# Errata Instructor s Solutions Manual Introduction to Electrodynamics, 3rd ed Author: David Griffiths Date: September 1, 2004

Save this PDF as:

Size: px
Start display at page:

Download "Errata Instructor s Solutions Manual Introduction to Electrodynamics, 3rd ed Author: David Griffiths Date: September 1, 2004"

## Transcription

1 Errata Instructor s Solutions Manual Introduction to Electrodynamics, 3rd ed Author: David Griffiths Date: September, 004 Page 4, Prob..5 (b): last expression should read y +z +3x. Page 4, Prob..6: at the beginning, insert the following figure Page 8, Prob..6: last line should read From Prob..8: v a = 6xz ˆx +zŷ +3z ẑ ( v a )= x ( 6xz)+ y (z)+ z (3z )= 6z +6z =0. Page 8, Prob..7, in the determinant for ( f), 3rd row, nd column: change y 3 to y. Page 8, Prob..9, line : the number in the box should be - (insert minus sign). Page 9, Prob..3, line : change x 3 to z 3 ; first line of part (c): insert comma between dx and dz. Page, Probl.39, line 5: remove comma after cos θ. Page 3, Prob..4(c), last line: insert ẑ after ). Page 4, Prob..46(b): change r to a. Page 4, Prob..48, second line of J: change the upper limit on the r integral from to R. Fix the last line to read: = 4π ( e r) R 0 +4πe R =4π ( e R + e 0) +4πe R =4π. Page 5, Prob..49(a), line 3: in the box, change x to x 3.

2 Page 5, Prob..49(b), last integration constant should be l(x, z), not l(x, y). Page 7, Prob..53, first expression in (4): insert θ, soda = r sin θdrdφˆθˆθˆθ. Page 7, Prob..55: Solution should read as follows: Problem.55 () x = z =0; dx = dz =0; y :0. v dl =(yz ) dy =0; v dl =0. () x =0; z = y; dz = dy; y : 0. v dl =(yz ) dy +(3y + z) dz = y( y) dy (3y + y) dy; v dl = 0 [ ] y (y 3 4y 4 + y ) dy = 4y3 3 + y 0 y = 4 3. (3) x = y =0; dx = dy =0; z : 0. v dl =(3y + z) dz = zdz. v dl = 0 zdz= z 0 =. Total: v dl = = 8 3. Meanwhile, Stokes thereom says v dl = ( v) da. Here da = dy dz ˆx, so all we need is ( v) x = y (3y + z) z (yz )=3 yz. Therefore { } y ( v) da = (3 yz) dy dz = (3 yz) dz dy 0 0 = [ 0 3( y) y ( y)] dy = 0 ( 4y3 +8y 0y +6) dy = [ y y3 5y +6y ] = = 8 3. Page 8, Prob..56: change (3) and (4) to read as follows: (3) φ = π ; r sin θ = y =, so r = sin θ, dr = sin θ cos θdθ, θ : π θ 0 tan ( ). v dl = ( r cos θ ) ( (dr) (r cos θ sin θ)(rdθ)= cos θ cos θ ) cos θ sin θ sin θ sin dθ θ sin dθ θ ( cos 3 θ = sin 3 θ + cos θ ) dθ = cos θ ( cos θ +sin ) θ sin θ sin θ sin dθ = cos θ θ sin 3 θ dθ. Therefore v dl = θ 0 π/ cos θ sin 3 θ dθ = sin θ θ 0 π/ = (/5) () = 5 =.

3 (4) θ = θ 0,φ= π ; r : 5 0. v dl = ( r cos θ ) (dr) = 4 5 rdr. 0 v dl = 4 rdr= r 0 = =. Total: v dl =0+ 3π + = 3π. Page, Probl.6(e), line : change = z ẑ to +z ẑ. Page 5, Prob..: last line should read Since Q tot = 4 3 πr3 ρ, E = Q 4πɛ 0 R r (as in Prob..8). 3 Page 6, Prob..5: last expression in first line of (ii) should be dφ, not d phi. Page 8, Prob.., at the end, insert the following figure V(r) r In the figure, r is in units of R, and V (r) is in units of Page 30, Prob..8: remove right angle sign in the figure. q 4πɛ 0R. Page 4, Prob. 3.5: subscript on V in last integral should be 3, not. Page 45, Prob. 3.0: after the first box, add: q F = 4πɛ 0 { (a) ˆx } (b) ŷ + ( [cos θ ˆx +sin θ ŷ], a + b ) where cos θ = a/ a + b, sin θ = b/ a + b. {[ F = q a 6πɛ 0 (a + b ) ] 3/ a [ ˆx + b (a + b ) 3/ b ] } ŷ. 3

4 W = 4 4πɛ 0 [ q (a) + q (b) + q ] ( = q a + b ) 3πɛ 0 [ a + b a b Page 45, Prob. 3.0: in the second box, change and to an. Page 46, Probl 3.3, at the end, insert the following: [Comment: Technically, the series solution for σ is defective, since term-by-term differentiation has produced a (naively) non-convergent sum. More sophisticated definitions of convergence permit one to work with series of this form, but it is better to sum the series first and then differentiate (the second method).] Page 5, Prob. 3.8, midpage: the reference to Eq. 3.7 should be 3.7. Page 53, Prob. 3.(b), line 5: A should be R. σ 4ɛ 0R ; next line, insert r after Page 55, Prob. 3.3, third displayed equation: remove the first Φ. Page 58, Prob. 3.8(a), second line, first integral: R 3 should read R. Page 59, Prob. 3.3(c): change first V to W. Page 64, Prob. 3.4(a), lines and 3: remove ɛ 0 in the first factor in the expressions for E ave ; in the second expression change ρ to q. Page 69, Prob. 3.47, at the end add the following: Alternatively, start with the separable solution V (x, y) =(C sin kx + D cos kx) ( Ae ky + Be ky). Note that the configuration is symmetric in x, soc = 0, and V (x, 0) = 0 B = A, so (combining the constants) V (x, y) =A cos kx sinh ky. But V (b, y) =0,socoskb = 0, which means that kb = ±π/, ±3π/,, or k =(n )π/b α n, with n =,, 3,... (negative k does not yield a different solution the sign can be absorbed into A). The general linear combination is V (x, y) = A n cos α n x sinh α n y, n= and it remains to fit the final boundary condition: V (x, a) =V 0 = A n cos α n x sinh α n a. n= ]. 4

5 Use Fourier s trick, multiplying by cos α n x and integrating: b b V 0 cos α n xdx= A n sinh α n a cos α n x cos α n xdx b So A n = V 0 b V 0 sin α n b α n = n= b A n sinh α n a (bδ n n)=ba n sinh α n a; n= sin α n b α n sinh α n a. But sin α nb = sin V (x, y) = V 0 b ( n ( ) n sinh α ny α n sinh α n a cos α nx. n= ) π = ( ) n, so Page 74, Prob. 4.4: exponent on r in boxed equation should be 5, not 3. Page 75, Prob. 4.7: replace the (defective) solution with the following: If the potential is zero at infinity, the energy of a point charge Q is (Eq..39) W = QV (r). For a physical dipole, with q at r and +q at r+d, [ ] r+d U = qv (r + d) qv (r) =q [V (r + d) V (r)] = q E dl. For an ideal dipole the integral reduces to E d, and U = qe d = p E, since p = qd. If you do not (or cannot) use infinity as the reference point, the result still holds, as long as you bring the two charges in from the same point, r 0 (or two points at the same potential). In that case W = Q [V (r) V (r 0 )], and U = q [V (r + d) V (r 0 )] q [V (r) V (r 0 )] = q [V (r + d) V (r)], as before. Page 75, Prob. 4.0(a): r should be 3 r. Page 79, Prob. 4.9: in the upper right box of the Table (σ f for air) there is a missing factor of ɛ 0. Page 9, Problem 5.0(b): in the first line µ 0 I /π should read µ 0 I a/πs; in the final boxed equation the first should be a s. Page 9, Prob. 5.5: the signs are all wrong. The end of line should read right (ẑ), the middle of the next line should read left ( ẑ). In the first box it should be (n n ), and in the second box the minus sign does not belong. r 5

6 Page 4, Prob. 6.4: last term in second expression for F should be +ẑ Bx z (plus, not minus). Page 9, Prob. 6.(a): replace with the following: The magnetic force on the dipole is given by Eq. 6.3; to move the dipole in from infinity we must exert an opposite force, so the work done is U = r F dl = r (m B) dl = m B(r)+m B( ) (I used the gradient theorem, Eq..55). As long as the magnetic field goes to zero at infinity, then, U = m B. If the magnetic field does not go to zero at infinity, one must stipulate that the dipole starts out oriented perpendicular to the field. Page 5, Prob. 7.(b): in the box, c should be C. Page 9, Prob. 7.8: change first two lines to read: Φ= B da; B = µ 0I πs ˆφˆφˆφ; Φ= µ s+a 0Ia ds π s s = µ ( ) 0Ia s + a π ln ; s E = I loop R = dq dt R = dφ dt = µ 0a ln( + a/s)di π dt. dq = µ 0a ln( + a/s) di πr Q = µ 0aI ln( + a/s). πr Page 3, Prob. 7.7: in the second integral, r should be s. Page 3, Prob. 7.3(c), last line: in the final two equations, insert an I immediately after µ 0. Page 40, Prob. 7.47: in the box, the top equation should have a minus sign in front, and in the bottom equation the plus sign should be minus. Page 4, Prob. 7.50, final answer: R should read R. Page 43, Prob. 7.55, penultimate displayed equation: tp should be. Page 47, Prob. 8., top line, penultimate expression: change a to a 4 ;in (c), in the first box, change 6 to 8. Page 49, Prob. 8.5(c): there should be a minus sign in front of σ in the box. Page 49, Prob. 8.7: almost all the r s here should be s s. In line change a <r<r to s<r ; in the same line change dr to ds; in the next line change dr to ds (twice), and change ˆr to ŝ; in the last line change r to s, dr to ds, and ˆr to ŝ (but leave r as is). 6

7 Page 53, Prob. 8., last line of equations: in the numerator of the expression for R change.0 to.0. Page 75, Prob. 9.34, penultimate line: α = n 3 /n (not n 3 /n 3 ). Page 77, Prob. 9.38: half-way down, remove minus sign in k x + k y + k z = (ω/c). Page 8, Prob. 0.8: first line: remove. Page 84, Prob. 0.4: in the first line, change (9.98) to (0.4). Page 03, Prob..4: at beginning of second paragraph, remove. Page, Prob..5, end of first sentence: change comma to period. Page 5, Prob..3. The figure contains two errors: the slopes are for v/c = / (not 3/), and the intervals are incorrect. The correct solution is as follows: Page 7, Prob..33: first expression in third line, change c to c. 7

### xy 2 e 2z dx dy dz = 8 3 (1 e 4 ) = 2.62 mc. 12 x2 y 3 e 2z 2 m 2 m 2 m Figure P4.1: Cube of Problem 4.1.

Problem 4.1 A cube m on a side is located in the first octant in a Cartesian coordinate system, with one of its corners at the origin. Find the total charge contained in the cube if the charge density

### Homework Assignment 4 Solution Set

Homework Assignment 4 Solution Set PHYCS 442 7 February, 24 Problem (Griffiths 2.37 If the plates are sufficiently large the field near them does not depend on d. The field between the plates is zero (the

### Corrections and Minor Revisions of Mathematical Methods in the Physical Sciences, third edition, by Mary L. Boas (deceased)

Corrections and Minor Revisions of Mathematical Methods in the Physical Sciences, third edition, by Mary L. Boas (deceased) Updated December 6, 2017 by Harold P. Boas This list includes all errors known

### Curvilinear Coordinates

University of Alabama Department of Physics and Astronomy PH 106-4 / LeClair Fall 2008 Curvilinear Coordinates Note that we use the convention that the cartesian unit vectors are ˆx, ŷ, and ẑ, rather than

### UNIVERSITY OF CALIFORNIA - SANTA CRUZ DEPARTMENT OF PHYSICS PHYS 110A. Homework #7. Benjamin Stahl. March 3, 2015

UNIVERSITY OF CALIFORNIA - SANTA CRUZ DEPARTMENT OF PHYSICS PHYS A Homework #7 Benjamin Stahl March 3, 5 GRIFFITHS, 5.34 It will be shown that the magnetic field of a dipole can written in the following

### 3. Calculating Electrostatic Potential

3. Calculating Electrostatic Potential 3. Laplace s Equation 3. The Method of Images 3.3 Separation of Variables 3.4 Multipole Expansion 3.. Introduction The primary task of electrostatics is to study

### UNIVERSITY OF CALIFORNIA - SANTA CRUZ DEPARTMENT OF PHYSICS PHYS 110A. Homework #6. Benjamin Stahl. February 17, 2015

UNIVERSITY OF CALIFORNIA - SANTA CRUZ DEPARTMENT OF PHYSICS PHYS A Homework #6 Benjamin Stahl February 7, 5 GRIFFITHS, 5.9 The magnetic field at a point, P, will be found for each of the steady current

### Single Particle Motion

Single Particle Motion C ontents Uniform E and B E = - guiding centers Definition of guiding center E gravitation Non Uniform B 'grad B' drift, B B Curvature drift Grad -B drift, B B invariance of µ. Magnetic

### Physics 505 Fall 2005 Homework Assignment #7 Solutions

Physics 505 Fall 005 Homework Assignment #7 Solutions Textbook problems: Ch. 4: 4.10 Ch. 5: 5.3, 5.6, 5.7 4.10 Two concentric conducting spheres of inner and outer radii a and b, respectively, carry charges

### Quiz 4 (Discussion Session) Phys 1302W.400 Spring 2018

Quiz 4 (Discussion ession) Phys 1302W.400 pring 2018 This group quiz consists of one problem that, together with the individual problems on Friday, will determine your grade for quiz 4. For the group problem,

### Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9

MATH 32B-2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B-2 (8W)

### SOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours)

SOLUTIONS TO THE 18.02 FINAL EXAM BJORN POONEN December 14, 2010, 9:00am-12:00 (3 hours) 1) For each of (a)-(e) below: If the statement is true, write TRUE. If the statement is false, write FALSE. (Please

### MATH503 - HOMEWORK #2. v k. + v i. (fv i ) = f v i. + v k. ) + u j (ɛ ijk. ) u j (ɛ jik

1 KO UNIVERITY Mon April 18, ollege of Arts and ciences Handout # Department of Physics Instructor: Alkan Kabakço lu MATH - HOMEWORK # 1 ( f) = Using Einstein notation and expressing x, y, x as e 1, e,

### 7 Curvilinear coordinates

7 Curvilinear coordinates Read: Boas sec. 5.4, 0.8, 0.9. 7. Review of spherical and cylindrical coords. First I ll review spherical and cylindrical coordinate systems so you can have them in mind when

### Electric fields in matter

Electric fields in matter November 2, 25 Suppose we apply a constant electric field to a block of material. Then the charges that make up the matter are no longer in equilibrium: the electrons tend to

### Supporting Information

Supporting Information A: Calculation of radial distribution functions To get an effective propagator in one dimension, we first transform 1) into spherical coordinates: x a = ρ sin θ cos φ, y = ρ sin

### Maxima/minima with constraints

Maxima/minima with constraints Very often we want to find maxima/minima but subject to some constraint Example: A wire is bent to a shape y = 1 x 2. If a string is stretched from the origin to the wire,

### MATHS 267 Answers to Stokes Practice Dr. Jones

MATH 267 Answers to tokes Practice Dr. Jones 1. Calculate the flux F d where is the hemisphere x2 + y 2 + z 2 1, z > and F (xz + e y2, yz, z 2 + 1). Note: the surface is open (doesn t include any of the

### S12.1 SOLUTIONS TO PROBLEMS 12 (ODD NUMBERS)

OLUTION TO PROBLEM 2 (ODD NUMBER) 2. The electric field is E = φ = 2xi + 2y j and at (2, ) E = 4i + 2j. Thus E = 2 5 and its direction is 2i + j. At ( 3, 2), φ = 6i + 4 j. Thus the direction of most rapid

### The Central Force Problem: Hydrogen Atom

The Central Force Problem: Hydrogen Atom B. Ramachandran Separation of Variables The Schrödinger equation for an atomic system with Z protons in the nucleus and one electron outside is h µ Ze ψ = Eψ, r

### PHY481 - Outline of solutions to Homework 3

1 PHY481 - Outline of solutions to Homework 3 Problem 3.8: We consider a charge outside a conducting sphere that is neutral. In order that the sphere be neutral, we have to introduce a new image charge

### Polymer Theory: Freely Jointed Chain

Polymer Theory: Freely Jointed Chain E.H.F. February 2, 23 We ll talk today about the most simple model for a single polymer in solution. It s called the freely jointed chain model. Each monomer occupies

### PHYS 110B - HW #6 Spring 2004, Solutions by David Pace Any referenced equations are from Griffiths Problem statements are paraphrased

PHYS B - HW #6 Spring 4, Solution by David Pace Any referenced equation are from Griffith Problem tatement are paraphraed. Problem. from Griffith Show that the following, A µo ɛ o A V + A ρ ɛ o Eq..4 A

### Introduction and Vectors Lecture 1

1 Introduction Introduction and Vectors Lecture 1 This is a course on classical Electromagnetism. It is the foundation for more advanced courses in modern physics. All physics of the modern era, from quantum

### r 4 r 2 q 2 r 3 V () r En dln (r) 1 q 1 Negative sign from integration of E field cancels the negative sign from the original equation.

Question from last class Potential due to a Group of Point Charges rr V () r E dl r r 1 q 1 X rr N V () r E dl r n n N V(r) V n (r) 1 n1 4 o 1/30/2018 2 q 2 N r r N r r q n V () r En dln dr 2 n n r n r

### Summary: Curvilinear Coordinates

Physics 2460 Electricity and Magnetism I, Fall 2007, Lecture 10 1 Summary: Curvilinear Coordinates 1. Summary of Integral Theorems 2. Generalized Coordinates 3. Cartesian Coordinates: Surfaces of Constant

### University of California, Berkeley Physics H7B Spring 1999 (Strovink) SOLUTION TO PROBLEM SET 12 Solutions by P. Pebler. ( 5 gauss) ẑ 1+[k(x + ct)] 2

University of California, Berkeley Physics H7B Spring 1999 (Strovink) SOLUTION TO PROBLEM SET 12 Solutions by P. Pebler 1 Purcell 9.3 A free proton was at rest at the origin before the wave E = (5 statvolt/cm)

### Without fully opening the exam, check that you have pages 1 through 10.

MTH 234 Solutions to Exam 2 April 11th 216 Name: Section: Recitation Instructor: INSTRUTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through

### Electric Potential. David J. Starling Penn State Hazleton PHYS 212. Electricity is really just organized lightning. - George Carlin.

Electricity is really just organized lightning. - George Carlin David J. Starling Penn State Hazleton PHYS 212 Since the electric force is so similar to gravity, might it be conservative? Yes! If we move

### Chapter 28 Sources of Magnetic Field

Chapter 28 Sources of Magnetic Field In this chapter we investigate the sources of magnetic of magnetic field, in particular, the magnetic field produced by moving charges (i.e., currents). Ampere s Law

### The Spin (continued). February 8, 2012

The Spin continued. Magnetic moment of an electron Particle wave functions including spin Stern-Gerlach experiment February 8, 2012 1 Magnetic moment of an electron. The coordinates of a particle include

### Magnetic Materials. 1. Magnetization 2. Potential and field of a magnetized object

Magnetic Materials 1. Magnetization 2. Potential and field of a magnetized object 3. H-field 4. Susceptibility and permeability 5. Boundary conditions 6. Magnetic field energy and magnetic pressure 1 Magnetic

### Chapter 29: Maxwell s Equation and EM Waves. Slide 29-1

Chapter 29: Maxwell s Equation and EM Waves Slide 29-1 Equations of electromagnetism: a review We ve now seen the four fundamental equations of electromagnetism, here listed together for the first time.

### 7a3 2. (c) πa 3 (d) πa 3 (e) πa3

1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin

### Probability with Applications and R: Errata Updated: October 27, 2016

Probability with Applications and R: Errata Updated: October 7, 06 Page xii, lines -4: The first sentence should be changed as: The book references numerous R script files which are available for download.

### TUTORIAL 4. Proof. Computing the potential at the center and pole respectively,

TUTORIAL 4 Problem 1 An inverted hemispherical bowl of radius R carries a uniform surface charge density σ. Find the potential difference between the north pole and the center. Proof. Computing the potential

### The dependence of the cross-sectional shape on the hydraulic resistance of microchannels

3-weeks course report, s0973 The dependence of the cross-sectional shape on the hydraulic resistance of microchannels Hatim Azzouz a Supervisor: Niels Asger Mortensen and Henrik Bruus MIC Department of

### Vector Potential for the Magnetic Field

Vector Potential for the Magnetic Field Let me start with two two theorems of Vector Calculus: Theorem 1: If a vector field has zero curl everywhere in space, then that field is a gradient of some scalar

### Expansion of 1/r potential in Legendre polynomials

Expansion of 1/r potential in Legendre polynomials In electrostatics and gravitation, we see scalar potentials of the form V = K d Take d = R r = R 2 2Rr cos θ + r 2 = R 1 2 r R cos θ + r R )2 Use h =

### Currents (1) Line charge λ (C/m) with velocity v : in time t, This constitutes a current I = λv (vector). Magnetic force on a segment of length dl is

Magnetostatics 1. Currents 2. Relativistic origin of magnetic field 3. Biot-Savart law 4. Magnetic force between currents 5. Applications of Biot-Savart law 6. Ampere s law in differential form 7. Magnetic

### Solid State Physics FREE ELECTRON MODEL. Lecture 17. A.H. Harker. Physics and Astronomy UCL

Solid State Physics FREE ELECTRON MODEL Lecture 17 A.H. Harker Physics and Astronomy UCL Magnetic Effects 6.7 Plasma Oscillations The picture of a free electron gas and a positive charge background offers

### Solutions to Exam 1, Math Solution. Because f(x) is one-to-one, we know the inverse function exists. Recall that (f 1 ) (a) =

Solutions to Exam, Math 56 The function f(x) e x + x 3 + x is one-to-one (there is no need to check this) What is (f ) ( + e )? Solution Because f(x) is one-to-one, we know the inverse function exists

### 1. If the line l has symmetric equations. = y 3 = z+2 find a vector equation for the line l that contains the point (2, 1, 3) and is parallel to l.

. If the line l has symmetric equations MA 6 PRACTICE PROBLEMS x = y = z+ 7, find a vector equation for the line l that contains the point (,, ) and is parallel to l. r = ( + t) i t j + ( + 7t) k B. r

### Physics 9 Spring 2012 Midterm 1 Solutions

Physics 9 Spring 22 NAME: TA: Physics 9 Spring 22 Midterm s For the midterm, you may use one sheet of notes with whatever you want to put on it, front and back. Please sit every other seat, and please

### Foundation Course Unit. Fields and Waves. SOLUTIONS. Examiners: Dr. A. G. Polnarev Prof. F. Di Lodovico

Foundation Course Unit SEF006 Fields and Waves. SOLUTIONS Examiners: Dr. A. G. Polnarev Prof. F. Di Lodovico c Queen Mary University of London, 2016 Page 2 SEF006 (2016) SECTION A SOLUTIONS TO SECTION

### M273Q Multivariable Calculus Spring 2017 Review Problems for Exam 3

M7Q Multivariable alculus Spring 7 Review Problems for Exam Exam covers material from Sections 5.-5.4 and 6.-6. and 7.. As you prepare, note well that the Fall 6 Exam posted online did not cover exactly

### Tangent Plane. Linear Approximation. The Gradient

Calculus 3 Lia Vas Tangent Plane. Linear Approximation. The Gradient The tangent plane. Let z = f(x, y) be a function of two variables with continuous partial derivatives. Recall that the vectors 1, 0,

### Introduction to Differentials

Introduction to Differentials David G Radcliffe 13 March 2007 1 Increments Let y be a function of x, say y = f(x). The symbol x denotes a change or increment in the value of x. Note that a change in the

### Review for the Final Exam

Math 171 Review for the Final Exam 1 Find the limits (4 points each) (a) lim 4x 2 3; x x (b) lim ( x 2 x x 1 )x ; (c) lim( 1 1 ); x 1 ln x x 1 sin (x 2) (d) lim x 2 x 2 4 Solutions (a) The limit lim 4x

### Electromagnetism Answers to Problem Set 8 Spring Jackson Prob. 4.1: Multipole expansion for various charge distributions

Electromagnetism 76 Answers to Problem Set 8 Spring 6. Jackson Prob. 4.: Multipole expansion for various charge distributions (a) In the first case, we have 4 charges in the xy plane at distance a from

### Physics 116C Solutions to the Practice Final Exam Fall The method of Frobenius fails because it works only for Fuchsian equations, of the type

Physics 6C Solutions to the Practice Final Exam Fall 2 Consider the differential equation x y +x(x+)y y = () (a) Explain why the method of Frobenius method fails for this problem. The method of Frobenius

### on an open connected region D, then F is conservative on D. (c) If curl F=curl G on R 3, then C F dr = C G dr for all closed path C.

. (5%) Determine the statement is true ( ) or false ( ). 微甲 -4 班期末考解答和評分標準 (a) If f(x, y) is continuous on the rectangle R = {(x, y) a x b, c y d} except for finitely many points, then f(x, y) is integrable

### Calculus II Practice Test Problems for Chapter 7 Page 1 of 6

Calculus II Practice Test Problems for Chapter 7 Page of 6 This is a set of practice test problems for Chapter 7. This is in no way an inclusive set of problems there can be other types of problems on

### Qualifying Examination Electricity and Magnetism Solutions January 12, 2006

1 Qualifying Examination Electicity and Magnetism Solutions Januay 12, 2006 PROBLEM EA. a. Fist, we conside a unit length of cylinde to find the elationship between the total chage pe unit length λ and

### PHY132 Lecture 13 02/24/2010. Lecture 13 1

Classical Physics II PHY132 Lecture 13 Magnetism II: Magnetic torque Lecture 13 1 Magnetic Force MAGNETISM is yet another force that has been known since a very long time. Its name stems from the mineral

### Jim Lambers MAT 280 Fall Semester Practice Final Exam Solution

Jim Lambers MAT 8 Fall emester 6-7 Practice Final Exam olution. Use Lagrange multipliers to find the point on the circle x + 4 closest to the point (, 5). olution We have f(x, ) (x ) + ( 5), the square

### MTH 234 Exam 2 November 21st, Without fully opening the exam, check that you have pages 1 through 12.

Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 12. Show all your work on the standard

### AAPT UNITED STATES PHYSICS TEAM AIP 2007

007 Semifinal Exam Solutions 1 AAPT UNITED STATES PHYSICS TEAM AIP 007 Solutions to Problems Part A Question 1 a. There is a high degree of symmetry present. Points b, c, and e are at the same potential;

### Problem Solving 1: Line Integrals and Surface Integrals

A. Line Integrals MASSACHUSETTS INSTITUTE OF TECHNOLOY Department of Physics Problem Solving 1: Line Integrals and Surface Integrals The line integral of a scalar function f ( xyz),, along a path C is

### Chapter-2 2.1)Page-28, Eq (2-9): should read (add parentheses around the terms involving the x-end-points):

The following is an errata for the text Quantum mechanics and Path Integrals by R. Feynman edited by A. Hibbs (1965) Mc Graw Hill, New York. For the latest version of this document visit: http://mathematicuslabs.com/pi

### Integral Theorems. September 14, We begin by recalling the Fundamental Theorem of Calculus, that the integral is the inverse of the derivative,

Integral Theorems eptember 14, 215 1 Integral of the gradient We begin by recalling the Fundamental Theorem of Calculus, that the integral is the inverse of the derivative, F (b F (a f (x provided f (x

### Chapter 30 Solutions

Chapter 30 Solutions 30.1 B µ 0I R µ 0q(v/π R) R 1.5 T *30. We use the Biot-Savart law. For bits of wire along the straight-line sections, ds is at 0 or 180 to ~, so ds ~ 0. Thus, only the curved section

### abc Mathematics Further Pure General Certificate of Education SPECIMEN UNITS AND MARK SCHEMES

abc General Certificate of Education Mathematics Further Pure SPECIMEN UNITS AND MARK SCHEMES ADVANCED SUBSIDIARY MATHEMATICS (56) ADVANCED SUBSIDIARY PURE MATHEMATICS (566) ADVANCED SUBSIDIARY FURTHER

### Jim Lambers MAT 280 Summer Semester Practice Final Exam Solution. dy + xz dz = x(t)y(t) dt. t 3 (4t 3 ) + e t2 (2t) + t 7 (3t 2 ) dt

Jim Lambers MAT 28 ummer emester 212-1 Practice Final Exam olution 1. Evaluate the line integral xy dx + e y dy + xz dz, where is given by r(t) t 4, t 2, t, t 1. olution From r (t) 4t, 2t, t 2, we obtain

### Quantum Mechanics in Three Dimensions

Physics 342 Lecture 21 Quantum Mechanics in Three Dimensions Lecture 21 Physics 342 Quantum Mechanics I Monday, March 22nd, 21 We are used to the temporal separation that gives, for example, the timeindependent

### LESSON 2 PHYSICS NOTES

LESSON 2 ELECTROSTATIC POTENTIAL AND CAPACITANCE SECTION I ELECTROSTATIC POTENTIAL ELECTRIC FIELD IS CONSERVATIVE In an electric field work done by the electric field in moving a unit positive charge from

### MATH 104 MID-TERM EXAM SOLUTIONS. (1) Find the area of the region enclosed by the curves y = x 1 and y = x 1

MATH MID-TERM EXAM SOLUTIONS CLAY SHONKWILER ( Find the area of the region enclosed by the curves y and y. Answer: First, we find the points of intersection by setting the two functions equal to eachother:.

### Physics 214 Solution Set 4 Winter 2017

Physics 14 Solution Set 4 Winter 017 1. [Jackson, problem 1.3] A particle with mass m and charge e moves in a uniform, static, electric field E 0. (a) Solve for the velocity and position of the particle

### Physics 2212 GH Quiz #4 Solutions Spring 2016

Physics 2212 GH Quiz #4 Solutions Spring 2016 I. (18 points) A bar (mass m, length L) is connected to two frictionless vertical conducting rails with loops of wire, in the presence of a uniform magnetic

### Problem Max. Possible Points Total

MA 262 Exam 1 Fall 2011 Instructor: Raphael Hora Name: Max Possible Student ID#: 1234567890 1. No books or notes are allowed. 2. You CAN NOT USE calculators or any electronic devices. 3. Show all work

### Handout 8: Sources of magnetic field. Magnetic field of moving charge

1 Handout 8: Sources of magnetic field Magnetic field of moving charge Moving charge creates magnetic field around it. In Fig. 1, charge q is moving at constant velocity v. The magnetic field at point

### Faraday s Law. Faraday s Law of Induction Motional emf. Lenz s Law. Motors and Generators. Eddy Currents

Faraday s Law Faraday s Law of Induction Motional emf Motors and Generators Lenz s Law Eddy Currents Induced EMF A current flows through the loop when a magnet is moved near it, without any batteries!

### Calculation of Reflection and Transmission Coefficients in scuff-transmission

Calculation of Reflection and Transmission Coefficients in scuff-transmission Homer Reid May 9, 2015 Contents 1 The Setup 2 2 Scattering coefficients from surface currents 4 2.1 Computation of b(q).........................

### ERRATA ET ADDENDA DIOPHANTINE APPROXIMATIONS AND VALUE DISTRIBUTION THEORY (SLN 1239) Paul Vojta 1 July 2003; last revised 11 August 2017

ERRATA ET ADDENDA DIOPHANTINE APPROXIMATIONS AND VALUE DISTRIBUTION THEORY (SLN 1239) Paul Vojta 1 July 2003; last revised 11 August 2017 Page iii, line 5 Change number number field to number field. Page

### Chapter 6: Vector Analysis

Chapter 6: Vector Analysis We use derivatives and various products of vectors in all areas of physics. For example, Newton s 2nd law is F = m d2 r. In electricity dt 2 and magnetism, we need surface and

### e x3 dx dy. 0 y x 2, 0 x 1.

Problem 1. Evaluate by changing the order of integration y e x3 dx dy. Solution:We change the order of integration over the region y x 1. We find and x e x3 dy dx = y x, x 1. x e x3 dx = 1 x=1 3 ex3 x=

### Core Electrodynamics. Sandra C. Chapman. June 29, 2010 CORE ELECTRODYNAMICS COPYRIGHT SANDRA C CHAPMAN

Core Electrodynamics Sandra C. Chapman June 29, 2010 Preface This monograph is based on a final year undergraduate course in Electrodynamics that I introduced at Warwick as part of the new four year physics

### arxiv:physics/ v1 [physics.ed-ph] 24 May 2006

arxiv:physics/6521v1 [physics.ed-ph] 24 May 26 Comparing a current-carrying circular wire with polygons of equal perimeter: Magnetic field versus magnetic flux J P Silva and A J Silvestre Instituto Superior

### Here are some solutions to the sample problems assigned for Chapter 6.8 to 6.11.

Lecture 3 Appendi B: Some sample problems from Boas Here are some solutions to the sample problems assigned for Chapter 6.8 to 6.. 6.8: Solution: We want to practice doing closed line integrals of the

### Calculus III. Math 233 Spring Final exam May 3rd. Suggested solutions

alculus III Math 33 pring 7 Final exam May 3rd. uggested solutions This exam contains twenty problems numbered 1 through. All problems are multiple choice problems, and each counts 5% of your total score.

### SOLUTIONS TO HOMEWORK ASSIGNMENT #2, Math 253

SOLUTIONS TO HOMEWORK ASSIGNMENT #, Math 5. Find the equation of a sphere if one of its diameters has end points (, 0, 5) and (5, 4, 7). The length of the diameter is (5 ) + ( 4 0) + (7 5) = =, so the

### Hertz potentials in curvilinear coordinates

Hertz potentials in curvilinear coordinates Jeff Bouas Texas A&M University July 9, 2010 Quantum Vacuum Workshop Jeff Bouas (Texas A&M University) Hertz potentials in curvilinear coordinates July 9, 2010

### MTH 234 Solutions to Exam 2 April 10th, Without fully opening the exam, check that you have pages 1 through 12.

MTH 34 Solutions to am April 1th, 17 Name: Section: Recitation Instructor: INSTRUTIONS Fill in your name, etc. on this first page. Without fully opening the eam, check that you have pages 1 through 1.

### Physics 236a assignment, Week 2:

Physics 236a assignment, Week 2: (October 8, 2015. Due on October 15, 2015) 1. Equation of motion for a spin in a magnetic field. [10 points] We will obtain the relativistic generalization of the nonrelativistic

### Faraday's Law ds B B G G ΦB B ds Φ ε = d B dt

Faraday's Law ds ds ε= d Φ dt Φ Global Review Electrostatics» motion of q in external E-field» E-field generated by Σq i Magnetostatics» motion of q and i in external -field» -field generated by I Electrodynamics»

### Lecture 10 - Moment of Inertia

Lecture 10 - oment of Inertia A Puzzle... Question For any object, there are typically many ways to calculate the moment of inertia I = r 2 dm, usually by doing the integration by considering different

### Electrodynamics HW Problems 06 EM Waves

Electrodynamics HW Problems 06 EM Waves 1. Energy in a wave on a string 2. Traveling wave on a string 3. Standing wave 4. Spherical traveling wave 5. Traveling EM wave 6. 3- D electromagnetic plane wave

### Electricity. Revision Notes. R.D.Pilkington

Electricity Revision Notes R.D.Pilkington DIRECT CURRENTS Introduction Current: Rate of charge flow, I = dq/dt Units: amps Potential and potential difference: work done to move unit +ve charge from point

### Exam 2 Solutions. ε 3. ε 1. Problem 1

Exam 2 Solutions Problem 1 In the circuit shown, R1=100 Ω, R2=25 Ω, and the ideal batteries have EMFs of ε1 = 6.0 V, ε2 = 3.0 V, and ε3 = 1.5 V. What is the magnitude of the current flowing through resistor

### A Mathematical Trivium

A Mathematical Trivium V.I. Arnold 1991 1. Sketch the graph of the derivative and the graph of the integral of a function given by a freehand graph. 2. Find the limit lim x 0 sin tan x tan sin x arcsin

### Complex Differentials and the Stokes, Goursat and Cauchy Theorems

Complex Differentials and the Stokes, Goursat and Cauchy Theorems Benjamin McKay June 21, 2001 1 Stokes theorem Theorem 1 (Stokes) f(x, y) dx + g(x, y) dy = U ( g y f ) dx dy x where U is a region of the

### Solutions for the Practice Final - Math 23B, 2016

olutions for the Practice Final - Math B, 6 a. True. The area of a surface is given by the expression d, and since we have a parametrization φ x, y x, y, f x, y with φ, this expands as d T x T y da xy

### Aperture Antennas 1 Introduction

1 Introduction Very often, we have antennas in aperture forms, for example, the antennas shown below: Pyramidal horn antenna Conical horn antenna 1 Paraboloidal antenna Slot antenna Analysis Method for.1

### Cambridge International Examinations Cambridge International General Certificate of Secondary Education. Published

Cambridge International Eaminations Cambridge International General Certificate of Secondary Education ADDITIONAL MATHEMATICS 0606/ Paper October/November 06 MARK SCHEME Maimum Mark: 80 Published This

### The dielectrophoretic force on dielectric spherical objects

10064 - Physics Project The dielectrophoretic force on dielectric spherical objects Andreas Ammitzbøll s042191 Janosch Michael Cristoph Rauba s042606 Marco Haller Schultz s042357 Supervisors: Niels Asger

### SOLUTIONS 1 (27) 2 (18) 3 (18) 4 (15) 5 (22) TOTAL (100) PROBLEM NUMBER SCORE MIDTERM 2. Form A. Recitation Instructor : Recitation Time :

Math 5 March 8, 206 Form A Page of 8 Name : OSU Name.# : Lecturer:: Recitation Instructor : SOLUTIONS Recitation Time : SHOW ALL WORK in problems, 2, and 3. Incorrect answers with work shown may receive

### PHY2049 Fall11. Final Exam Solutions (1) 700 N (2) 350 N (3) 810 N (4) 405 N (5) 0 N

Exam Solutions 1. Three charges form an equilateral triangle of side length d = 2 cm. The top charge is q3 = 3 μc, while the bottom two are q1 = q2 = - 6 μc. What is the magnitude of the net force acting

### Problem Set #5: 5.2, 5.4, 5.8, 5.12, 5.15, 5.19, 5.24, 5.27, 5.35 (Due Tuesday, April 8th)

Chapter 5 Magnetostatics Problem Set #5: 5.2, 5.4, 5.8, 5.12, 5.15, 5.19, 5.24, 5.27, 5.35 (Due Tuesday, April 8th 5.1 Lorentz Force So far we were concerned with forces on test charges Q due to static