertion. Also some of the lsm informtion cn e deduced from this rmeter, such s lsm toroidl current rofile. Mgnetic dignostics, in rticulr dimgnetic loo (toroidl flux loo) re commonly used in tokmks to mesure the vrition of toroidl flux induced y the lsm nd then the oloidl Bet. On the other hnd, the mgnetic fields dis triution outs ide the ls m rovides the mesurement of the comintion of oloid l Bet nd internl inductnce, vi the Shfrnov rmeter ( ). Then mesurement of from the mgnetic roes nd oloidl * Corresonding uthor: Slri_hy@yhoo.com (A. Slr Elhi) Pulished online t htt://journl.su.org/jn Coyright 13 Scientific & Acdemic Pulishing. All Rights Reserved Bet from dimgnetic loo gives vlue of internl inductnce[1-65]. n this er we resent exerimentl investigtion of the TF rile on this rmeter. Becuse of deendence of the toroidl field on the TF rile mlitude, therefore we exect tht this rmeter is lso deending on TF rile mlitude. Brief roch for determintions of the TF rile nd Shfrnov rmeter using the discrete mgnetic roes will e resent in section. imgnetic loo method for mesurement of the oloidl Bet nd internl inductnce will resent in section 3. Exerimentl results of effects of TF rile on the lsm internl inductnce will discuss in section 4. Summry nd discussion will resent in section 5. Tle 1. Min Prmeters of the R-T1 Tokmk Prmeters Mjor Rdius Minor Rdius Toroidl Field Plsm Current ischrge urtion Vlue 45 cm 1.5 cm 1. T 4 ka 35 ms Electron ensity cm 3 Toroidl Field Coils 16. etermintions of the TF Rile nd Shfrnov Prmeter Using the iscrete Mgnetic Proes A simle nlytic model of the toroidl mgnetic field strength widely used in the nlysis is[1]: B, B 1 cos - cos N, (1)

2 1 A. Slr Elhi et l.: Plsm nternl nductnce in Presence of Toroidl Field Rile of Tokmk where B is the toroidl mgnetic field t center of the tokmk chmer, nd re oloidl nd toroidl ngles resectively, is the inverse sect rtio, N is the numer of the toroidl field coils, nd is the mlitude of the TF rile where defined s: B Bmx Bmin. () B B B n the R-T1 the numer of TF coils is 16, then the eriod of the TF rile ws.5, nd the inverse sect rtio is.78. From the Eq. (1) we cn write: 1 B, / N B, 4 B (3) 1 B, / N B,, 4 B where these vlues of the toroidl mgnetic fields cn e determined using the mgnetic roes t ove oloidl nd toroidl ngles. Our mesurements show tht the mlitude of the TF rile in R-T1 is.1, s shown in Fig. (1). mx min where R is the mjo r rdius of the vcuum vessel, the Shfrnov shift, is s is the lsm current, nd re the minor lsm rdius nd minor chmer rdius resectively. These equtions ccurte for low lsm, lrge sect rtio, nd circulr cross section tokmks s R-T1, nd where: where nd where l B B ln i / 1 R B B ( B ( ) B ( ) B B ), ( ), 3 ( ), (6) (7) is the oloidl Bet nd l i is the ls m internl inductnce. We cn otin B nd B fter comensting nd integrting of outut signls of the mgnetic roes. The comenstion done y fields dischrge without lsm nd receives outut signls of the mgnetic roes nd sutrct those from totl outut signls. Exerimentl results will resent in the section 4. Figure (1). eendence of the Toroidl Mgnetic Field on the Poloidl nd Toroidl Angles, TF Rile is lso oservle Also the Shfrnov rmeter relte to the distriution of mgnetic fields round the lsm current. Therefore, those cn e written in terms of the tngentil nd norml comonents of the mgnetic field on the contour (see Fig. ()). istriutions of the tngentil nd norml mgnetic fields re lso cn e written in the first order of the inverse sect rtio s follows, resectively[,3,5]: B 4 R ln 1 1 B 4 R R 1 1 R ln 1 s sin s (4) cos, (5) 3. etermintion of the Plsm nternl nductnce Using the imgnetic Loo The toroidl flux tht roduced y the lsm is relted to the totl erendiculr therml energy of the lsm. This dimgnetic flux is usully mesured with the dimgnetic loo. n cses of the ohmiclly heted tokmks (low et) where the lsm energy density is smll comred to the energy density of the mgnetic field, the chnge in the totl toroidl mgnetic flux is smll. Therefore reference signl equl to the vcuum toroidl mgnetic flux is usully sutrcted from it, giving net toroidl flux equl to the dimgnetic flu x roduced y the circulr lsm. Reltion etween the dimgnetic flux nd the oloidl et derived from simlified equilirium reltion[-7]: totl vcuum 1, (8) 8 B y sustituting the Eq. (1) in the Eq. (8) we hve: 8 B (1 cos - cos N ) 1, (9) where vcuum T O V E, nd where B is the toroidl mgnetic field in the sence of the lsm wh ich cn e otined y the mgnetic roe, is the lsm current which cn e otined y the Rogowski coil, T is the toroidl flux ecuse of toroidl field coils,

3 Journl of Nucler nd Prticle Physics 13, 3(4): O nd V re the ssing flux through loo due to ossile mislignment etween ohmic field nd verticl field nd the dimgnetic loo, nd is the toroidl field due E to eddy current on the vcuum chmer. These fluxes cn e comensted either with comenstion coil or fields dischrge without lsm. t must e noted tht comensting coil for dimgnetic loo is wred out of the lsm current, nd only the toroidl flux (which is induced y the chnge of toroidl field coil current when lsm dischrges) cn e received. So the dimgnetic flu x cused y lsm current cn e mesured from the dimgnetic nd comensting coil using sutrction. Therefore, ccording to ove two sections we cn find the internl inductnce. From Eq. (6) we hve: li 1 (1) By sustituting the Eq. (6) nd (9) in Eq. (1), we cn write: li ln 16 B R ( B 1 cos - cos N B ), (11) where the effect of the TF rile introduced in the Shfrnov rmeter. Exerimentl results of effects of the TF rile on the internl inductnce will resent in next section. 4. Exerimentl Results According to ove discussion, we determined the lsm internl inductnce nd the effects of TF rile on it. Results resent in Figs. () nd (3). As shown, the difference etween the internl inductnce in resence of the TF rile nd in sence of the TF rile is in order of the 1, nd in the high field side region of tokmk chmer, the TF rile effect is incresing of the lsm internl inductnce, wheres the low field side hs inverse sitution. 5. Summry nd iscussion n this reserch we investigted the effects of TF rile on the lsm internl inductnce in R-T1 Tokmk. For this urose, rry of mgnetic roes nd lso dimgnetic loo with its comenstion coil were designed, constructed, nd instlled on the outer surfce of the R-T1. Amlitude of the TF rile is otined.1, nd lso the effects of the TF rile on the ls m internl inductnce resented. One of the results is tht the difference etween the internl inductnce in resence of the TF rile nd in sence of the TF rile is in order of the 1, nd lso in the high field side region the difference is ositive, wheres in low field side the difference is negtive. n other words, in the high field side region of tokmk chmer, the TF ri le effect is incresing of the ls m internl inductnce, wheres the low field side hs inverse sitution. Figure (). Prmeters in sence of the TF rile, () lsm current, () oloidl Bet, (c) internl inductnce, nd (d) Horizontl islcement (H..)

4 14 A. Slr Elhi et l.: Plsm nternl nductnce in Presence of Toroidl Field Rile of Tokmk Figure (3). Effects of the TF rile mlitude on the difference of internl inductnce with nd without TF rile (li) t different oloidl ngles. As shown, difference etween the internl inductnce in resent of the TF rile nd in sent of the TF rile is in order of the1. Also in the high field o side region ( 18 ) the difference is ositive, ut in low field side ( ) the difference is negtive o REFERENCES [1] K. Nkmur nd M. Ghornneviss, Fusion Eng. es (3) [] V. S. Mukhovtov nd V.. Shfrnov, Nucl. Fusion 11, (1971), 65 [3]. P. Shkrofsky, Phys. Fluids 5 (1) 89-96, (198). [4] G.S. Lee nd nd M. Ghornneviss, Nucl. Fusion 41, 1515 (1) [5] S.H. Seo, Phys. Plsms 16, 351 (9) [6] A. Slr Elhi nd M. Ghornneviss, EEE Trns. Plsm Science 38 (), , (1) [7] A. Slr Elhi nd M. Ghornneviss, EEE Trns. Plsm Science 38 (9), , (1) [8] A. Slr Elhi nd M. Ghornneviss, J. Plsm Physics 76 (1), 1-8, (9) [9] A. Slr Elhi nd M. Ghornneviss, Fusion Engineering nd esign 85, 74 77, (1) [1] A. Slr Elhi nd M. Ghornneviss, Phys. Scrit 8, 4551, (9) [11] A. Slr Elhi nd M. Ghornneviss, Phys. Scrit 8, 555, (9) [1] A. Slr Elhi nd M. Ghornneviss, Phys. Scrit 81 (5), 5551, (1) [13] A. Slr Elhi nd M. Ghornneviss, Phys. Scrit 8, 55, (1) [14] M. Ghornneviss, A. Slr Elhi, Phys. Scrit 8 (3), 355, (1) [15] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 8 (4), , (9) [16] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 8 (4), , (9) [17] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 8 (4), , (9) [18] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 8 (4), , (9) [19] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 8 (4), , (9) [] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 8 (4), 44-47, (9) [1] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 8 (4), , (9) [] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 8 (4), , (9) [3] A. Rhimi Rd, M. Ghornneviss, M. Emmi, nd A. Slr Elhi, J. Fusion Energy 8 (4), 4-46, (9) [4] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 9 (1), 1-4, (1) [5] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 9 (1), -5, (1)

5 Journl of Nucler nd Prticle Physics 13, 3(4): [6] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 9 (1), 9-31, (1) [7] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 9 (1), 6-8, (1) [8] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 9 (1), 3-35, (1) [9] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 9 (1), 36-4, (1) [3] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 9 (1), 6-64, (1) [31] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 9 (1), 76-8, (1) [3] A. Rhimi Rd, M. Emmi, M. Ghornneviss, A. Slr Elhi, J. Fusion Energy 9 (1), 73-75, (1) [33] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 9 (1), 83-87, (1) [34] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 9 (1), 88-93, (1) [35] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 9 (3), 9-14, (1) [36] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 9 (3), 3-36, (1) [37] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 9 (3), 51-55, (1) [38] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 9 (3), 79-84, (1) [39] M. Ghornneviss, A. Slr Elhi, J. Fusion Energy 9 (5), , (1) [4] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 9 (5), , (1) [41] A. Slr Elhi nd M. Ghornneviss, Brzilin J. Physics 4 (3), 33-36, (1) [4] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 3 (), 116-1, (11) [43] M.R. Ghnri, M. Ghornneviss, A. Slr Elhi, Phys. Scrit 83, 5551, (11) [44] A. Slr Elhi, J. Fusion Energy 3 (6), , (11), [45] A. Slr Elhi nd M. Ghornneviss, Fusion Engineering nd esign 86, , (11) [46] A. Slr Elhi nd M. Ghornneviss, J. Fusion Energy 31 (), , (1) [47] M.R. Ghnri, M. Ghornneviss, A. Slr Elhi, R. Arvin nd S. Mohmmdi, Rdition Effects & efects in Solids 166 (1), , (11) [48] A. Slr Elhi nd M. Ghornneviss, EEE Trns. Plsm Science 4 (3), , (1) [49] Z. Goodrzi, M. Ghornneviss nd A. Slr Elhi, J. Fusion Energy 3 (1), 13-16, (13) [5] M.R. Ghnri, M. Ghornneviss, A. Slr Elhi, Phys. Scrit 85 (5), 555, (1) [51] A. Slr Elhi nd M. Ghornneviss, Rdition Effects nd efects in Solids 168(1), 4-47, (13) [5] K. Mikili Agh, M. Ghornneviss, A. Slr Elhi, J. Fusion Energy 3 (), 68-7, (13) [53] A. Slr Elhi nd M. Ghornneviss, Fusion Engineering nd esign 88 (), 94-99, (13) [54] A. Slr Elhi nd M. Ghornneviss, EEE Trnsctions on Plsm Science 41 (), , (13) [55] A. Slr Elhi nd M. Ghornneviss, J. of Fusion Energy 3, 496-5, (13) [56] A. Slr Elhi nd M. Ghornneviss, Review of Scientific nstruments 84 (5), 5354 (13) [57] A. Slr Elhi nd M. Ghornneviss, J. Nucler nd Prticle Physics 1(1), 1-15, (11) [58] A. Slr Elhi nd M. Ghornneviss, J. Nucler nd Prticle Physics (), 1-5, (1) [59] A. Slr Elhi nd M. Ghornneviss, J. Nucler nd Prticle Physics (), -5, (1) [6] A. Slr Elhi nd M. Ghornneviss, J. Nucler nd Prticle Physics (4), 91-97, (1) [61] A. Slr Elhi nd M. Ghornneviss, J. Nucler nd Prticle Physics (4), 11-16, (1) [6] A. Slr Elhi nd M. Ghornneviss, J. Nucler nd Prticle Physics (5), , (1) [63] A. Slr Elhi nd M. Ghornneviss, J. Nucler nd Prticle Physics (6), , (1) [64] A. Slr Elhi nd M. Ghornneviss, J. Nucler nd Prticle Physics 3(1), 1-7, (13) [65] A. Slr Elhi nd M. Ghornneviss, J. Nucler nd Prticle Physics 3(1), 14-19, (13)

Minimum Energy State of Plasmas with an Internal Transport Barrier

Minimum Energy State of Plasmas with an Internal Transport Barrier Minimum Energy Stte of Plsms with n Internl Trnsport Brrier T. Tmno ), I. Ktnum ), Y. Skmoto ) ) Formerly, Plsm Reserch Center, University of Tsukub, Tsukub, Ibrki, Jpn ) Plsm Reserch Center, University

More information

22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 7: The First Order Grad Shafranov Equation. dp 1 dp

22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 7: The First Order Grad Shafranov Equation. dp 1 dp First Order Eqution.65, MHD Theory of Fusion Systems Prof. Freiderg Lecture 7: The First Order Grd Shfrnov Eqution The first order Grd Shfrnov eqution is given y d p d dp d + μr + R B B = μr r cos + cos

More information

Joule-Thomson effect TEP

Joule-Thomson effect TEP Joule-homson effect EP elted oics el gs; intrinsic energy; Gy-Lussc theory; throttling; n der Wls eqution; n der Wls force; inverse Joule- homson effect; inversion temerture. Princile A strem of gs is

More information

CBSE Sample Paper 2. Question 6 The maximum KE of the electrons emitted in a photocell is 10eV. What is the stopping potential?

CBSE Sample Paper 2. Question 6 The maximum KE of the electrons emitted in a photocell is 10eV. What is the stopping potential? CBSE Smle Per 2 Generl Instruction:. Answer ll questions 2. Internl choices re rovided for some questions 3. Question numbers to 8 re very short nswer questions nd crry mrk ech. 4. Question numbers 8 to

More information

CONIC SECTIONS. Chapter 11

CONIC SECTIONS. Chapter 11 CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round

More information

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2 1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion

More information

PHYSICS ASSIGNMENT-9

PHYSICS ASSIGNMENT-9 MPS/PHY-XII-11/A9 PHYSICS ASSIGNMENT-9 *********************************************************************************************************** 1. A wire kept long the north-south direction is llowed

More information

Studies on Nuclear Fuel Rod Thermal Performance

Studies on Nuclear Fuel Rod Thermal Performance Avilble online t www.sciencedirect.com Energy Procedi 1 (1) 1 17 Studies on Nucler Fuel od herml Performnce Eskndri, M.1; Bvndi, A ; Mihndoost, A3* 1 Deprtment of Physics, Islmic Azd University, Shirz

More information

7.3 Problem 7.3. ~B(~x) = ~ k ~ E(~x)=! but we also have a reected wave. ~E(~x) = ~ E 2 e i~ k 2 ~x i!t. ~B R (~x) = ~ k R ~ E R (~x)=!

7.3 Problem 7.3. ~B(~x) = ~ k ~ E(~x)=! but we also have a reected wave. ~E(~x) = ~ E 2 e i~ k 2 ~x i!t. ~B R (~x) = ~ k R ~ E R (~x)=! 7. Problem 7. We hve two semi-innite slbs of dielectric mteril with nd equl indices of refrction n >, with n ir g (n ) of thickness d between them. Let the surfces be in the x; y lne, with the g being

More information

Reference. Vector Analysis Chapter 2

Reference. Vector Analysis Chapter 2 Reference Vector nlsis Chpter Sttic Electric Fields (3 Weeks) Chpter 3.3 Coulomb s Lw Chpter 3.4 Guss s Lw nd pplictions Chpter 3.5 Electric Potentil Chpter 3.6 Mteril Medi in Sttic Electric Field Chpter

More information

Machine Design II Prof. K.Gopinath & Prof. M.M.Mayuram. Drum Brakes. Among the various types of devices to be studied, based on their practical use,

Machine Design II Prof. K.Gopinath & Prof. M.M.Mayuram. Drum Brakes. Among the various types of devices to be studied, based on their practical use, chine Design II Prof. K.Gointh & Prof...yurm Drum Brkes Among the vrious tyes of devices to be studied, bsed on their rcticl use, the discussion will be limited to Drum brkes of the following tyes which

More information

MASKING OF FERROMAGNETIC ELLIPTICAL SHELL IN TRANSVERSE MAGNETIC FIELD

MASKING OF FERROMAGNETIC ELLIPTICAL SHELL IN TRANSVERSE MAGNETIC FIELD POZNAN UNVE RSTY OF TE HNOLOGY AADE M JOURNALS No 7 Electricl Engineering Kzimierz JAKUUK* Mirosł WOŁOSZYN* Peł ZMNY* MASKNG OF FERROMAGNET ELLPTAL SHELL N TRANSVERSE MAGNET FELD A ferromgnetic oject,

More information

Homework Assignment 6 Solution Set

Homework Assignment 6 Solution Set Homework Assignment 6 Solution Set PHYCS 440 Mrch, 004 Prolem (Griffiths 4.6 One wy to find the energy is to find the E nd D fields everywhere nd then integrte the energy density for those fields. We know

More information

Fully Kinetic Simulations of Ion Beam Neutralization

Fully Kinetic Simulations of Ion Beam Neutralization Fully Kinetic Simultions of Ion Bem Neutrliztion Joseph Wng University of Southern Cliforni Hideyuki Usui Kyoto University E-mil: josephjw@usc.edu; usui@rish.kyoto-u.c.jp 1. Introduction Ion em emission/neutrliztion

More information

Problem Solving 7: Faraday s Law Solution

Problem Solving 7: Faraday s Law Solution MASSACHUSETTS NSTTUTE OF TECHNOLOGY Deprtment of Physics: 8.02 Prolem Solving 7: Frdy s Lw Solution Ojectives 1. To explore prticulr sitution tht cn led to chnging mgnetic flux through the open surfce

More information

Industrial Electrical Engineering and Automation

Industrial Electrical Engineering and Automation CODEN:LUTEDX/(TEIE-719)/1-7/(7) Industril Electricl Engineering nd Automtion Estimtion of the Zero Sequence oltge on the D- side of Dy Trnsformer y Using One oltge Trnsformer on the D-side Frncesco Sull

More information

Edexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks

Edexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1

More information

Motion of Electrons in Electric and Magnetic Fields & Measurement of the Charge to Mass Ratio of Electrons

Motion of Electrons in Electric and Magnetic Fields & Measurement of the Charge to Mass Ratio of Electrons n eperiment of the Electron topic Motion of Electrons in Electric nd Mgnetic Fields & Mesurement of the Chrge to Mss Rtio of Electrons Instructor: 梁生 Office: 7-318 Emil: shling@bjtu.edu.cn Purposes 1.

More information

Physics 202, Lecture 14

Physics 202, Lecture 14 Physics 202, Lecture 14 Tody s Topics Sources of the Mgnetic Field (Ch. 28) Biot-Svrt Lw Ampere s Lw Mgnetism in Mtter Mxwell s Equtions Homework #7: due Tues 3/11 t 11 PM (4th problem optionl) Mgnetic

More information

Applications of Bernoulli s theorem. Lecture - 7

Applications of Bernoulli s theorem. Lecture - 7 Applictions of Bernoulli s theorem Lecture - 7 Prcticl Applictions of Bernoulli s Theorem The Bernoulli eqution cn be pplied to gret mny situtions not just the pipe flow we hve been considering up to now.

More information

7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?

7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement? 7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge

More information

Ch AP Problems

Ch AP Problems Ch. 7.-7. AP Prolems. Willy nd his friends decided to rce ech other one fternoon. Willy volunteered to rce first. His position is descried y the function f(t). Joe, his friend from school, rced ginst him,

More information

AN IMPROVED SMALL CLOSED DRIFT THRUSTER WITH BOTH CONDUCTING AND DIELECT RIC CHANNELS

AN IMPROVED SMALL CLOSED DRIFT THRUSTER WITH BOTH CONDUCTING AND DIELECT RIC CHANNELS AN IMPROVED SMALL CLOSED DRIFT THRUSTER WITH BOTH CONDUCTING AND DIELECT RIC CHANNELS A.I.Bugrov, A.D.Desitskov, H.R.Kufmn, V.K.Khrchevnikov, A.I.Morozov c, V.V.Zhurin d Moscow Institute of Rdioelectronics,

More information

#6A&B Magnetic Field Mapping

#6A&B Magnetic Field Mapping #6A& Mgnetic Field Mpping Gol y performing this lb experiment, you will: 1. use mgnetic field mesurement technique bsed on Frdy s Lw (see the previous experiment),. study the mgnetic fields generted by

More information

Families of Solutions to Bernoulli ODEs

Families of Solutions to Bernoulli ODEs In the fmily of solutions to the differentil eqution y ry dx + = it is shown tht vrition of the initil condition y( 0 = cuses horizontl shift in the solution curve y = f ( x, rther thn the verticl shift

More information

Physics 1402: Lecture 7 Today s Agenda

Physics 1402: Lecture 7 Today s Agenda 1 Physics 1402: Lecture 7 Tody s gend nnouncements: Lectures posted on: www.phys.uconn.edu/~rcote/ HW ssignments, solutions etc. Homework #2: On Msterphysics tody: due Fridy Go to msteringphysics.com Ls:

More information

Section 6: Area, Volume, and Average Value

Section 6: Area, Volume, and Average Value Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

More information

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus 7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e

More information

Phys 7221, Fall 2006: Homework # 6

Phys 7221, Fall 2006: Homework # 6 Phys 7221, Fll 2006: Homework # 6 Gbriel González October 29, 2006 Problem 3-7 In the lbortory system, the scttering ngle of the incident prticle is ϑ, nd tht of the initilly sttionry trget prticle, which

More information

in a uniform magnetic flux density B = Boa z. (a) Show that the electron moves in a circular path. (b) Find the radius r o

in a uniform magnetic flux density B = Boa z. (a) Show that the electron moves in a circular path. (b) Find the radius r o 6. THE TATC MAGNETC FELD 6- LOENTZ FOCE EQUATON Lorent force eqution F = Fe + Fm = q ( E + v B ) Exmple 6- An electron hs n initil velocity vo = vo y in uniform mgnetic flux density B = Bo. () how tht

More information

Electromagnetism Answers to Problem Set 10 Spring 2006

Electromagnetism Answers to Problem Set 10 Spring 2006 Electromgnetism 76 Answers to Problem Set 1 Spring 6 1. Jckson Prob. 5.15: Shielded Bifilr Circuit: Two wires crrying oppositely directed currents re surrounded by cylindricl shell of inner rdius, outer

More information

Thomas Whitham Sixth Form

Thomas Whitham Sixth Form Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos

More information

Method of Localisation and Controlled Ejection of Swarms of Likely Charged Particles

Method of Localisation and Controlled Ejection of Swarms of Likely Charged Particles Method of Loclistion nd Controlled Ejection of Swrms of Likely Chrged Prticles I. N. Tukev July 3, 17 Astrct This work considers Coulom forces cting on chrged point prticle locted etween the two coxil,

More information

KINEMATICS OF RIGID BODIES

KINEMATICS OF RIGID BODIES KINEMTICS OF RIGID ODIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. Description

More information

Phys. 506 Electricity and Magnetism Winter 2004 Prof. G. Raithel Problem Set 1 Total 30 Points. 1. Jackson Points

Phys. 506 Electricity and Magnetism Winter 2004 Prof. G. Raithel Problem Set 1 Total 30 Points. 1. Jackson Points Phys. 56 Electricity nd Mgnetism Winter 4 Prof. G. Rithel Prolem Set Totl 3 Points. Jckson 8. Points : The electric field is the sme s in the -dimensionl electrosttic prolem of two concentric cylinders,

More information

k ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola.

k ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola. Stndrd Eqution of Prol with vertex ( h, k ) nd directrix y = k p is ( x h) p ( y k ) = 4. Verticl xis of symmetry Stndrd Eqution of Prol with vertex ( h, k ) nd directrix x = h p is ( y k ) p( x h) = 4.

More information

Homework Assignment 3 Solution Set

Homework Assignment 3 Solution Set Homework Assignment 3 Solution Set PHYCS 44 6 Ferury, 4 Prolem 1 (Griffiths.5(c The potentil due to ny continuous chrge distriution is the sum of the contriutions from ech infinitesiml chrge in the distriution.

More information

ECE 3318 Applied Electricity and Magnetism. Spring Prof. David R. Jackson Dept. of ECE. Notes 31 Inductance

ECE 3318 Applied Electricity and Magnetism. Spring Prof. David R. Jackson Dept. of ECE. Notes 31 Inductance ECE 3318 Applied Electricity nd Mgnetism Spring 018 Prof. Dvid R. Jckson Dept. of ECE Notes 31 nductnce 1 nductnce ˆn S Single turn coil The current produces flux though the loop. Definition of inductnce:

More information

Mathematics. Area under Curve.

Mathematics. Area under Curve. Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding

More information

332:221 Principles of Electrical Engineering I Fall Hourly Exam 2 November 6, 2006

332:221 Principles of Electrical Engineering I Fall Hourly Exam 2 November 6, 2006 2:221 Principles of Electricl Engineering I Fll 2006 Nme of the student nd ID numer: Hourly Exm 2 Novemer 6, 2006 This is closed-ook closed-notes exm. Do ll your work on these sheets. If more spce is required,

More information

Trigonometric Functions

Trigonometric Functions Exercise. Degrees nd Rdins Chpter Trigonometric Functions EXERCISE. Degrees nd Rdins 4. Since 45 corresponds to rdin mesure of π/4 rd, we hve: 90 = 45 corresponds to π/4 or π/ rd. 5 = 7 45 corresponds

More information

1B40 Practical Skills

1B40 Practical Skills B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need

More information

(9) P (x)u + Q(x)u + R(x)u =0

(9) P (x)u + Q(x)u + R(x)u =0 STURM-LIOUVILLE THEORY 7 2. Second order liner ordinry differentil equtions 2.1. Recll some sic results. A second order liner ordinry differentil eqution (ODE) hs the form (9) P (x)u + Q(x)u + R(x)u =0

More information

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero

More information

Bend Forms of Circular Saws and Evaluation of their Mechanical Properties

Bend Forms of Circular Saws and Evaluation of their Mechanical Properties ISSN 139 13 MATERIALS SCIENCE (MEDŽIAGOTYRA). Vol. 11, No. 1. 5 Bend Forms of Circulr s nd Evlution of their Mechnicl Properties Kristin UKVALBERGIENĖ, Jons VOBOLIS Deprtment of Mechnicl Wood Technology,

More information

u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph.

u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph. nlyzing Dmped Oscilltions Prolem (Medor, exmple 2-18, pp 44-48): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + $

More information

ragsdale (zdr82) HW2 ditmire (58335) 1

ragsdale (zdr82) HW2 ditmire (58335) 1 rgsdle (zdr82) HW2 ditmire (58335) This print-out should hve 22 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. 00 0.0 points A chrge of 8. µc

More information

4 The dynamical FRW universe

4 The dynamical FRW universe 4 The dynmicl FRW universe 4.1 The Einstein equtions Einstein s equtions G µν = T µν (7) relte the expnsion rte (t) to energy distribution in the universe. On the left hnd side is the Einstein tensor which

More information

Chapter 9 Definite Integrals

Chapter 9 Definite Integrals Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

More information

A ROTATING DISC IN CONSTANT PURE SHEAR BY S. KUMAR AND C. V. JOGA RAO

A ROTATING DISC IN CONSTANT PURE SHEAR BY S. KUMAR AND C. V. JOGA RAO A ROTATING DISC IN CONSTANT PURE SHEAR BY S. KUMAR AND C. V. JOGA RAO (Deprtment of Aeronuticl Engineering, Indin Institute of Science, Bnglore-3) Received April 25, 1954 SUMMARY The disc of constnt pure

More information

ADVANCEMENT OF THE CLOSELY COUPLED PROBES POTENTIAL DROP TECHNIQUE FOR NDE OF SURFACE CRACKS

ADVANCEMENT OF THE CLOSELY COUPLED PROBES POTENTIAL DROP TECHNIQUE FOR NDE OF SURFACE CRACKS ADVANCEMENT OF THE CLOSELY COUPLED PROBES POTENTIAL DROP TECHNIQUE FOR NDE OF SURFACE CRACKS F. Tkeo 1 nd M. Sk 1 Hchinohe Ntionl College of Technology, Hchinohe, Jpn; Tohoku University, Sendi, Jpn Abstrct:

More information

AMPERE CONGRESS AMPERE on Magnetic Resonance and Related Phenomena. Under the auspices of The GROUPEMENT AMPERE

AMPERE CONGRESS AMPERE on Magnetic Resonance and Related Phenomena. Under the auspices of The GROUPEMENT AMPERE AMPERE 2000 th 30 CONGRESS AMPERE on Mgnetic Resonnce nd Relted Phenomen Lison, Portugl, 23-2 July 2000 Under the uspices of The GROUPEMENT AMPERE Edited y: A.F. MARTINS, A.G. FEIO nd J.G. MOURA Sponsoring

More information

Creating A New Planck s Formula of Spectral Density of Black-body Radiation by Means of AF(A) Diagram

Creating A New Planck s Formula of Spectral Density of Black-body Radiation by Means of AF(A) Diagram nd Jogj Interntionl Physics Conference Enhncing Network nd Collortion Developing Reserch nd Eduction in Physics nd Nucler Energy Septemer 6-9, 007, Yogykrt-Indonesi Creting A New Plnck s Formul of Spectrl

More information

arxiv:hep-ex/ v1 12 Sep 1998

arxiv:hep-ex/ v1 12 Sep 1998 Evidence of the φ ηπ γ decy rxiv:hep-ex/9891v1 12 Sep 1998 Astrct M.N.Achsov, V.M.Aulchenko, S.E.Bru, A.V.Berdyugin, A.V.Bozhenok, A.D.Bukin, D.A.Bukin, S.V.Burdin, T.V.Dimov, S.I.Dolinski, V.P.Druzhinin,

More information

Thermal Stability of Ti-C-Ni-Cr and Ti-C-Ni-Cr-Al-Si Nanocomposite Coatings

Thermal Stability of Ti-C-Ni-Cr and Ti-C-Ni-Cr-Al-Si Nanocomposite Coatings Journl of Physics: Conference Series PAPER OPEN ACCESS Therml Stility of Ti-C-Ni-Cr nd Ti-C-Ni-Cr-Al-Si Nnocomposite Cotings To cite this rticle: A V Andreev et l 2015 J. Phys.: Conf. Ser. 652 012057 View

More information

KINEMATICS OF RIGID BODIES

KINEMATICS OF RIGID BODIES KINEMTICS OF RIGI OIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. escription

More information

AN020. a a a. cos. cos. cos. Orientations and Rotations. Introduction. Orientations

AN020. a a a. cos. cos. cos. Orientations and Rotations. Introduction. Orientations AN020 Orienttions nd Rottions Introduction The fct tht ccelerometers re sensitive to the grvittionl force on the device llows them to be used to determine the ttitude of the sensor with respect to the

More information

Network Analysis and Synthesis. Chapter 5 Two port networks

Network Analysis and Synthesis. Chapter 5 Two port networks Network Anlsis nd Snthesis hpter 5 Two port networks . ntroduction A one port network is completel specified when the voltge current reltionship t the terminls of the port is given. A generl two port on

More information

INTRODUCTION. The three general approaches to the solution of kinetics problems are:

INTRODUCTION. The three general approaches to the solution of kinetics problems are: INTRODUCTION According to Newton s lw, prticle will ccelerte when it is subjected to unblnced forces. Kinetics is the study of the reltions between unblnced forces nd the resulting chnges in motion. The

More information

A5682: Introduction to Cosmology Course Notes. 4. Cosmic Dynamics: The Friedmann Equation. = GM s

A5682: Introduction to Cosmology Course Notes. 4. Cosmic Dynamics: The Friedmann Equation. = GM s 4. Cosmic Dynmics: The Friedmnn Eqution Reding: Chpter 4 Newtonin Derivtion of the Friedmnn Eqution Consider n isolted sphere of rdius R s nd mss M s, in uniform, isotropic expnsion (Hubble flow). The

More information

ANALYSIS OF FAST REACTORS SYSTEMS

ANALYSIS OF FAST REACTORS SYSTEMS ANALYSIS OF FAST REACTORS SYSTEMS M. Rghe 4/7/006 INTRODUCTION Fst rectors differ from therml rectors in severl spects nd require specil tretment. The prsitic cpture cross sections in the fuel, coolnt

More information

Reversible magnetization processes in scalar Preisachtype models of hysteresis

Reversible magnetization processes in scalar Preisachtype models of hysteresis JOURNAL O OPTOELECTRONIC AND ADVANCED ATERIAL Vol. 8, No. 5, Octoer 26, p. 171-1714 Reversile mgnetiztion processes in sclr Preischtype models of hysteresis L. TOLERIU *, A. TANCU Deprtment of olid tte

More information

A little harder example. A block sits at rest on a flat surface. The block is held down by its weight. What is the interaction pair for the weight?

A little harder example. A block sits at rest on a flat surface. The block is held down by its weight. What is the interaction pair for the weight? Neton s Ls of Motion (ges 9-99) 1. An object s velocit vector v remins constnt if nd onl if the net force cting on the object is zero.. hen nonzero net force cts on n object, the object s velocit chnges.

More information

The Properties of Stars

The Properties of Stars 10/11/010 The Properties of Strs sses Using Newton s Lw of Grvity to Determine the ss of Celestil ody ny two prticles in the universe ttrct ech other with force tht is directly proportionl to the product

More information

Things to Memorize: A Partial List. January 27, 2017

Things to Memorize: A Partial List. January 27, 2017 Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved

More information

Exploring parametric representation with the TI-84 Plus CE graphing calculator

Exploring parametric representation with the TI-84 Plus CE graphing calculator Exploring prmetric representtion with the TI-84 Plus CE grphing clcultor Richrd Prr Executive Director Rice University School Mthemtics Project rprr@rice.edu Alice Fisher Director of Director of Technology

More information

PHYS Summer Professor Caillault Homework Solutions. Chapter 2

PHYS Summer Professor Caillault Homework Solutions. Chapter 2 PHYS 1111 - Summer 2007 - Professor Cillult Homework Solutions Chpter 2 5. Picture the Problem: The runner moves long the ovl trck. Strtegy: The distnce is the totl length of trvel, nd the displcement

More information

k and v = v 1 j + u 3 i + v 2

k and v = v 1 j + u 3 i + v 2 ORTHOGONAL FUNCTIONS AND FOURIER SERIES Orthogonl functions A function cn e considered to e generliztion of vector. Thus the vector concets like the inner roduct nd orthogonlity of vectors cn e extended

More information

The Shortest Confidence Interval for the Mean of a Normal Distribution

The Shortest Confidence Interval for the Mean of a Normal Distribution Interntionl Journl of Sttistics nd Proility; Vol. 7, No. 2; Mrch 208 ISSN 927-7032 E-ISSN 927-7040 Pulished y Cndin Center of Science nd Eduction The Shortest Confidence Intervl for the Men of Norml Distriution

More information

Unit 2 Exponents Study Guide

Unit 2 Exponents Study Guide Unit Eponents Stud Guide 7. Integer Eponents Prt : Zero Eponents Algeric Definition: 0 where cn e n non-zero vlue 0 ecuse 0 rised to n power less thn or equl to zero is n undefined vlue. Eple: 0 If ou

More information

FEM ANALYSIS OF ROGOWSKI COILS COUPLED WITH BAR CONDUCTORS

FEM ANALYSIS OF ROGOWSKI COILS COUPLED WITH BAR CONDUCTORS XIX IMEKO orld Congress Fundmentl nd Applied Metrology September 6 11, 2009, Lisbon, Portugl FEM ANALYSIS OF ROGOSKI COILS COUPLED ITH BAR CONDUCTORS Mirko Mrrcci, Bernrdo Tellini, Crmine Zppcost University

More information

THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES

THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES 1. TRANSLATION Figure shows rigid body trnslting in three-dimensionl spce. Any two points in the body, such s A nd B, will move long prllel stright lines if

More information

FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81

FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81 FORM FIVE ADDITIONAL MATHEMATIC NOTE CHAPTER : PROGRESSION Arithmetic Progression T n = + (n ) d S n = n [ + (n )d] = n [ + Tn ] S = T = T = S S Emple : The th term of n A.P. is 86 nd the sum of the first

More information

Haplotype Frequencies and Linkage Disequilibrium. Biostatistics 666

Haplotype Frequencies and Linkage Disequilibrium. Biostatistics 666 Hlotye Frequencies nd Linkge isequilirium iosttistics 666 Lst Lecture Genotye Frequencies llele Frequencies Phenotyes nd Penetrnces Hrdy-Weinerg Equilirium Simle demonstrtion Exercise: NO2 nd owel isese

More information

EMF Notes 9; Electromagnetic Induction ELECTROMAGNETIC INDUCTION

EMF Notes 9; Electromagnetic Induction ELECTROMAGNETIC INDUCTION EMF Notes 9; Electromgnetic nduction EECTOMAGNETC NDUCTON (Y&F Chpters 3, 3; Ohnin Chpter 3) These notes cover: Motionl emf nd the electric genertor Electromgnetic nduction nd Frdy s w enz s w nduced electric

More information

Properties of Lorenz Curves for Transformed Income Distributions

Properties of Lorenz Curves for Transformed Income Distributions Theoreticl Economics etters 22 2 487-493 htt://ddoiorg/4236/tel22259 Published Online December 22 (htt://wwwscirporg/journl/tel) Proerties of orenz Curves for Trnsformed Income Distributions John Fellmn

More information

Generalized Surface Area of Revolution

Generalized Surface Area of Revolution Generlized Surfce Are of Revolution Richrd Winton, Ph.D. Michel. Wrren Astrct Suose curve in the lne R is defined y continuous function over closed ounded intervl. A forul is develoed for the rdius of

More information

critical number where f '(x) = 0 or f '(x) is undef (where denom. of f '(x) = 0)

critical number where f '(x) = 0 or f '(x) is undef (where denom. of f '(x) = 0) Decoding AB Clculus Voculry solute mx/min x f(x) (sometimes do sign digrm line lso) Edpts C.N. ccelertion rte of chnge in velocity or x''(t) = v'(t) = (t) AROC Slope of secnt line, f () f () verge vlue

More information

1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.

1.2. Linear Variable Coefficient Equations. y + b ! = a y + b  Remark: The case b = 0 and a non-constant can be solved with the same idea as above. 1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt

More information

Homework Assignment 9 Solution Set

Homework Assignment 9 Solution Set Homework Assignment 9 Solution Set PHYCS 44 3 Mrch, 4 Problem (Griffiths 77) The mgnitude of the current in the loop is loop = ε induced = Φ B = A B = π = π µ n (µ n) = π µ nk According to Lense s Lw this

More information

Some circular summation formulas for theta functions

Some circular summation formulas for theta functions Ci et l. Boundr Vlue Prolems 013, 013:59 R E S E A R C H Open Access Some circulr summtion formuls for thet functions Yi Ci, Si Chen nd Qiu-Ming Luo * * Correspondence: luomth007@163.com Deprtment of Mthemtics,

More information

A, Electromagnetic Fields Final Exam December 14, 2001 Solution

A, Electromagnetic Fields Final Exam December 14, 2001 Solution 304-351, Electrognetic Fiels Finl Ex Deceer 14, 2001 Solution 1. e9.8. In chpter9.proles.extr.two loops, e of thin wire crry equl n opposite currents s shown in the figure elow. The rius of ech loop is

More information

On the Linear Stability of Compound Capillary Jets

On the Linear Stability of Compound Capillary Jets ILASS Americs, th Annul Conference on Liquid Atomiztion nd Spry Systems, Chicgo, IL, My 7 On the Liner Stbility of Compound Cpillry Jets Mksud (Mx) Ismilov, Stephen D Heister School of Aeronutics nd Astronutics,

More information

CHAPTER 20: Second Law of Thermodynamics

CHAPTER 20: Second Law of Thermodynamics CHAER 0: Second Lw of hermodynmics Responses to Questions 3. kg of liquid iron will hve greter entropy, since it is less ordered thn solid iron nd its molecules hve more therml motion. In ddition, het

More information

Simple Harmonic Motion I Sem

Simple Harmonic Motion I Sem Simple Hrmonic Motion I Sem Sllus: Differentil eqution of liner SHM. Energ of prticle, potentil energ nd kinetic energ (derivtion), Composition of two rectngulr SHM s hving sme periods, Lissjous figures.

More information

Chapter 7: Applications of Integrals

Chapter 7: Applications of Integrals Chpter 7: Applictions of Integrls 78 Chpter 7 Overview: Applictions of Integrls Clculus, like most mthemticl fields, egn with tring to solve everd prolems. The theor nd opertions were formlized lter. As

More information

CAPACITORS AND DIELECTRICS

CAPACITORS AND DIELECTRICS Importnt Definitions nd Units Cpcitnce: CAPACITORS AND DIELECTRICS The property of system of electricl conductors nd insultors which enbles it to store electric chrge when potentil difference exists between

More information

Effects of peripheral drilling moment on delamination using special drill bits

Effects of peripheral drilling moment on delamination using special drill bits journl of mterils processing technology 01 (008 471 476 journl homepge: www.elsevier.com/locte/jmtprotec Effects of peripherl illing moment on delmintion using specil ill bits C.C. Tso,, H. Hocheng b Deprtment

More information

MA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations

MA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations LESSON 0 Chpter 7.2 Trigonometric Integrls. Bsic trig integrls you should know. sin = cos + C cos = sin + C sec 2 = tn + C sec tn = sec + C csc 2 = cot + C csc cot = csc + C MA 6200 Em 2 Study Guide, Fll

More information

Available online at ScienceDirect. Procedia Engineering 172 (2017 )

Available online at  ScienceDirect. Procedia Engineering 172 (2017 ) Aville online t www.sciencedirect.com ScienceDirect Procedi Engineering 172 (2017 ) 218 225 Modern Building Mterils, Structures nd Techniques, MBMST 2016 Experimentl nd Numericl Anlysis of Direct Sher

More information

S56 (5.3) Vectors.notebook January 29, 2016

S56 (5.3) Vectors.notebook January 29, 2016 Dily Prctice 15.1.16 Q1. The roots of the eqution (x 1)(x + k) = 4 re equl. Find the vlues of k. Q2. Find the rte of chnge of 剹 x when x = 1 / 8 Tody we will e lerning out vectors. Q3. Find the eqution

More information

, MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF

, MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF DOWNLOAD FREE FROM www.tekoclsses.com, PH.: 0 903 903 7779, 98930 5888 Some questions (Assertion Reson tpe) re given elow. Ech question contins Sttement (Assertion) nd Sttement (Reson). Ech question hs

More information

Travelling Profile Solutions For Nonlinear Degenerate Parabolic Equation And Contour Enhancement In Image Processing

Travelling Profile Solutions For Nonlinear Degenerate Parabolic Equation And Contour Enhancement In Image Processing Applied Mthemtics E-Notes 8(8) - c IN 67-5 Avilble free t mirror sites of http://www.mth.nthu.edu.tw/ men/ Trvelling Profile olutions For Nonliner Degenerte Prbolic Eqution And Contour Enhncement In Imge

More information

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3 2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is

More information

2. VECTORS AND MATRICES IN 3 DIMENSIONS

2. VECTORS AND MATRICES IN 3 DIMENSIONS 2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

More information

Lecture 13 - Linking E, ϕ, and ρ

Lecture 13 - Linking E, ϕ, and ρ Lecture 13 - Linking E, ϕ, nd ρ A Puzzle... Inner-Surfce Chrge Density A positive point chrge q is locted off-center inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on

More information

8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.

8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers. 8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the

More information

Improper Integrals with Infinite Limits of Integration

Improper Integrals with Infinite Limits of Integration 6_88.qd // : PM Pge 578 578 CHAPTER 8 Integrtion Techniques, L Hôpitl s Rule, nd Improper Integrls Section 8.8 f() = d The unounded region hs n re of. Figure 8.7 Improper Integrls Evlute n improper integrl

More information

Some basic concepts of fluid dynamics derived from ECE theory

Some basic concepts of fluid dynamics derived from ECE theory Some sic concepts of fluid dynmics 363 Journl of Foundtions of Physics nd Chemistry, 2, vol. (4) 363 374 Some sic concepts of fluid dynmics derived from ECE theory M.W. Evns Alph Institute for Advnced

More information

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007 A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

More information