MAGNETISM MAGNETIC DIPOLES

Size: px
Start display at page:

Download "MAGNETISM MAGNETIC DIPOLES"

Transcription

1 MAGNETISM We now turn to magnetsm. Ths has actually been used for longer than electrcty. People were usng compasses to sal around the Medterranean Sea several hundred years BC. However t was not understood untl the frst half of the 19 th century. The trouble was that t appears to have two sources: magnetc materals and electrc currents. The frst was responsble for compasses etc, whle the second was not even observed untl the early part of the 19 th century. Ultmately, of course, they are the same, but that s hndsght. We wll start wth the effect of magnetc materals. MAGNETIC DIPOLES As seen n class the fundamental source of magnetsm appears to be not a pont charge but rather a pont dpole. We have already worked out the effect of dpoles n electrostatcs. Expermentally we fnd that magnetc dpoles work n exactly the same way except that the force constant, k, has a dfferent value: 7 km 110 n m /q where q s the unt of magnetc charge whch we wll defne later. It s mportant to realze that so far as we know THERE IS NO MAGNETIC SINGLE CHARGE. Such an entty would be called a magnetc monopole and has never been observed. A bt later we wll fnd the physcal source of the dpole. In order to determne the sze of the effects we should expect we have to know the sze of the dpoles. To fnd ths we have to consder the second source currents. FIELD PRODUCED BY CURRENT In the 180 s Ampere studed the forces between current carryng wres and found the followng result. If you thnk about t ths s qute an achevement. Just magne how hard t would have been to deduce these equatons based on the equpment he had to work wth. It s really true that we stand on the shoulders of gants! Imagne two lttle current elements as shown: He found that the force on element two due to element one was gven by: df Id k Id 1 m r r

2 Followng what we dd for electrc forces we break ths up nto two parts: the dsturbed envronment produced by current one, and the force the dsturbance produces on current two: k Id r df Id Id B r r 1 m wth k mid1 r db r r We then get the total feld by summng over the entre current I 1 B r k m1 I d r r r r where Ths s the basc result for the feld produced by a current. We can compare ths wth the correspondng result for electrostatcs: de r kq r r r r

3 E r kq r r r r The smlartes are obvous. We now connect these two dfferent approaches to magnetc feld by consderng the feld produced by a small current loop. Suppose the loop s a crcle and les n the xy plane wth center at the orgn. Let s fnd the feld on the axs of the loop at the pont (0,0,z). Assume the loop has radus R and carres a current I. Then: Consder a tny pece of the loop Idl IN s perpendcular to r r. Thus Id IN r r Id r r c ˆ As we go around the crcle, all components wll cancel except those n ẑ. But the z component s where r r' dbz kmid sn r r' sn R r r'

4 Hence db k Id R m z r r' Addng these up around the crcle we get Now suppose we make z >> R. Then and we get B B0,0,z kmir z r r' r r' z k m z IR But ths s exactly the feld we found for a pont dpole wth p IR IA The drecton s gven by the rght-hand rule. Curl the fngers of rght hand n drecton of current and the thumb wll pont n the drecton of dpole. In our case p IAzˆ Ths provdes the couplng between the two sources of magnetc feld. Note that any current loop can be thought of as a collecton of dpoles: Note that all the nteror currents cancel so that all that s left s the current around the loop. But we have now replaced t wth an nfnte set of dpoles. To fnd the feld we can use ether the current or the dpole approach.

5 DIPOLE MOMENT OF BAR MAGNET We can now estmate the dpole moment of the bar magnets shown n class. We frst realze that the magnet s not really a dpole unless we are far away from t. If we are, we can consder t to be made up of a collecton of ndvdual atomc dpoles, each due to a current loop caused by the electron movng around the nucleus. Then we need the number of atoms where Then We have qv qv qvr I p R zˆ R R volume A # N (A#) r Aw Aw (A#) = Avogadros number = /kg mole (Aw) = atomc weght = 57 for ron ρ = densty = 8000 kg/m for ron l = 7 =.18 m r = m Q = coul R m v 10 6 m/sec qvr A # p r zˆ Aw p amp m zˆ

6 p amp / m vol We can now fnd the force between two such magnets placed a dstance d apart. From our prevous result n electrostatcs we expect: 7 4 6kmm1m F n d d d d1m Ths s clearly a small force. Estmate what t would do No wonder we can t see t!! mg mg T cost cos FM Tsnmgtan 4 F rad.05 mg MAGNET AGAINST PLATE We now consder the famlar case of a magnet aganst a steel plate. To do ths we recall the effect of an electrc feld on a delectrc. The result was to polarze the delectrc resultng n a net charge at the surface of delectrc:

7 If the dpole moment/volume s P then the charge/area on the surface s P ˆn where ˆn was a unt vector pontng outward from the surface. To see ths let l be the length of the dpole. Then Then the dpole contaned n the volume V A where A s the area of the surface, s PA Thus the charge s PA q PA Hence the charge/area s P Ths means that we can pcture the bar magnet as Now consder the ron plate. The atoms can t move, but ther magnets can lne up, just as for the delectrc. Hence we expect

8 We expect ths to occur untl the charge on the surface of the ron s equal and opposte to that on the magnet. Note that these are fcttous magnetc charges. However, ths s ok snce we have seen that the math s the same as f we had looked at current loops. Hence Fnally, we recall that the force on a conductor was F k k M A 7 6 m 7 6 F n attractve. To do ths we have assumed equal and opposte charges, and neglected the effect of the charges at the other end of the magnet. The frst approxmaton s very good for ferromagnetc materals such as ron. The second depends on the magnet beng long. In our case, snce the force falls off as 1/R 4 t s a pretty good approxmaton. AMPERE S LAW In general to fnd the feld produced by currents we have to perform the sum found above. Ths equaton s known as the Bot Savart Law. Unfortunately ths n general nvolves calculus just as for electrostatcs. However there s an analogue to Gauss s Law whch can be used n specal symmetres to get the result easly. Ths result s known as Ampere s Law. To derve t from the Bot Savart Law requres calculus so I wll smply state the result: Bd 4kmout I where I out s the current comng out through the cap coverng the loop. The sgns matter. You must go around the loop keepng t to your left. Then the current s postve and out of the page. Just as wth Gauss s Law, ths s always true, but normally only useful f we can get B out of the sum. It s nterestng to compare the two laws: Gauss Law Qn E da 4kQn 0

9 Ampere s Law B d 4 I I out 0 out We can now use ths to fnd the magnetc feld n two very mportant specal cases.

Electricity and Magnetism - Physics 121 Lecture 10 - Sources of Magnetic Fields (Currents) Y&F Chapter 28, Sec. 1-7

Electricity and Magnetism - Physics 121 Lecture 10 - Sources of Magnetic Fields (Currents) Y&F Chapter 28, Sec. 1-7 Electrcty and Magnetsm - Physcs 11 Lecture 10 - Sources of Magnetc Felds (Currents) Y&F Chapter 8, Sec. 1-7 Magnetc felds are due to currents The Bot-Savart Law Calculatng feld at the centers of current

More information

CONDUCTORS AND INSULATORS

CONDUCTORS AND INSULATORS CONDUCTORS AND INSULATORS We defne a conductor as a materal n whch charges are free to move over macroscopc dstances.e., they can leave ther nucle and move around the materal. An nsulator s anythng else.

More information

PHYS 1101 Practice problem set 12, Chapter 32: 21, 22, 24, 57, 61, 83 Chapter 33: 7, 12, 32, 38, 44, 49, 76

PHYS 1101 Practice problem set 12, Chapter 32: 21, 22, 24, 57, 61, 83 Chapter 33: 7, 12, 32, 38, 44, 49, 76 PHYS 1101 Practce problem set 1, Chapter 3: 1,, 4, 57, 61, 83 Chapter 33: 7, 1, 3, 38, 44, 49, 76 3.1. Vsualze: Please reer to Fgure Ex3.1. Solve: Because B s n the same drecton as the ntegraton path s

More information

Physics 114 Exam 3 Spring Name:

Physics 114 Exam 3 Spring Name: Physcs 114 Exam 3 Sprng 015 Name: For gradng purposes (do not wrte here): Queston 1. 1... 3. 3. Problem 4. Answer each of the followng questons. Ponts for each queston are ndcated n red. Unless otherwse

More information

ˆ (0.10 m) E ( N m /C ) 36 ˆj ( j C m)

ˆ (0.10 m) E ( N m /C ) 36 ˆj ( j C m) 7.. = = 3 = 4 = 5. The electrc feld s constant everywhere between the plates. Ths s ndcated by the electrc feld vectors, whch are all the same length and n the same drecton. 7.5. Model: The dstances to

More information

NEWTON S LAWS. These laws only apply when viewed from an inertial coordinate system (unaccelerated system).

NEWTON S LAWS. These laws only apply when viewed from an inertial coordinate system (unaccelerated system). EWTO S LAWS Consder two partcles. 1 1. If 1 0 then 0 wth p 1 m1v. 1 1 2. 1.. 3. 11 These laws only apply when vewed from an nertal coordnate system (unaccelerated system). consder a collecton of partcles

More information

PES 1120 Spring 2014, Spendier Lecture 6/Page 1

PES 1120 Spring 2014, Spendier Lecture 6/Page 1 PES 110 Sprng 014, Spender Lecture 6/Page 1 Lecture today: Chapter 1) Electrc feld due to charge dstrbutons -> charged rod -> charged rng We ntroduced the electrc feld, E. I defned t as an nvsble aura

More information

Section 8.3 Polar Form of Complex Numbers

Section 8.3 Polar Form of Complex Numbers 80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the

More information

PHY2049 Exam 2 solutions Fall 2016 Solution:

PHY2049 Exam 2 solutions Fall 2016 Solution: PHY2049 Exam 2 solutons Fall 2016 General strategy: Fnd two resstors, one par at a tme, that are connected ether n SERIES or n PARALLEL; replace these two resstors wth one of an equvalent resstance. Now

More information

Frequency dependence of the permittivity

Frequency dependence of the permittivity Frequency dependence of the permttvty February 7, 016 In materals, the delectrc constant and permeablty are actually frequency dependent. Ths does not affect our results for sngle frequency modes, but

More information

A particle in a state of uniform motion remain in that state of motion unless acted upon by external force.

A particle in a state of uniform motion remain in that state of motion unless acted upon by external force. The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty,

More information

PHYSICS - CLUTCH CH 28: INDUCTION AND INDUCTANCE.

PHYSICS - CLUTCH CH 28: INDUCTION AND INDUCTANCE. !! www.clutchprep.com CONCEPT: ELECTROMAGNETIC INDUCTION A col of wre wth a VOLTAGE across each end wll have a current n t - Wre doesn t HAVE to have voltage source, voltage can be INDUCED V Common ways

More information

kq r 2 2kQ 2kQ (A) (B) (C) (D)

kq r 2 2kQ 2kQ (A) (B) (C) (D) PHYS 1202W MULTIPL CHOIC QUSTIONS QUIZ #1 Answer the followng multple choce questons on the bubble sheet. Choose the best answer, 5 pts each. MC1 An uncharged metal sphere wll (A) be repelled by a charged

More information

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle

More information

Fields, Charges, and Field Lines

Fields, Charges, and Field Lines Felds, Charges, and Feld Lnes Electrc charges create electrc felds. (Gauss Law) Electrc feld lnes begn on + charges and end on - charges. Lke charges repel, oppostes attract. Start wth same dea for magnetc

More information

Physics 114 Exam 2 Fall 2014 Solutions. Name:

Physics 114 Exam 2 Fall 2014 Solutions. Name: Physcs 114 Exam Fall 014 Name: For gradng purposes (do not wrte here): Queston 1. 1... 3. 3. Problem Answer each of the followng questons. Ponts for each queston are ndcated n red. Unless otherwse ndcated,

More information

PHYSICS - CLUTCH 1E CH 28: INDUCTION AND INDUCTANCE.

PHYSICS - CLUTCH 1E CH 28: INDUCTION AND INDUCTANCE. !! www.clutchprep.com CONCEPT: ELECTROMAGNETIC INDUCTION A col of wre wth a VOLTAGE across each end wll have a current n t - Wre doesn t HAVE to have voltage source, voltage can be INDUCED V Common ways

More information

One Dimension Again. Chapter Fourteen

One Dimension Again. Chapter Fourteen hapter Fourteen One Dmenson Agan 4 Scalar Lne Integrals Now we agan consder the dea of the ntegral n one dmenson When we were ntroduced to the ntegral back n elementary school, we consdered only functons

More information

Economics 101. Lecture 4 - Equilibrium and Efficiency

Economics 101. Lecture 4 - Equilibrium and Efficiency Economcs 0 Lecture 4 - Equlbrum and Effcency Intro As dscussed n the prevous lecture, we wll now move from an envronment where we looed at consumers mang decsons n solaton to analyzng economes full of

More information

Electricity and Magnetism Review Faraday s Law

Electricity and Magnetism Review Faraday s Law Electrcty and Magnetsm Revew Faraday s Law Lana Sherdan De Anza College Dec 3, 2015 Overvew Faraday s law Lenz s law magnetc feld from a movng charge Gauss s law Remnder: (30.18) Magnetc Flux feld S that

More information

A capacitor is simply two pieces of metal near each other, separated by an insulator or air. A capacitor is used to store charge and energy.

A capacitor is simply two pieces of metal near each other, separated by an insulator or air. A capacitor is used to store charge and energy. -1 apactors A capactor s smply two peces of metal near each other, separate by an nsulator or ar. A capactor s use to store charge an energy. A parallel-plate capactor conssts of two parallel plates separate

More information

Electricity and Magnetism Gauss s Law

Electricity and Magnetism Gauss s Law Electrcty and Magnetsm Gauss s Law Ampère s Law Lana Sherdan De Anza College Mar 1, 2018 Last tme magnetc feld of a movng charge magnetc feld of a current the Bot-Savart law magnetc feld around a straght

More information

Conservation of Angular Momentum = "Spin"

Conservation of Angular Momentum = Spin Page 1 of 6 Conservaton of Angular Momentum = "Spn" We can assgn a drecton to the angular velocty: drecton of = drecton of axs + rght hand rule (wth rght hand, curl fngers n drecton of rotaton, thumb ponts

More information

8.1 Arc Length. What is the length of a curve? How can we approximate it? We could do it following the pattern we ve used before

8.1 Arc Length. What is the length of a curve? How can we approximate it? We could do it following the pattern we ve used before .1 Arc Length hat s the length of a curve? How can we approxmate t? e could do t followng the pattern we ve used before Use a sequence of ncreasngly short segments to approxmate the curve: As the segments

More information

Physics 111: Mechanics Lecture 11

Physics 111: Mechanics Lecture 11 Physcs 111: Mechancs Lecture 11 Bn Chen NJIT Physcs Department Textbook Chapter 10: Dynamcs of Rotatonal Moton q 10.1 Torque q 10. Torque and Angular Acceleraton for a Rgd Body q 10.3 Rgd-Body Rotaton

More information

THE CURRENT BALANCE Physics 258/259

THE CURRENT BALANCE Physics 258/259 DSH 1988, 005 THE CURRENT BALANCE Physcs 58/59 The tme average force between two parallel conductors carryng an alternatng current s measured by balancng ths force aganst the gravtatonal force on a set

More information

1 Matrix representations of canonical matrices

1 Matrix representations of canonical matrices 1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:

More information

Moments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.

Moments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is. Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these

More information

Temperature. Chapter Heat Engine

Temperature. Chapter Heat Engine Chapter 3 Temperature In prevous chapters of these notes we ntroduced the Prncple of Maxmum ntropy as a technque for estmatng probablty dstrbutons consstent wth constrants. In Chapter 9 we dscussed the

More information

Rotational Dynamics. Physics 1425 Lecture 19. Michael Fowler, UVa

Rotational Dynamics. Physics 1425 Lecture 19. Michael Fowler, UVa Rotatonal Dynamcs Physcs 1425 Lecture 19 Mchael Fowler, UVa Rotatonal Dynamcs Newton s Frst Law: a rotatng body wll contnue to rotate at constant angular velocty as long as there s no torque actng on t.

More information

VEKTORANALYS GAUSS THEOREM STOKES THEOREM. and. Kursvecka 3. Kapitel 6 7 Sidor 51 82

VEKTORANALYS GAUSS THEOREM STOKES THEOREM. and. Kursvecka 3. Kapitel 6 7 Sidor 51 82 VEKTORANAY Kursvecka 3 GAU THEOREM and TOKE THEOREM Kaptel 6 7 dor 51 82 TARGET PROBEM Do magnetc monopoles est? EECTRIC FIED MAGNETIC FIED N +? 1 TARGET PROBEM et s consder some EECTRIC CHARGE 2 - + +

More information

Physics 2113 Lecture 14: WED 18 FEB

Physics 2113 Lecture 14: WED 18 FEB Physcs 2113 Jonathan Dowlng Physcs 2113 Lecture 14: WED 18 FEB Electrc Potental II Danger! Electrc Potental Energy, Unts : Electrc Potental Potental Energy = U = [J] = Joules Electrc Potental = V = U/q

More information

Dr. Fritz Wilhelm, Physics 230 E:\Excel files\230 lecture\ch26 capacitance.docx 1 of 13 Last saved: 12/27/2008; 8:40 PM. Homework: See website.

Dr. Fritz Wilhelm, Physics 230 E:\Excel files\230 lecture\ch26 capacitance.docx 1 of 13 Last saved: 12/27/2008; 8:40 PM. Homework: See website. Dr. Frtz Wlhelm, Physcs 3 E:\Excel fles\3 lecture\ch6 capactance.docx of 3 Last saved: /7/8; 8:4 PM Homework: See webste. Table of ontents: h.6. Defnton of apactance, 6. alculatng apactance, 6.a Parallel

More information

The Dirac Equation for a One-electron atom. In this section we will derive the Dirac equation for a one-electron atom.

The Dirac Equation for a One-electron atom. In this section we will derive the Dirac equation for a one-electron atom. The Drac Equaton for a One-electron atom In ths secton we wll derve the Drac equaton for a one-electron atom. Accordng to Ensten the energy of a artcle wth rest mass m movng wth a velocty V s gven by E

More information

Chapter 18 MAGNETIC FIELDS

Chapter 18 MAGNETIC FIELDS Ch. 18--Magnetc Felds Chapter 18 MAGNETIC FIELDS A.) A Small Matter of Specal Relatvty: 1.) Assume we have a partcle of charge q movng wth an ntal velocty v q parallel to a currentcarryng wre as shown

More information

So far: simple (planar) geometries

So far: simple (planar) geometries Physcs 06 ecture 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap. to 3 Rotatonal quanttes as vectors Cross product Torque epressed as a vector Angular momentum defned Angular momentum as a vector

More information

COMPLEX NUMBERS AND QUADRATIC EQUATIONS

COMPLEX NUMBERS AND QUADRATIC EQUATIONS COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not

More information

Solutions to exam in SF1811 Optimization, Jan 14, 2015

Solutions to exam in SF1811 Optimization, Jan 14, 2015 Solutons to exam n SF8 Optmzaton, Jan 4, 25 3 3 O------O -4 \ / \ / The network: \/ where all lnks go from left to rght. /\ / \ / \ 6 O------O -5 2 4.(a) Let x = ( x 3, x 4, x 23, x 24 ) T, where the varable

More information

From Biot-Savart Law to Divergence of B (1)

From Biot-Savart Law to Divergence of B (1) From Bot-Savart Law to Dvergence of B (1) Let s prove that Bot-Savart gves us B (r ) = 0 for an arbtrary current densty. Frst take the dvergence of both sdes of Bot-Savart. The dervatve s wth respect to

More information

Gravitational Acceleration: A case of constant acceleration (approx. 2 hr.) (6/7/11)

Gravitational Acceleration: A case of constant acceleration (approx. 2 hr.) (6/7/11) Gravtatonal Acceleraton: A case of constant acceleraton (approx. hr.) (6/7/11) Introducton The gravtatonal force s one of the fundamental forces of nature. Under the nfluence of ths force all objects havng

More information

VEKTORANALYS. GAUSS s THEOREM and STOKES s THEOREM. Kursvecka 3. Kapitel 6-7 Sidor 51-82

VEKTORANALYS. GAUSS s THEOREM and STOKES s THEOREM. Kursvecka 3. Kapitel 6-7 Sidor 51-82 VEKTORANAY Kursvecka 3 GAU s THEOREM and TOKE s THEOREM Kaptel 6-7 dor 51-82 TARGET PROBEM EECTRIC FIED MAGNETIC FIED N + Magnetc monopoles do not est n nature. How can we epress ths nformaton for E and

More information

Physics 207: Lecture 20. Today s Agenda Homework for Monday

Physics 207: Lecture 20. Today s Agenda Homework for Monday Physcs 207: Lecture 20 Today s Agenda Homework for Monday Recap: Systems of Partcles Center of mass Velocty and acceleraton of the center of mass Dynamcs of the center of mass Lnear Momentum Example problems

More information

Kernel Methods and SVMs Extension

Kernel Methods and SVMs Extension Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general

More information

CHAPTER 14 GENERAL PERTURBATION THEORY

CHAPTER 14 GENERAL PERTURBATION THEORY CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves

More information

ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM

ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look

More information

Confirmation of Gauss s law

Confirmation of Gauss s law Gauss s law Consder charge n a generc surface urround charge wth sphercal surface 1 concentrc to charge Consder cone of sold angle dω from charge to surface through the lttle sphere Electrc flu through

More information

Module 1 : The equation of continuity. Lecture 1: Equation of Continuity

Module 1 : The equation of continuity. Lecture 1: Equation of Continuity 1 Module 1 : The equaton of contnuty Lecture 1: Equaton of Contnuty 2 Advanced Heat and Mass Transfer: Modules 1. THE EQUATION OF CONTINUITY : Lectures 1-6 () () () (v) (v) Overall Mass Balance Momentum

More information

Spring 2002 Lecture #13

Spring 2002 Lecture #13 44-50 Sprng 00 ecture # Dr. Jaehoon Yu. Rotatonal Energy. Computaton of oments of nerta. Parallel-as Theorem 4. Torque & Angular Acceleraton 5. Work, Power, & Energy of Rotatonal otons Remember the md-term

More information

Inductance Calculation for Conductors of Arbitrary Shape

Inductance Calculation for Conductors of Arbitrary Shape CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors

More information

Thermodynamics General

Thermodynamics General Thermodynamcs General Lecture 1 Lecture 1 s devoted to establshng buldng blocks for dscussng thermodynamcs. In addton, the equaton of state wll be establshed. I. Buldng blocks for thermodynamcs A. Dmensons,

More information

Module 14: THE INTEGRAL Exploring Calculus

Module 14: THE INTEGRAL Exploring Calculus Module 14: THE INTEGRAL Explorng Calculus Part I Approxmatons and the Defnte Integral It was known n the 1600s before the calculus was developed that the area of an rregularly shaped regon could be approxmated

More information

Lecture 12: Discrete Laplacian

Lecture 12: Discrete Laplacian Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly

More information

Formulas for the Determinant

Formulas for the Determinant page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use

More information

MEASUREMENT OF MOMENT OF INERTIA

MEASUREMENT OF MOMENT OF INERTIA 1. measurement MESUREMENT OF MOMENT OF INERTI The am of ths measurement s to determne the moment of nerta of the rotor of an electrc motor. 1. General relatons Rotatng moton and moment of nerta Let us

More information

(b) This is about one-sixth the magnitude of the Earth s field. It will affect the compass reading.

(b) This is about one-sixth the magnitude of the Earth s field. It will affect the compass reading. Chapter 9 (a) The magntude of the magnetc feld due to the current n the wre, at a pont a dstance r from the wre, s gven by r Wth r = ft = 6 m, we have c4 T m AhbAg 6 33 T 33 T m b g (b) Ths s about one-sxth

More information

NUMERICAL DIFFERENTIATION

NUMERICAL DIFFERENTIATION NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the

More information

Open Systems: Chemical Potential and Partial Molar Quantities Chemical Potential

Open Systems: Chemical Potential and Partial Molar Quantities Chemical Potential Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,

More information

Part C Dynamics and Statics of Rigid Body. Chapter 5 Rotation of a Rigid Body About a Fixed Axis

Part C Dynamics and Statics of Rigid Body. Chapter 5 Rotation of a Rigid Body About a Fixed Axis Part C Dynamcs and Statcs of Rgd Body Chapter 5 Rotaton of a Rgd Body About a Fxed Axs 5.. Rotatonal Varables 5.. Rotaton wth Constant Angular Acceleraton 5.3. Knetc Energy of Rotaton, Rotatonal Inerta

More information

Difference Equations

Difference Equations Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1

More information

PHYS 705: Classical Mechanics. Newtonian Mechanics

PHYS 705: Classical Mechanics. Newtonian Mechanics 1 PHYS 705: Classcal Mechancs Newtonan Mechancs Quck Revew of Newtonan Mechancs Basc Descrpton: -An dealzed pont partcle or a system of pont partcles n an nertal reference frame [Rgd bodes (ch. 5 later)]

More information

Physics 53. Rotational Motion 3. Sir, I have found you an argument, but I am not obliged to find you an understanding.

Physics 53. Rotational Motion 3. Sir, I have found you an argument, but I am not obliged to find you an understanding. Physcs 53 Rotatonal Moton 3 Sr, I have found you an argument, but I am not oblged to fnd you an understandng. Samuel Johnson Angular momentum Wth respect to rotatonal moton of a body, moment of nerta plays

More information

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number

More information

Physics 2A Chapter 3 HW Solutions

Physics 2A Chapter 3 HW Solutions Phscs A Chapter 3 HW Solutons Chapter 3 Conceptual Queston: 4, 6, 8, Problems: 5,, 8, 7, 3, 44, 46, 69, 70, 73 Q3.4. Reason: (a) C = A+ B onl A and B are n the same drecton. Sze does not matter. (b) C

More information

N S. 4/4/2006 Magnetic Fields ( F.Robilliard) 1

N S. 4/4/2006 Magnetic Fields ( F.Robilliard) 1 y F N +q + θ S v x z 4/4/6 Magnetc Felds ( F.obllard) 1 4/4/6 Magnetc Felds ( F.obllard) Introducton: It has been known, snce antquty, that when certan peces of rock are hung by a thread, from ther centre

More information

Level Crossing Spectroscopy

Level Crossing Spectroscopy Level Crossng Spectroscopy October 8, 2008 Contents 1 Theory 1 2 Test set-up 4 3 Laboratory Exercses 4 3.1 Hanle-effect for fne structure.................... 4 3.2 Hanle-effect for hyperfne structure.................

More information

MATH 5630: Discrete Time-Space Model Hung Phan, UMass Lowell March 1, 2018

MATH 5630: Discrete Time-Space Model Hung Phan, UMass Lowell March 1, 2018 MATH 5630: Dscrete Tme-Space Model Hung Phan, UMass Lowell March, 08 Newton s Law of Coolng Consder the coolng of a well strred coffee so that the temperature does not depend on space Newton s law of collng

More information

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons

More information

20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The first idea is connectedness.

20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The first idea is connectedness. 20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The frst dea s connectedness. Essentally, we want to say that a space cannot be decomposed

More information

Interconnect Modeling

Interconnect Modeling Interconnect Modelng Modelng of Interconnects Interconnect R, C and computaton Interconnect models umped RC model Dstrbuted crcut models Hgher-order waveform n dstrbuted RC trees Accuracy and fdelty Prepared

More information

Lecture Note 3. Eshelby s Inclusion II

Lecture Note 3. Eshelby s Inclusion II ME340B Elastcty of Mcroscopc Structures Stanford Unversty Wnter 004 Lecture Note 3. Eshelby s Incluson II Chrs Wenberger and We Ca c All rghts reserved January 6, 004 Contents 1 Incluson energy n an nfnte

More information

8.022 (E&M) Lecture 4

8.022 (E&M) Lecture 4 Topcs: 8.0 (E&M) Lecture 4 More applcatons of vector calculus to electrostatcs: Laplacan: Posson and Laplace equaton url: concept and applcatons to electrostatcs Introducton to conductors Last tme Electrc

More information

Physics 207: Lecture 27. Announcements

Physics 207: Lecture 27. Announcements Physcs 07: ecture 7 Announcements ake-up labs are ths week Fnal hwk assgned ths week, fnal quz next week Revew sesson on Thursday ay 9, :30 4:00pm, Here Today s Agenda Statcs recap Beam & Strngs» What

More information

Problem Set 9 Solutions

Problem Set 9 Solutions Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem

More information

First Law: A body at rest remains at rest, a body in motion continues to move at constant velocity, unless acted upon by an external force.

First Law: A body at rest remains at rest, a body in motion continues to move at constant velocity, unless acted upon by an external force. Secton 1. Dynamcs (Newton s Laws of Moton) Two approaches: 1) Gven all the forces actng on a body, predct the subsequent (changes n) moton. 2) Gven the (changes n) moton of a body, nfer what forces act

More information

More metrics on cartesian products

More metrics on cartesian products More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of

More information

Note on the Electron EDM

Note on the Electron EDM Note on the Electron EDM W R Johnson October 25, 2002 Abstract Ths s a note on the setup of an electron EDM calculaton and Schff s Theorem 1 Basc Relatons The well-known relatvstc nteracton of the electron

More information

Experiment 1 Mass, volume and density

Experiment 1 Mass, volume and density Experment 1 Mass, volume and densty Purpose 1. Famlarze wth basc measurement tools such as verner calper, mcrometer, and laboratory balance. 2. Learn how to use the concepts of sgnfcant fgures, expermental

More information

DECOUPLING THEORY HW2

DECOUPLING THEORY HW2 8.8 DECOUPLIG THEORY HW2 DOGHAO WAG DATE:OCT. 3 207 Problem We shall start by reformulatng the problem. Denote by δ S n the delta functon that s evenly dstrbuted at the n ) dmensonal unt sphere. As a temporal

More information

Week 11: Chapter 11. The Vector Product. The Vector Product Defined. The Vector Product and Torque. More About the Vector Product

Week 11: Chapter 11. The Vector Product. The Vector Product Defined. The Vector Product and Torque. More About the Vector Product The Vector Product Week 11: Chapter 11 Angular Momentum There are nstances where the product of two vectors s another vector Earler we saw where the product of two vectors was a scalar Ths was called the

More information

APPENDIX A Some Linear Algebra

APPENDIX A Some Linear Algebra APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,

More information

j) = 1 (note sigma notation) ii. Continuous random variable (e.g. Normal distribution) 1. density function: f ( x) 0 and f ( x) dx = 1

j) = 1 (note sigma notation) ii. Continuous random variable (e.g. Normal distribution) 1. density function: f ( x) 0 and f ( x) dx = 1 Random varables Measure of central tendences and varablty (means and varances) Jont densty functons and ndependence Measures of assocaton (covarance and correlaton) Interestng result Condtonal dstrbutons

More information

Chapter 12 Equilibrium & Elasticity

Chapter 12 Equilibrium & Elasticity Chapter 12 Equlbrum & Elastcty If there s a net force, an object wll experence a lnear acceleraton. (perod, end of story!) If there s a net torque, an object wll experence an angular acceleraton. (perod,

More information

Physics 2A Chapters 6 - Work & Energy Fall 2017

Physics 2A Chapters 6 - Work & Energy Fall 2017 Physcs A Chapters 6 - Work & Energy Fall 017 These notes are eght pages. A quck summary: The work-energy theorem s a combnaton o Chap and Chap 4 equatons. Work s dened as the product o the orce actng on

More information

Lecture 3. Ax x i a i. i i

Lecture 3. Ax x i a i. i i 18.409 The Behavor of Algorthms n Practce 2/14/2 Lecturer: Dan Spelman Lecture 3 Scrbe: Arvnd Sankar 1 Largest sngular value In order to bound the condton number, we need an upper bound on the largest

More information

Foundations of Arithmetic

Foundations of Arithmetic Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an

More information

Spin-rotation coupling of the angularly accelerated rigid body

Spin-rotation coupling of the angularly accelerated rigid body Spn-rotaton couplng of the angularly accelerated rgd body Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 E-mal: louaelzen@gmal.com November 1, 2017 All Rghts Reserved. Abstract Ths paper s

More information

ECEN 5005 Crystals, Nanocrystals and Device Applications Class 19 Group Theory For Crystals

ECEN 5005 Crystals, Nanocrystals and Device Applications Class 19 Group Theory For Crystals ECEN 5005 Crystals, Nanocrystals and Devce Applcatons Class 9 Group Theory For Crystals Dee Dagram Radatve Transton Probablty Wgner-Ecart Theorem Selecton Rule Dee Dagram Expermentally determned energy

More information

measurement and the charge the electric field will be calculated using E =. The direction of the field will be the . But, so

measurement and the charge the electric field will be calculated using E =. The direction of the field will be the . But, so THE ELECTRIC FIELD 6 Conceptual Questons 6.. A tny, postve test charge wll be placed at the pont n space and the force wll be measured. From the force F measurement and the charge the electrc feld wll

More information

χ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body

χ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown

More information

MTH 263 Practice Test #1 Spring 1999

MTH 263 Practice Test #1 Spring 1999 Pat Ross MTH 6 Practce Test # Sprng 999 Name. Fnd the area of the regon bounded by the graph r =acos (θ). Observe: Ths s a crcle of radus a, for r =acos (θ) r =a ³ x r r =ax x + y =ax x ax + y =0 x ax

More information

Analytical Chemistry Calibration Curve Handout

Analytical Chemistry Calibration Curve Handout I. Quck-and Drty Excel Tutoral Analytcal Chemstry Calbraton Curve Handout For those of you wth lttle experence wth Excel, I ve provded some key technques that should help you use the program both for problem

More information

Math1110 (Spring 2009) Prelim 3 - Solutions

Math1110 (Spring 2009) Prelim 3 - Solutions Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 1 Queston 1. (16 ponts) Short answer. Math1110 (Sprng 2009) Prelm 3 - Solutons x a 1 (a) (4 ponts) Please evaluate lm, where a and b are postve numbers.

More information

CSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography

CSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography CSc 6974 and ECSE 6966 Math. Tech. for Vson, Graphcs and Robotcs Lecture 21, Aprl 17, 2006 Estmatng A Plane Homography Overvew We contnue wth a dscusson of the major ssues, usng estmaton of plane projectve

More information

This model contains two bonds per unit cell (one along the x-direction and the other along y). So we can rewrite the Hamiltonian as:

This model contains two bonds per unit cell (one along the x-direction and the other along y). So we can rewrite the Hamiltonian as: 1 Problem set #1 1.1. A one-band model on a square lattce Fg. 1 Consder a square lattce wth only nearest-neghbor hoppngs (as shown n the fgure above): H t, j a a j (1.1) where,j stands for nearest neghbors

More information

Problem Do any of the following determine homomorphisms from GL n (C) to GL n (C)?

Problem Do any of the following determine homomorphisms from GL n (C) to GL n (C)? Homework 8 solutons. Problem 16.1. Whch of the followng defne homomomorphsms from C\{0} to C\{0}? Answer. a) f 1 : z z Yes, f 1 s a homomorphsm. We have that z s the complex conjugate of z. If z 1,z 2

More information

One Dimensional Axial Deformations

One Dimensional Axial Deformations One Dmensonal al Deformatons In ths secton, a specfc smple geometr s consdered, that of a long and thn straght component loaded n such a wa that t deforms n the aal drecton onl. The -as s taken as the

More information

Statistical Inference. 2.3 Summary Statistics Measures of Center and Spread. parameters ( population characteristics )

Statistical Inference. 2.3 Summary Statistics Measures of Center and Spread. parameters ( population characteristics ) Ismor Fscher, 8//008 Stat 54 / -8.3 Summary Statstcs Measures of Center and Spread Dstrbuton of dscrete contnuous POPULATION Random Varable, numercal True center =??? True spread =???? parameters ( populaton

More information

CHAPTER 10 ROTATIONAL MOTION

CHAPTER 10 ROTATIONAL MOTION CHAPTER 0 ROTATONAL MOTON 0. ANGULAR VELOCTY Consder argd body rotates about a fxed axs through pont O n x-y plane as shown. Any partcle at pont P n ths rgd body rotates n a crcle of radus r about O. The

More information

Phys102 General Physics II

Phys102 General Physics II Electrc Potental/Energy Phys0 General Physcs II Electrc Potental Topcs Electrc potental energy and electrc potental Equpotental Surace Calculaton o potental rom eld Potental rom a pont charge Potental

More information

DO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED.

DO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED. EE 539 Homeworks Sprng 08 Updated: Tuesday, Aprl 7, 08 DO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED. For full credt, show all work. Some problems requre hand calculatons.

More information