Physics Courseware Electromagnetism

Size: px
Start display at page:

Download "Physics Courseware Electromagnetism"

Transcription

1 Pysics Cousewae lectomagnetism lectic field Poblem.- a) Find te electic field at point P poduced by te wie sown in te figue. Conside tat te wie as a unifom linea cage distibution of λ.5µ C / m b) Find te electic field at point P poduced by te squae of unifom suface cage distibution σ.µc / m

2 Solution: Pat a) We conside te electic field poduced by a diffeential of wie as sown in te figue. Te diffeential of cage is dq λdy and te distance to point P is d + magnitude of te electic field is: y, so te d kλdy + y Te two components can be found by multiplying by te cosine and sine of te angle wit te oizontal: kλdy d + y + y and kλdy d y + y + y y Now, all we need to do is integate te epessions above to find te electic field: Fo te -component te integal is: Cange of vaiable y tanθ kλ kλdy + y + y dy ( + y ) ( + y ) / sec θ, dy sec θdθ, y θ, y θ tan / tan ( / ) tan ( / ) sec θdθ kλ cos d sec θ θ θ kλ kλ sinθ kλ + tan ( y / )

3 And fo te y-component te integal is: kλdy y y kλ + y + y kλ + kλ y + kλ + ydy ( + y ) / kλ + y Wit te values of te poblem: 6.5.5, V/m y , V/m Pat b) If you conside te contibution of a diffeential of aea of te squae, te electic field will be: kσddy d + y + Ten, to find te z-component we multiply by te cosine of te angle between te z-ais and te electic field vecto: d kσddy z + y + + y +

4 You sould ealize tat te epession ddy + y + + y + solid angle poduced at point P by ddy, so te electic field will be: z kσ S.A. is te diffeential of Wee S.A. is te solid angle poduced by te squae. You can use te epession L / deived elsewee: S. A. cos to calculate te electic field: + L / z kσ cos L / + L / Wit te values of te poblem: 6 z. cos. /.5 +. / 6, V/m Poblem.- Two identical conducting spees S and S ave adii.m and ae sepaated by a long distance dm. Initially one spee is caged wit Q µc and te ote is neutal (Q ). Ten you bing a tid identical conducting spee (initially neutal) and make contact wit te caged spee fist and ten wit te neutal one. Afte tat you take te tid spee vey fa away. Wat is now te foce between S and S? Solution: Wen two identical spees touc tey will ecange cage until te potential is te same on bot spees, so until te cage is saed by in equal quantities. Afte te tid spee touces S, te cage on S will be Q µc (te ote µc will be in te tid spee). Afte te tid spee touces S, te cage on S will be Q /µc Te foce is: F 6 6 QQ ( )(.5 ).5 ε o d N

5 Poblem.- Find te foce on te positive cage Q5µC at te oigin of coodinates due to te seven negative cages q-µc located aound te cicle of adius.5m at te positions sown in te figue. [Hint: Tee is an easy way and a ad way of doing tis] Solution: Notice tat all te foces cancel eac ote ecept fo te one due to te cage on te positive -ais. Te foce is: 6 6 Qq (5 )( ) F. N ε o d.5 Poblem.- Find te electic field at point P, wic is at a distance d.m fom te cente of two squaes of side L.m and unifom suface cage density σ.µc / m Solution: Te electic field poduced by eac squae is: kσ 6, V/m 6 Adding te two vectos we get: +,6 V/m

6 Poblem 5.- A spee of adius m is located wit its cente at te oigin of coodinates and as a cage of Q nc unifomly distibuted ove its suface. Anote spee of adius.m as its cente on te -ais at a distance of D.m fom te oigin of coodinates and as a cage of Q -nc also unifomly distibuted ove its suface. Calculate te electic field at point B(.5m,.5m,) Solution: Tee ae two vectos tat contibute to te total electic field, te magnitude of electic field due to te lage spee is kq 8V/m and te one due to te smalle spee kq 7.8V/m. +.5 To add te vectos it sould be done component by component, wit te esult: o o (8cos 5,8sin 5 ) o o ( 7.8cos 5.7, 7.8sin 5.7 ) (.,-.66) V/m

7 Poblem 6.- A spee of adius cm is located wit te cente at te oigin of coodinates and as a cage of nc unifomly distibuted ove its suface. Anote spee of adius 6cm as its cente on te -ais at a distance of D5cm fom te oigin of coodinates and as a cage of -nc also unifomly distibuted ove its suface. Calculate te electic field at point A(,,) Solution: Te electic field at point A(,,): Notice tat te contibution to te electic field fom te cage of te lage spee is zeo. Tis is because te cage is unifomly distibuted on its suface and point A is inside te spee. On te ote and, te smalle spee does poduce an electic field at point A. Since te cage distibution is speical and point A is outside te smalle spee, its electic field is te same as if all te cage wee located at its cente, so: q k d (.) 6 V/m Notice tat in te calculation of electic field we ignoed te sign of te cage. Wat te sign indicates is wete te vecto is towad te cage o away fom it. In tis case te electic field vecto is diected in te positive -diection. Poblem 7.- Calculate te electic field at point C due to te unifomly distibuted cage Q on te semicicle of adius.

8 Solution: In poblems wee tee is a continuous distibution of cages te standad pocedue to find electic field o potential is: - Divide te continuous distibution in small pieces (diffeentials) tat can be teated as point cages. - Calculate te field o potential poduced by te diffeential. Be caeful tat in te case of electic field you will need to calculate te components independently. - All distances, cages and angles sould be witten in tems of te vaiables cosen. - Integate ove tose vaiables. Te poblem above falls pecisely in te categoy tat we just mentioned. So let s divide te cage in small pieces. To do tis, notice tat te cage is unifomly distibuted, so te linea density of cage is: cage Q λ lengt Ten, we divide te ac in diffeentials as sown in te figue: Te lengt of te diffeential of ac is Q Q dq λdl λd θ dθ dθ d l dθ and te diffeential of cage is We fist calculate te magnitude of te electic field poduced by te diffeential of Q k dθ kdq kqdθ cage, wic is as if it wee a point: d d But tis is te magnitude of te vecto. It as an -component and a y-component. By te symmety of te poblem we notice tat we only need to calculate te -component.

9 Te -component of te electic field is: kqdθ d dcosθ cosθ Now we need to integate tis epession to find te electic field: / d / kqdθ cosθ kq Poblem 8.- A speical sell as intenal adius and etenal adius and contains a unifom distibution of cage wit density ρ Calculate te electic field fo a point at a distance fom te cente of te sell. Conside cases: a) < b) << c) > Solution: Inside te sell, wen < te electic field is zeo because if you conside a speical suface and apply Gauss law, te cage enclosed is zeo. At a point in te sell, wee << we can use Gauss s law, but te cage enclosed will only be te one in te sell between and. Te cage can be calculated by multiplying te volume of te sell (between and ) by te density of cage. Tis woks because te density is constant, if it wee a function

10 of te adius we would ave to divide te sell in layes (like an onion) and integate laye by laye. So, te cage enclosed is: Q enclosed ρ And using Gauss s law: ( ) / kq k k enclosed ρ ρ Finally, outside te spee you can calculate te electic field as if all te cage wee located at te cente of te sell. Te total cage is: Q ρ And te electic field: ( ) kq k ρ

Electric Potential and Energy

Electric Potential and Energy Electic Potential and Enegy Te polem: A solid spee of te adius R is omogeneously caged wit te cage Q and put inside an infinite ollow cylinde. Te cylinde inne and oute adii ae a and, R < a

More information

Continuous Charge Distributions: Electric Field and Electric Flux

Continuous Charge Distributions: Electric Field and Electric Flux 8/30/16 Quiz 2 8/25/16 A positive test chage qo is eleased fom est at a distance away fom a chage of Q and a distance 2 away fom a chage of 2Q. How will the test chage move immediately afte being eleased?

More information

Review. Electrostatic. Dr. Ray Kwok SJSU

Review. Electrostatic. Dr. Ray Kwok SJSU Review Electostatic D. Ray Kwok SJSU Paty Balloons Coulomb s Law F e q q k 1 Coulomb foce o electical foce. (vecto) Be caeful on detemining the sign & diection. k 9 10 9 (N m / C ) k 1 4πε o k is the Coulomb

More information

Physics 2212 GH Quiz #2 Solutions Spring 2016

Physics 2212 GH Quiz #2 Solutions Spring 2016 Physics 2212 GH Quiz #2 Solutions Sping 216 I. 17 points) Thee point chages, each caying a chage Q = +6. nc, ae placed on an equilateal tiangle of side length = 3. mm. An additional point chage, caying

More information

Physics Courseware Physics II Electric Field and Force

Physics Courseware Physics II Electric Field and Force Physics Cousewae Physics II lectic iel an oce Coulomb s law, whee k Nm /C test Definition of electic fiel. This is a vecto. test Q lectic fiel fo a point chage. This is a vecto. Poblem.- chage of µc is

More information

CHAPTER 25 ELECTRIC POTENTIAL

CHAPTER 25 ELECTRIC POTENTIAL CHPTE 5 ELECTIC POTENTIL Potential Diffeence and Electic Potential Conside a chaged paticle of chage in a egion of an electic field E. This filed exets an electic foce on the paticle given by F=E. When

More information

Problem Set 5: Universal Law of Gravitation; Circular Planetary Orbits

Problem Set 5: Universal Law of Gravitation; Circular Planetary Orbits Poblem Set 5: Univesal Law of Gavitation; Cicula Planetay Obits Design Engineeing Callenge: Te Big Dig.007 Contest Evaluation of Scoing Concepts: Spinne vs. Plowe PROMBLEM 1: Daw a fee-body-diagam of a

More information

cos kd kd 2 cosθ = π 2 ± nπ d λ cosθ = 1 2 ± n N db

cos kd kd 2 cosθ = π 2 ± nπ d λ cosθ = 1 2 ± n N db . (Balanis 6.43) You can confim tat AF = e j kd cosθ + e j kd cosθ N = cos kd cosθ gives te same esult as (6-59) and (6-6), fo a binomial aay wit te coefficients cosen as in section 6.8.. Tis single expession

More information

Chapter 22 The Electric Field II: Continuous Charge Distributions

Chapter 22 The Electric Field II: Continuous Charge Distributions Chapte The lectic Field II: Continuous Chage Distibutions A ing of adius a has a chage distibution on it that vaies as l(q) l sin q, as shown in Figue -9. (a) What is the diection of the electic field

More information

Chapter 22: Electric Fields. 22-1: What is physics? General physics II (22102) Dr. Iyad SAADEDDIN. 22-2: The Electric Field (E)

Chapter 22: Electric Fields. 22-1: What is physics? General physics II (22102) Dr. Iyad SAADEDDIN. 22-2: The Electric Field (E) Geneal physics II (10) D. Iyad D. Iyad Chapte : lectic Fields In this chapte we will cove The lectic Field lectic Field Lines -: The lectic Field () lectic field exists in a egion of space suounding a

More information

(r) = 1. Example: Electric Potential Energy. Summary. Potential due to a Group of Point Charges 9/10/12 1 R V(r) + + V(r) kq. Chapter 23.

(r) = 1. Example: Electric Potential Energy. Summary. Potential due to a Group of Point Charges 9/10/12 1 R V(r) + + V(r) kq. Chapter 23. Eample: Electic Potential Enegy What is the change in electical potential enegy of a eleased electon in the atmosphee when the electostatic foce fom the nea Eath s electic field (diected downwad) causes

More information

Objectives: After finishing this unit you should be able to:

Objectives: After finishing this unit you should be able to: lectic Field 7 Objectives: Afte finishing this unit you should be able to: Define the electic field and explain what detemines its magnitude and diection. Wite and apply fomulas fo the electic field intensity

More information

Objects usually are charged up through the transfer of electrons from one object to the other.

Objects usually are charged up through the transfer of electrons from one object to the other. 1 Pat 1: Electic Foce 1.1: Review of Vectos Review you vectos! You should know how to convet fom pola fom to component fom and vice vesa add and subtact vectos multiply vectos by scalas Find the esultant

More information

Chapter 1 Functions and Graphs

Chapter 1 Functions and Graphs Capte Functions and Gaps Section.... 6 7. 6 8 8 6. 6 6 8 8.... 6.. 6. n n n n n n n 6 n 6 n n 7. 8 7 7..8..8 8.. 8. a b ± ± 6 c ± 6 ± 8 8 o 8 6. 8y 8y 7 8y y 8y y 8 o y y. 7 7 o 7 7 Capte : Functions and

More information

Chapter 23: GAUSS LAW 343

Chapter 23: GAUSS LAW 343 Chapte 23: GAUSS LAW 1 A total chage of 63 10 8 C is distibuted unifomly thoughout a 27-cm adius sphee The volume chage density is: A 37 10 7 C/m 3 B 69 10 6 C/m 3 C 69 10 6 C/m 2 D 25 10 4 C/m 3 76 10

More information

Look over Chapter 22 sections 1-8 Examples 2, 4, 5, Look over Chapter 16 sections 7-9 examples 6, 7, 8, 9. Things To Know 1/22/2008 PHYS 2212

Look over Chapter 22 sections 1-8 Examples 2, 4, 5, Look over Chapter 16 sections 7-9 examples 6, 7, 8, 9. Things To Know 1/22/2008 PHYS 2212 PHYS 1 Look ove Chapte sections 1-8 xamples, 4, 5, PHYS 111 Look ove Chapte 16 sections 7-9 examples 6, 7, 8, 9 Things To Know 1) What is an lectic field. ) How to calculate the electic field fo a point

More information

Tutorial on Strehl ratio, wavefront power series expansion, Zernike polynomials expansion in small aberrated optical systems By Sheng Yuan

Tutorial on Strehl ratio, wavefront power series expansion, Zernike polynomials expansion in small aberrated optical systems By Sheng Yuan Tutoial on Stel atio, wavefont powe seies expansion, Zenike polynomials expansion in small abeated optical systems By Seng Yuan. Stel Ratio Te wave abeation function, (x,y, is defined as te distance, in

More information

Electromagnetic Theory 1

Electromagnetic Theory 1 / lectomagnetic Theoy uestion : lectostatic Potential negy A sphee of adius caies a positive chage density ρ constant Obviously the spheical coodinates system is appopiate hee Take - C m - and cm τ a)

More information

JURONG JUNIOR COLLEGE Physics Department Tutorial: Electric Fields (solutions)

JURONG JUNIOR COLLEGE Physics Department Tutorial: Electric Fields (solutions) JJ 5 H Physics (646) Electic Fields_tutsoln JURONG JUNIOR COLLEGE Physics Depatment Tutoial: Electic Fields (solutions) No Solution LO Electic field stength at a point in an electic field is defined as

More information

Electromagnetism Physics 15b

Electromagnetism Physics 15b lectomagnetism Physics 15b Lectue #20 Dielectics lectic Dipoles Pucell 10.1 10.6 What We Did Last Time Plane wave solutions of Maxwell s equations = 0 sin(k ωt) B = B 0 sin(k ωt) ω = kc, 0 = B, 0 ˆk =

More information

Electrostatics (Electric Charges and Field) #2 2010

Electrostatics (Electric Charges and Field) #2 2010 Electic Field: The concept of electic field explains the action at a distance foce between two chaged paticles. Evey chage poduces a field aound it so that any othe chaged paticle expeiences a foce when

More information

EM-2. 1 Coulomb s law, electric field, potential field, superposition q. Electric field of a point charge (1)

EM-2. 1 Coulomb s law, electric field, potential field, superposition q. Electric field of a point charge (1) EM- Coulomb s law, electic field, potential field, supeposition q ' Electic field of a point chage ( ') E( ) kq, whee k / 4 () ' Foce of q on a test chage e at position is ee( ) Electic potential O kq

More information

Welcome to Physics 272

Welcome to Physics 272 Welcome to Physics 7 Bob Mose mose@phys.hawaii.edu http://www.phys.hawaii.edu/~mose/physics7.html To do: Sign into Masteing Physics phys-7 webpage Registe i-clickes (you i-clicke ID to you name on class-list)

More information

B da = 0. Q E da = ε. E da = E dv

B da = 0. Q E da = ε. E da = E dv lectomagnetic Theo Pof Ruiz, UNC Asheville, doctophs on YouTube Chapte Notes The Maxwell quations in Diffeential Fom 1 The Maxwell quations in Diffeential Fom We will now tansfom the integal fom of the

More information

Faraday s Law (continued)

Faraday s Law (continued) Faaday s Law (continued) What causes cuent to flow in wie? Answe: an field in the wie. A changing magnetic flux not only causes an MF aound a loop but an induced electic field. Can wite Faaday s Law: ε

More information

PHYS 2135 Exam I February 13, 2018

PHYS 2135 Exam I February 13, 2018 Exam Total /200 PHYS 2135 Exam I Febuay 13, 2018 Name: Recitation Section: Five multiple choice questions, 8 points each Choose the best o most nealy coect answe Fo questions 6-9, solutions must begin

More information

2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum

2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum 2. Electostatics D. Rakhesh Singh Kshetimayum 1 2.1 Intoduction In this chapte, we will study how to find the electostatic fields fo vaious cases? fo symmetic known chage distibution fo un-symmetic known

More information

Gauss Law. Physics 231 Lecture 2-1

Gauss Law. Physics 231 Lecture 2-1 Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

More information

Your Comments. Do we still get the 80% back on homework? It doesn't seem to be showing that. Also, this is really starting to make sense to me!

Your Comments. Do we still get the 80% back on homework? It doesn't seem to be showing that. Also, this is really starting to make sense to me! You Comments Do we still get the 8% back on homewok? It doesn't seem to be showing that. Also, this is eally stating to make sense to me! I am a little confused about the diffeences in solid conductos,

More information

PHYS 110B - HW #7 Spring 2004, Solutions by David Pace Any referenced equations are from Griffiths Problem statements are paraphrased

PHYS 110B - HW #7 Spring 2004, Solutions by David Pace Any referenced equations are from Griffiths Problem statements are paraphrased PHYS 0B - HW #7 Sping 2004, Solutions by David Pace Any efeenced euations ae fom Giffiths Poblem statements ae paaphased. Poblem 0.3 fom Giffiths A point chage,, moves in a loop of adius a. At time t 0

More information

$ i. !((( dv vol. Physics 8.02 Quiz One Equations Fall q 1 q 2 r 2 C = 2 C! V 2 = Q 2 2C F = 4!" or. r ˆ = points from source q to observer

$ i. !((( dv vol. Physics 8.02 Quiz One Equations Fall q 1 q 2 r 2 C = 2 C! V 2 = Q 2 2C F = 4! or. r ˆ = points from source q to observer Physics 8.0 Quiz One Equations Fall 006 F = 1 4" o q 1 q = q q ˆ 3 4" o = E 4" o ˆ = points fom souce q to obseve 1 dq E = # ˆ 4" 0 V "## E "d A = Q inside closed suface o d A points fom inside to V =

More information

Module 05: Gauss s s Law a

Module 05: Gauss s s Law a Module 05: Gauss s s Law a 1 Gauss s Law The fist Maxwell Equation! And a vey useful computational technique to find the electic field E when the souce has enough symmety. 2 Gauss s Law The Idea The total

More information

Physics 181. Assignment 4

Physics 181. Assignment 4 Physics 181 Assignment 4 Solutions 1. A sphee has within it a gavitational field given by g = g, whee g is constant and is the position vecto of the field point elative to the cente of the sphee. This

More information

Page 1 of 6 Physics II Exam 1 155 points Name Discussion day/time Pat I. Questions 110. 8 points each. Multiple choice: Fo full cedit, cicle only the coect answe. Fo half cedit, cicle the coect answe and

More information

Sources of Magnetic Fields (chap 28)

Sources of Magnetic Fields (chap 28) Souces of Magnetic Fields (chap 8) In chapte 7, we consideed the magnetic field effects on a moving chage, a line cuent and a cuent loop. Now in Chap 8, we conside the magnetic fields that ae ceated by

More information

Hopefully Helpful Hints for Gauss s Law

Hopefully Helpful Hints for Gauss s Law Hopefully Helpful Hints fo Gauss s Law As befoe, thee ae things you need to know about Gauss s Law. In no paticula ode, they ae: a.) In the context of Gauss s Law, at a diffeential level, the electic flux

More information

Section 8.2 Polar Coordinates

Section 8.2 Polar Coordinates Section 8. Pola Coodinates 467 Section 8. Pola Coodinates The coodinate system we ae most familia with is called the Catesian coodinate system, a ectangula plane divided into fou quadants by the hoizontal

More information

! E da = 4πkQ enc, has E under the integral sign, so it is not ordinarily an

! E da = 4πkQ enc, has E under the integral sign, so it is not ordinarily an Physics 142 Electostatics 2 Page 1 Electostatics 2 Electicity is just oganized lightning. Geoge Calin A tick that sometimes woks: calculating E fom Gauss s law Gauss s law,! E da = 4πkQ enc, has E unde

More information

Phys-272 Lecture 17. Motional Electromotive Force (emf) Induced Electric Fields Displacement Currents Maxwell s Equations

Phys-272 Lecture 17. Motional Electromotive Force (emf) Induced Electric Fields Displacement Currents Maxwell s Equations Phys-7 Lectue 17 Motional Electomotive Foce (emf) Induced Electic Fields Displacement Cuents Maxwell s Equations Fom Faaday's Law to Displacement Cuent AC geneato Magnetic Levitation Tain Review of Souces

More information

Lecture 8 - Gauss s Law

Lecture 8 - Gauss s Law Lectue 8 - Gauss s Law A Puzzle... Example Calculate the potential enegy, pe ion, fo an infinite 1D ionic cystal with sepaation a; that is, a ow of equally spaced chages of magnitude e and altenating sign.

More information

1 Spherical multipole moments

1 Spherical multipole moments Jackson notes 9 Spheical multipole moments Suppose we have a chage distibution ρ (x) wheeallofthechageiscontained within a spheical egion of adius R, as shown in the diagam. Then thee is no chage in the

More information

Chapter 21: Gauss s Law

Chapter 21: Gauss s Law Chapte : Gauss s Law Gauss s law : intoduction The total electic flux though a closed suface is equal to the total (net) electic chage inside the suface divided by ε Gauss s law is equivalent to Coulomb

More information

Magnetic fields (origins) CHAPTER 27 SOURCES OF MAGNETIC FIELD. Permanent magnets. Electric currents. Magnetic field due to a moving charge.

Magnetic fields (origins) CHAPTER 27 SOURCES OF MAGNETIC FIELD. Permanent magnets. Electric currents. Magnetic field due to a moving charge. Magnetic fields (oigins) CHAPTER 27 SOURCES OF MAGNETC FELD Magnetic field due to a moving chage. Electic cuents Pemanent magnets Magnetic field due to electic cuents Staight wies Cicula coil Solenoid

More information

Your Comments. Conductors and Insulators with Gauss's law please...so basically everything!

Your Comments. Conductors and Insulators with Gauss's law please...so basically everything! You Comments I feel like I watch a pe-lectue, and agee with eveything said, but feel like it doesn't click until lectue. Conductos and Insulatos with Gauss's law please...so basically eveything! I don't

More information

F Q E v B MAGNETOSTATICS. Creation of magnetic field B. Effect of B on a moving charge. On moving charges only. Stationary and moving charges

F Q E v B MAGNETOSTATICS. Creation of magnetic field B. Effect of B on a moving charge. On moving charges only. Stationary and moving charges MAGNETOSTATICS Ceation of magnetic field. Effect of on a moving chage. Take the second case: F Q v mag On moving chages only F QE v Stationay and moving chages dw F dl Analysis on F mag : mag mag Qv. vdt

More information

PHYS 1444 Section 501 Lecture #7

PHYS 1444 Section 501 Lecture #7 PHYS 1444 Section 51 Lectue #7 Wednesday, Feb. 8, 26 Equi-potential Lines and Sufaces Electic Potential Due to Electic Dipole E detemined fom V Electostatic Potential Enegy of a System of Chages Capacitos

More information

Rotatoy Motion Hoizontal Cicula Motion In tanslatoy motion, evey point in te body follows te pat of its pecedin one wit same velocity includin te cente of mass In otatoy motion, evey point move wit diffeent

More information

( ) F α. a. Sketch! r as a function of r for fixed θ. For the sketch, assume that θ is roughly the same ( )

( ) F α. a. Sketch! r as a function of r for fixed θ. For the sketch, assume that θ is roughly the same ( ) . An acoustic a eflecting off a wav bounda (such as the sea suface) will see onl that pat of the bounda inclined towad the a. Conside a a with inclination to the hoizontal θ (whee θ is necessail positive,

More information

Potential Energy. The change U in the potential energy. is defined to equal to the negative of the work. done by a conservative force

Potential Energy. The change U in the potential energy. is defined to equal to the negative of the work. done by a conservative force Potential negy The change U in the potential enegy is defined to equal to the negative of the wok done by a consevative foce duing the shift fom an initial to a final state. U = U U = W F c = F c d Potential

More information

B. Spherical Wave Propagation

B. Spherical Wave Propagation 11/8/007 Spheical Wave Popagation notes 1/1 B. Spheical Wave Popagation Evey antenna launches a spheical wave, thus its powe density educes as a function of 1, whee is the distance fom the antenna. We

More information

Review of Potential Energy. The Electric Potential. Plotting Fields and Potentials. Electric Potential of a Point Charge

Review of Potential Energy. The Electric Potential. Plotting Fields and Potentials. Electric Potential of a Point Charge eview of Potential negy Potential enegy U() can be used to descibe a consevative foce. efeence point (U) can be chosen fo convenience. Wok done by F : W F d s F d (1D) Change in P.. : U U f U i W Foce

More information

GCSE: Volumes and Surface Area

GCSE: Volumes and Surface Area GCSE: Volumes and Suface Aea D J Fost (jfost@tiffin.kingston.sc.uk) www.dfostmats.com GCSE Revision Pack Refeence:, 1, 1, 1, 1i, 1ii, 18 Last modified: 1 st August 01 GCSE Specification. Know and use fomulae

More information

Physics 2102 Lecture: 07 TUE 09 FEB

Physics 2102 Lecture: 07 TUE 09 FEB Physics 2102 Jonathan Dowling Physics 2102 Lectue: 07 TUE 09 FEB Electic Potential II PHYS2102 FIRST MIDTERM EXAM 6 7PM THU 11 FEB 2009 Dowling s Sec. 4 in??? YOU MUST BRING YOUR STUDENT ID YOU MUST WRITE

More information

UNIT 3:Electrostatics

UNIT 3:Electrostatics The study of electic chages at est, the foces between them and the electic fields associated with them. UNIT 3:lectostatics S7 3. lectic Chages and Consevation of chages The electic chage has the following

More information

Magnetic Dipoles Challenge Problem Solutions

Magnetic Dipoles Challenge Problem Solutions Magnetic Dipoles Challenge Poblem Solutions Poblem 1: Cicle the coect answe. Conside a tiangula loop of wie with sides a and b. The loop caies a cuent I in the diection shown, and is placed in a unifom

More information

3.6 Applied Optimization

3.6 Applied Optimization .6 Applied Optimization Section.6 Notes Page In this section we will be looking at wod poblems whee it asks us to maimize o minimize something. Fo all the poblems in this section you will be taking the

More information

Chapter 13 Gravitation

Chapter 13 Gravitation Chapte 13 Gavitation In this chapte we will exploe the following topics: -Newton s law of gavitation, which descibes the attactive foce between two point masses and its application to extended objects

More information

(Sample 3) Exam 1 - Physics Patel SPRING 1998 FORM CODE - A (solution key at end of exam)

(Sample 3) Exam 1 - Physics Patel SPRING 1998 FORM CODE - A (solution key at end of exam) (Sample 3) Exam 1 - Physics 202 - Patel SPRING 1998 FORM CODE - A (solution key at end of exam) Be sue to fill in you student numbe and FORM lette (A, B, C) on you answe sheet. If you foget to include

More information

Flux. Area Vector. Flux of Electric Field. Gauss s Law

Flux. Area Vector. Flux of Electric Field. Gauss s Law Gauss s Law Flux Flux in Physics is used to two distinct ways. The fist meaning is the ate of flow, such as the amount of wate flowing in a ive, i.e. volume pe unit aea pe unit time. O, fo light, it is

More information

Gauss s Law Simulation Activities

Gauss s Law Simulation Activities Gauss s Law Simulation Activities Name: Backgound: The electic field aound a point chage is found by: = kq/ 2 If thee ae multiple chages, the net field at any point is the vecto sum of the fields. Fo a

More information

Physics 122, Fall October 2012

Physics 122, Fall October 2012 hsics 1, Fall 1 3 Octobe 1 Toda in hsics 1: finding Foce between paallel cuents Eample calculations of fom the iot- Savat field law Ampèe s Law Eample calculations of fom Ampèe s law Unifom cuents in conductos?

More information

3.1 Extreme Values of a Function

3.1 Extreme Values of a Function .1 Etreme Values of a Function Section.1 Notes Page 1 One application of te derivative is finding minimum and maimum values off a grap. In precalculus we were only able to do tis wit quadratics by find

More information

ELECTRIC FIELD. decos. 1 dq x.. Example:

ELECTRIC FIELD. decos. 1 dq x.. Example: ELECTRIC FIELD Example: Solution: A ing-shaped conducto with adius a caies a total positive chage Q unifomly distibuted on it. Find the electic field at a point P that lies on the axis of the ing at a

More information

4.[1pt] Two small spheres with charges -4 C and -9 C are held 9.5 m apart. Find the magnitude of the force between them.

4.[1pt] Two small spheres with charges -4 C and -9 C are held 9.5 m apart. Find the magnitude of the force between them. . [pt] A peson scuffing he feet on a wool ug on a y ay accumulates a net chage of - 4.uC. How many ecess electons oes this peson get? Coect, compute gets:.63e+4. [pt] By how much oes he mass incease? Coect,

More information

7.2. Coulomb s Law. The Electric Force

7.2. Coulomb s Law. The Electric Force Coulomb s aw Recall that chaged objects attact some objects and epel othes at a distance, without making any contact with those objects Electic foce,, o the foce acting between two chaged objects, is somewhat

More information

[Griffiths Ch.1-3] 2008/11/18, 10:10am 12:00am, 1. (6%, 7%, 7%) Suppose the potential at the surface of a hollow hemisphere is specified, as shown

[Griffiths Ch.1-3] 2008/11/18, 10:10am 12:00am, 1. (6%, 7%, 7%) Suppose the potential at the surface of a hollow hemisphere is specified, as shown [Giffiths Ch.-] 8//8, :am :am, Useful fomulas V ˆ ˆ V V V = + θ+ φ ˆ and v = ( v ) + (sin θvθ ) + v θ sinθ φ sinθ θ sinθ φ φ. (6%, 7%, 7%) Suppose the potential at the suface of a hollow hemisphee is specified,

More information

Magnetic field due to a current loop.

Magnetic field due to a current loop. Example using spheical hamonics Sp 18 Magnetic field due to a cuent loop. A cicula loop of adius a caies cuent I. We place the oigin at the cente of the loop, with pola axis pependicula to the plane of

More information

18.1 Origin of Electricity 18.2 Charged Objects and Electric Force

18.1 Origin of Electricity 18.2 Charged Objects and Electric Force 1 18.1 Oigin of lecticity 18. Chaged Objects and lectic Foce Thee ae two kinds of electic chage: positive and negative. The SI unit of electic chage is the coulomb (C). The magnitude of the chage on an

More information

Physics 2B Chapter 22 Notes - Magnetic Field Spring 2018

Physics 2B Chapter 22 Notes - Magnetic Field Spring 2018 Physics B Chapte Notes - Magnetic Field Sping 018 Magnetic Field fom a Long Staight Cuent-Caying Wie In Chapte 11 we looked at Isaac Newton s Law of Gavitation, which established that a gavitational field

More information

( ) Make-up Tests. From Last Time. Electric Field Flux. o The Electric Field Flux through a bit of area is

( ) Make-up Tests. From Last Time. Electric Field Flux. o The Electric Field Flux through a bit of area is Mon., 3/23 Wed., 3/25 Thus., 3/26 Fi., 3/27 Mon., 3/30 Tues., 3/31 21.4-6 Using Gauss s & nto to Ampee s 21.7-9 Maxwell s, Gauss s, and Ampee s Quiz Ch 21, Lab 9 Ampee s Law (wite up) 22.1-2,10 nto to

More information

Today s Plan. Electric Dipoles. More on Gauss Law. Comment on PDF copies of Lectures. Final iclicker roll-call

Today s Plan. Electric Dipoles. More on Gauss Law. Comment on PDF copies of Lectures. Final iclicker roll-call Today s Plan lectic Dipoles Moe on Gauss Law Comment on PDF copies of Lectues Final iclicke oll-call lectic Dipoles A positive (q) and negative chage (-q) sepaated by a small distance d. lectic dipole

More information

Solutions to Problems : Chapter 19 Problems appeared on the end of chapter 19 of the Textbook

Solutions to Problems : Chapter 19 Problems appeared on the end of chapter 19 of the Textbook Solutions to Poblems Chapte 9 Poblems appeae on the en of chapte 9 of the Textbook 8. Pictue the Poblem Two point chages exet an electostatic foce on each othe. Stategy Solve Coulomb s law (equation 9-5)

More information

( )( )( ) ( ) + ( ) ( ) ( )

( )( )( ) ( ) + ( ) ( ) ( ) 3.7. Moel: The magnetic fiel is that of a moving chage paticle. Please efe to Figue Ex3.7. Solve: Using the iot-savat law, 7 19 7 ( ) + ( ) qvsinθ 1 T m/a 1.6 1 C. 1 m/s sin135 1. 1 m 1. 1 m 15 = = = 1.13

More information

? this lecture. ? next lecture. What we have learned so far. a Q E F = q E a. F = q v B a. a Q in motion B. db/dt E. de/dt B.

? this lecture. ? next lecture. What we have learned so far. a Q E F = q E a. F = q v B a. a Q in motion B. db/dt E. de/dt B. PHY 249 Lectue Notes Chapte 32: Page 1 of 12 What we have leaned so fa a a F q a a in motion F q v a a d/ Ae thee othe "static" chages that can make -field? this lectue d/? next lectue da dl Cuve Cuve

More information

A moving charged particle creates a magnetic field vector at every point in space except at its position.

A moving charged particle creates a magnetic field vector at every point in space except at its position. 1 Pat 3: Magnetic Foce 3.1: Magnetic Foce & Field A. Chaged Paticles A moving chaged paticle ceates a magnetic field vecto at evey point in space ecept at its position. Symbol fo Magnetic Field mks units

More information

e.g: If A = i 2 j + k then find A. A = Ax 2 + Ay 2 + Az 2 = ( 2) = 6

e.g: If A = i 2 j + k then find A. A = Ax 2 + Ay 2 + Az 2 = ( 2) = 6 MOTION IN A PLANE 1. Scala Quantities Physical quantities that have only magnitude and no diection ae called scala quantities o scalas. e.g. Mass, time, speed etc. 2. Vecto Quantities Physical quantities

More information

Physics 2020, Spring 2005 Lab 5 page 1 of 8. Lab 5. Magnetism

Physics 2020, Spring 2005 Lab 5 page 1 of 8. Lab 5. Magnetism Physics 2020, Sping 2005 Lab 5 page 1 of 8 Lab 5. Magnetism PART I: INTRODUCTION TO MAGNETS This week we will begin wok with magnets and the foces that they poduce. By now you ae an expet on setting up

More information

arxiv: v1 [physics.pop-ph] 3 Jun 2013

arxiv: v1 [physics.pop-ph] 3 Jun 2013 A note on the electostatic enegy of two point chages axiv:1306.0401v1 [physics.pop-ph] 3 Jun 013 A C Tot Instituto de Física Univesidade Fedeal do io de Janeio Caixa Postal 68.58; CEP 1941-97 io de Janeio,

More information

13.10 Worked Examples

13.10 Worked Examples 13.10 Woked Examples Example 13.11 Wok Done in a Constant Gavitation Field The wok done in a unifom gavitation field is a faily staightfowad calculation when the body moves in the diection of the field.

More information

PHY2061 Enriched Physics 2 Lecture Notes. Gauss Law

PHY2061 Enriched Physics 2 Lecture Notes. Gauss Law PHY61 Eniched Physics Lectue Notes Law Disclaime: These lectue notes ae not meant to eplace the couse textbook. The content may be incomplete. ome topics may be unclea. These notes ae only meant to be

More information

Review for Midterm-1

Review for Midterm-1 Review fo Midtem-1 Midtem-1! Wednesday Sept. 24th at 6pm Section 1 (the 4:10pm class) exam in BCC N130 (Business College) Section 2 (the 6:00pm class) exam in NR 158 (Natual Resouces) Allowed one sheet

More information

Φ E = E A E A = p212c22: 1

Φ E = E A E A = p212c22: 1 Chapte : Gauss s Law Gauss s Law is an altenative fomulation of the elation between an electic field and the souces of that field in tems of electic flux. lectic Flux Φ though an aea A ~ Numbe of Field

More information

2 E. on each of these two surfaces. r r r r. Q E E ε. 2 2 Qencl encl right left 0

2 E. on each of these two surfaces. r r r r. Q E E ε. 2 2 Qencl encl right left 0 Ch : 4, 9,, 9,,, 4, 9,, 4, 8 4 (a) Fom the diagam in the textbook, we see that the flux outwad though the hemispheical suface is the same as the flux inwad though the cicula suface base of the hemisphee

More information

Introduction: Vectors and Integrals

Introduction: Vectors and Integrals Intoduction: Vectos and Integals Vectos a Vectos ae chaacteized by two paametes: length (magnitude) diection a These vectos ae the same Sum of the vectos: a b a a b b a b a b a Vectos Sum of the vectos:

More information

Chapter 25. Electric Potential

Chapter 25. Electric Potential Chapte 25 Electic Potential C H P T E R O U T L I N E 251 Potential Diffeence and Electic Potential 252 Potential Diffeences in a Unifom Electic Field 253 Electic Potential and Potential Enegy Due to Point

More information

ELECTROSTATICS::BHSEC MCQ 1. A. B. C. D.

ELECTROSTATICS::BHSEC MCQ 1. A. B. C. D. ELETROSTATIS::BHSE 9-4 MQ. A moving electic chage poduces A. electic field only. B. magnetic field only.. both electic field and magnetic field. D. neithe of these two fields.. both electic field and magnetic

More information

Chapter Sixteen: Electric Charge and Electric Fields

Chapter Sixteen: Electric Charge and Electric Fields Chapte Sixteen: Electic Chage and Electic Fields Key Tems Chage Conducto The fundamental electical popety to which the mutual attactions o epulsions between electons and potons ae attibuted. Any mateial

More information

MAGNETIC FIELD AROUND TWO SEPARATED MAGNETIZING COILS

MAGNETIC FIELD AROUND TWO SEPARATED MAGNETIZING COILS The 8 th Intenational Confeence of the Slovenian Society fo Non-Destuctive Testing»pplication of Contempoay Non-Destuctive Testing in Engineeing«Septembe 1-3, 5, Potoož, Slovenia, pp. 17-1 MGNETIC FIELD

More information

Question 1: The dipole

Question 1: The dipole Septembe, 08 Conell Univesity, Depatment of Physics PHYS 337, Advance E&M, HW #, due: 9/5/08, :5 AM Question : The dipole Conside a system as discussed in class and shown in Fig.. in Heald & Maion.. Wite

More information

MEI Structured Mathematics. Module Summary Sheets. Numerical Methods (Version B reference to new book)

MEI Structured Mathematics. Module Summary Sheets. Numerical Methods (Version B reference to new book) MEI Matematics in Education and Industy MEI Stuctued Matematics Module Summay Seets (Vesion B efeence to new book) Topic : Appoximations Topic : Te solution of equations Topic : Numeical integation Topic

More information

KEPLER S LAWS OF PLANETARY MOTION

KEPLER S LAWS OF PLANETARY MOTION EPER S AWS OF PANETARY MOTION 1. Intoduction We ae now in a position to apply what we have leaned about the coss poduct and vecto valued functions to deive eple s aws of planetay motion. These laws wee

More information

Physics 11 Chapter 20: Electric Fields and Forces

Physics 11 Chapter 20: Electric Fields and Forces Physics Chapte 0: Electic Fields and Foces Yesteday is not ous to ecove, but tomoow is ous to win o lose. Lyndon B. Johnson When I am anxious it is because I am living in the futue. When I am depessed

More information

Target Boards, JEE Main & Advanced (IIT), NEET Physics Gauss Law. H. O. D. Physics, Concept Bokaro Centre P. K. Bharti

Target Boards, JEE Main & Advanced (IIT), NEET Physics Gauss Law. H. O. D. Physics, Concept Bokaro Centre P. K. Bharti Page 1 CONCPT: JB-, Nea Jitenda Cinema, City Cente, Bokao www.vidyadishti.og Gauss Law Autho: Panjal K. Bhati (IIT Khaagpu) Mb: 74884484 Taget Boads, J Main & Advanced (IIT), NT 15 Physics Gauss Law Autho:

More information

13. The electric field can be calculated by Eq. 21-4a, and that can be solved for the magnitude of the charge N C m 8.

13. The electric field can be calculated by Eq. 21-4a, and that can be solved for the magnitude of the charge N C m 8. CHAPTR : Gauss s Law Solutions to Assigned Poblems Use -b fo the electic flux of a unifom field Note that the suface aea vecto points adially outwad, and the electic field vecto points adially inwad Thus

More information

Physics Electricity and Magnetism Lecture 3 - Electric Field. Y&F Chapter 21 Sec. 4 7

Physics Electricity and Magnetism Lecture 3 - Electric Field. Y&F Chapter 21 Sec. 4 7 Physics - lecticity and Magnetism Lectue 3 - lectic Field Y&F Chapte Sec. 4 7 Recap & Definition of lectic Field lectic Field Lines Chages in xtenal lectic Fields Field due to a Point Chage Field Lines

More information

( ) ( )( ) ˆ. Homework #8. Chapter 27 Magnetic Fields II.

( ) ( )( ) ˆ. Homework #8. Chapter 27 Magnetic Fields II. Homewok #8. hapte 7 Magnetic ields. 6 Eplain how ou would modif Gauss s law if scientists discoveed that single, isolated magnetic poles actuall eisted. Detemine the oncept Gauss law fo magnetism now eads

More information

MONTE CARLO SIMULATION OF FLUID FLOW

MONTE CARLO SIMULATION OF FLUID FLOW MONTE CARLO SIMULATION OF FLUID FLOW M. Ragheb 3/7/3 INTRODUCTION We conside the situation of Fee Molecula Collisionless and Reflective Flow. Collisionless flows occu in the field of aefied gas dynamics.

More information

Between any two masses, there exists a mutual attractive force.

Between any two masses, there exists a mutual attractive force. YEAR 12 PHYSICS: GRAVITATION PAST EXAM QUESTIONS Name: QUESTION 1 (1995 EXAM) (a) State Newton s Univesal Law of Gavitation in wods Between any two masses, thee exists a mutual attactive foce. This foce

More information

F (-4.32x x10 k) Newtons a = (-4.742x x10 )m/s 9.11x10 kg

F (-4.32x x10 k) Newtons a = (-4.742x x10 )m/s 9.11x10 kg P Physics d Quate Test Review KEY k 8.99 x 9 Nm /C 8.85 x - C /Nm e.6 x -9 C milli - mico -6 nano -9 pico - Mega 6 Electostatics. Chage, Field, and Potential a. poton is placed in a unifom electic field

More information

Section 26 The Laws of Rotational Motion

Section 26 The Laws of Rotational Motion Physics 24A Class Notes Section 26 The Laws of otational Motion What do objects do and why do they do it? They otate and we have established the quantities needed to descibe this motion. We now need to

More information