EE 333 Electricity and Magnetism, Fall 2009 Homework #9 solution
|
|
- Dana Ball
- 5 years ago
- Views:
Transcription
1 EE 333 Electricity and Magnetism, Fall 009 Homework #9 solution The two infinite conducting cones θ = θ 1, and θ = θ are maintained at the two potentials Φ 1 = 100, and Φ = 0, respectively, as shown in Figure P4.10. a) Use Laplace s equation in the spherical coordinates to solve for the potential variation between the two cones. b) Calculate the electric field vector in the region between the two cones and the charge density on each conductor. a) In spherical coordinates we have Φ = 1 ) r Φ + 1 r r r r sin θ sin θ Φ ) + θ θ 1 Φ r sin θ φ Now, first we realize that because of rotational symmetry around the z-axis θ = 0), the potential will not depend on φ, so the last term in the Laplace equation is zero. Next we realize that we can scale the cones up and down in size, and because they extend to infinite, the system is unchanged. Therefore, the potential also does not depend on r. We are then left only with the θ-variation: or Φ = 1 sin θ Φ ) = 0 r sin θ θ θ sin θ Φ ) = 0 θ θ sin θ Φ θ = α Φ θ = With the help of an integral table we get α sin θ Φ = α ln tan θ + β Since θ [0; 180 ], we can remove the absolute value sign to get Φ r, θ, φ) = α ln tan θ ) + β 1
2 Next we apply the boundary conditions: and Φ r, θ 1, φ) = 100 Φ r, θ, φ) = 0 We see that and then or β = α ln tan θ ) α ln tan θ ) 1 α ln tan θ ) = α = ln tan θ 1 ln tan θ = 100 ln tan θ 1 tan θ Note, α < 0) b) The electric field is the gradient of the potential. In spherical coordinates we have E = Φ = Φ r ˆr 1 Φ r θ ˆθ 1 Φ r sin θ φ ˆφ Because Φ only depends on θ it simplifies to E = 1 Φ r θ ˆθ = 1 α ln tan θ ) r θ + β ˆθ = 1 α r sin θ ˆθ The surface charge density on the upper conductor is
3 it is positive) σ 1 = ǫ 0 E 1θ = 1 α r sin θ 1 The surface charge density on the lower conductor is it is negative) σ = ǫ 0 E θ = 1 α r sin θ Consider the two parallel plates shown in Figure P4.11. The region between the parallel plates is filled with a nonunifom charge distribution of density ρy) = σ o y, where σ o is a constant. Solve Poisson s equation in the region between the parallel plates to show that the potential distribution Φy) is given by Φy) = d y + σ o 6ǫ o yd y 3) Poisson s equation in cartesian coordinates looks like this: Φ x + Φ y + Φ z = ρ ǫ 0 Because of symmetry only the y-derivative is non-zero, so and thus Φ y = σ oy ǫ 0 Φ y = σ o ǫ 0 y + α The boundary conditions are Φ = σ o y 3 + αy + β and thus Φ0) = 0 Φd) = and β = 0 = σ od 3 + αd 3
4 Inserting we get α = d + σ od Φ = σ oy 3 + d + σ ) od y = d y σ oy 3 + σ od y = d y + σ o d y y 3) 4.1. Use the expression of the potential distribution in problem 11 to obtain an expression for the electric field between the two plates. Show that the charge density at the lower plate is given by ρ s = ǫ 0 d + σ ) o d whereas the charge density at the upper plate is given by ρ s = ǫ 0 d σ ) o d 3ǫ 0 The capacitance is defined as C = Q and because in this case there are two different values of Q on the lower and upper plates for the same potential difference, there is no unique value for the capacitance under these circumstances. Compute the electric field as E = Φ Since the potential only varies with y, the electric field points in the ŷ direction, and E = ŷ Φ y = ŷ y = ŷ [ d y + σ o yd y 3) ] ] [ d + σ o d 3y ) This is the field pointing in the positive ŷ direction, so the charge density on the lower conductor is 4
5 ρ s =ǫ 0 E y y = 0) d σ o d 0 ) 6 d σ od 6 d + σ ) od and the charge density on the upper conductor is ρ s E y y = d) = ǫ 0 d + σ o d 3d ) 6 = ǫ 0 d σ od 6 =ǫ 0 d σ ) od 3 5
Lecture 13: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay. Poisson s and Laplace s Equations
Poisson s and Laplace s Equations Lecture 13: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay We will spend some time in looking at the mathematical foundations of electrostatics.
More informationFORMULA SHEET FOR QUIZ 2 Exam Date: November 8, 2017
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Electromagnetism II November 5, 207 Prof. Alan Guth FORMULA SHEET FOR QUIZ 2 Exam Date: November 8, 207 A few items below are marked
More informationSummary: 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
More informationHomework 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
More informationPHY481 - 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
More informationElectrostatics. Chapter Maxwell s Equations
Chapter 1 Electrostatics 1.1 Maxwell s Equations Electromagnetic behavior can be described using a set of four fundamental relations known as Maxwell s Equations. Note that these equations are observed,
More informationIndiana University Physics P331: Theory of Electromagnetism Review Problems #3
Indiana University Physics P331: Theory of Electromagnetism Review Problems #3 Note: The final exam (Friday 1/14 8:00-10:00 AM will be comprehensive, covering lecture and homework material pertaining to
More informationVector Integrals. Scott N. Walck. October 13, 2016
Vector Integrals cott N. Walck October 13, 16 Contents 1 A Table of Vector Integrals Applications of the Integrals.1 calar Line Integral.........................1.1 Finding Total Charge of a Line Charge..........1.
More informationE&M. 1 Capacitors. January 2009
E&M January 2009 1 Capacitors Consider a spherical capacitor which has the space between its plates filled with a dielectric of permittivity ɛ. The inner sphere has radius r 1 and the outer sphere has
More informationBoundary value problems
1 Introduction Boundary value problems Lecture 5 We have found that the electric potential is a solution of the partial differential equation; 2 V = ρ/ǫ 0 The above is Poisson s equation where ρ is the
More information(b) For the system in question, the electric field E, the displacement D, and the polarization P = D ɛ 0 E are as follows. r2 0 inside the sphere,
PHY 35 K. Solutions for the second midterm exam. Problem 1: a The boundary conditions at the oil-air interface are air side E oil side = E and D air side oil side = D = E air side oil side = ɛ = 1+χ E.
More informationJackson 2.3 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jackson.3 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell PROBLEM: A straight-line charge with constant linear charge λ is located perpendicular to the x-y plane in
More information1. Overview of the relations among charge, field and potential Gauss law Integrate charge to get potential More about energy Laplace and Poisson
1. Overview of the relations among charge, field and potential Gauss law Integrate charge to get potential More about energy Laplace and Poisson equations 2. Intro to conductors Field inside is zero BEFORE
More informationMATHEMATICS 200 April 2010 Final Exam Solutions
MATHEMATICS April Final Eam Solutions. (a) A surface z(, y) is defined by zy y + ln(yz). (i) Compute z, z y (ii) Evaluate z and z y in terms of, y, z. at (, y, z) (,, /). (b) A surface z f(, y) has derivatives
More informationElectromagnetism: Worked Examples. University of Oxford Second Year, Part A2
Electromagnetism: Worked Examples University of Oxford Second Year, Part A2 Caroline Terquem Department of Physics caroline.terquem@physics.ox.ac.uk Michaelmas Term 2017 2 Contents 1 Potentials 5 1.1 Potential
More informationCHAPTER 2. COULOMB S LAW AND ELECTRONIC FIELD INTENSITY. 2.3 Field Due to a Continuous Volume Charge Distribution
CONTENTS CHAPTER 1. VECTOR ANALYSIS 1. Scalars and Vectors 2. Vector Algebra 3. The Cartesian Coordinate System 4. Vector Cartesian Coordinate System 5. The Vector Field 6. The Dot Product 7. The Cross
More informationAnnouncements. From now on, the problem sets from each week s homework assignments will be the following Wednesday.
Announcements From now on, the problem sets from each week s homework assignments will be the following Wednesday. Late assignments will not be accepted. I will post the solutions on line after class on
More informationElectric 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
More informationExamples of Dielectric Problems and the Electric Susceptability. 2 A Dielectric Sphere in a Uniform Electric Field
Examples of Dielectric Problems and the Electric Susceptability Lecture 10 1 A Dielectric Filled Parallel Plate Capacitor Suppose an infinite, parallel plate capacitor filled with a dielectric of dielectric
More information7 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
More informationCurvilinear coordinates
C Curvilinear coordinates The distance between two points Euclidean space takes the simplest form (2-4) in Cartesian coordinates. The geometry of concrete physical problems may make non-cartesian coordinates
More informationSOLUTIONS 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
More informationxy 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
More informationPHYSICS 7B, Section 1 Fall 2013 Midterm 2, C. Bordel Monday, November 4, pm-9pm. Make sure you show your work!
PHYSICS 7B, Section 1 Fall 2013 Midterm 2, C. Bordel Monday, November 4, 2013 7pm-9pm Make sure you show your work! Problem 1 - Current and Resistivity (20 pts) a) A cable of diameter d carries a current
More informationPhysics 7B Midterm 2 Solutions - Fall 2017 Professor R. Birgeneau
Problem 1 Physics 7B Midterm 2 Solutions - Fall 217 Professor R. Birgeneau (a) Since the wire is a conductor, the electric field on the inside is simply zero. To find the electric field in the exterior
More informationPHYS463 Electricity& Magnetism III ( ) Solution #1
PHYS463 Electricity& Magnetism III (2003-04) lution #. Problem 3., p.5: Find the average potential over a spherical surface of radius R due to a point charge located inside (same as discussed in 3..4,
More informationMake sure you show all your work and justify your answers in order to get full credit.
PHYSICS 7B, Lectures & 3 Spring 5 Midterm, C. Bordel Monday, April 6, 5 7pm-9pm Make sure you show all your work and justify your answers in order to get full credit. Problem esistance & current ( pts)
More informationElectrostatics: Electrostatic Devices
Electrostatics: Electrostatic Devices EE331 Electromagnetic Field Theory Outline Laplace s Equation Derivation Meaning Solving Laplace s equation Resistors Capacitors Electrostatics -- Devices Slide 1
More informationChapter 4: Electrostatics
65 Chapter 4: Electrostatics Lesson # Chapter Section: 4- to 4-3 Topics: Charge and current distributions, Coulomb s law Highlights: Maxwell s Equations reduce to uncoupled electrostatics and magnetostatics
More informationUniversity of Saskatchewan Department of Electrical Engineering
University of Saskatchewan Department of Electrical Engineering December 9,2004 EE30 1 Electricity, Magnetism and Fields Final Examination Professor Robert E. Johanson Welcome to the EE301 Final. This
More informationProblem Set #4: 4.1,4.7,4.9 (Due Monday, March 25th)
Chapter 4 Multipoles, Dielectrics Problem Set #4: 4.,4.7,4.9 (Due Monday, March 5th 4. Multipole expansion Consider a localized distribution of charges described by ρ(x contained entirely in a sphere of
More informationUniversity of Illinois at Chicago Department of Physics
University of Illinois at Chicago Department of Physics Electromagnetism Qualifying Examination January 4, 2017 9.00 am - 12.00 pm Full credit can be achieved from completely correct answers to 4 questions.
More informationElectromagnetism HW 1 math review
Electromagnetism HW math review Problems -5 due Mon 7th Sep, 6- due Mon 4th Sep Exercise. The Levi-Civita symbol, ɛ ijk, also known as the completely antisymmetric rank-3 tensor, has the following properties:
More informationSI Units Coulomb s Law Gauss Law Voltage & Energy Poisson s Equation Capacitance Boundary Conditions MoI Summary Problems.
S. R. Zinka zinka@vit.ac.in School of Electronics Engineering Vellore Institute of Technology October 18, 2012 Outline 1 SI Units 2 Coulomb s Law 3 Gauss Law 4 Voltage & Energy 5 Poisson s Equation 6 Capacitance
More informationAdditional Mathematical Tools: Detail
Additional Mathematical Tools: Detail September 9, 25 The material here is not required, but gives more detail on the additional mathmatical tools: coordinate systems, rotations, the Dirac delta function
More informationCurvilinear 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
More informationPHYS 281: Midterm Exam
PHYS 28: Midterm Exam October 28, 200, 8:00-9:20 Last name (print): Initials: No calculator or other aids allowed PHYS 28: Midterm Exam Instructor: B. R. Sutherland Date: October 28, 200 Time: 8:00-9:20am
More informationLAPLACE EQUATION. = 2 is call the Laplacian or del-square operator. In two dimensions, the expression of in rectangular and polar coordinates are
LAPLACE EQUATION If a diffusion or wave problem is stationary (time independent), the pde reduces to the Laplace equation u = u =, an archetype of second order elliptic pde. = 2 is call the Laplacian or
More informationPhysics 7B, Speliotopoulos Final Exam, Fall 2014 Berkeley, CA
Physics 7B, Speliotopoulos Final Exam, Fall 4 Berkeley, CA Rules: This final exam is closed book and closed notes. In particular, calculators are not allowed during this exam. Cell phones must be turned
More informationwhen viewed from the top, the objects should move as if interacting gravitationally
2 Elastic Space 2 Elastic Space The dynamics and apparent interactions of massive balls rolling on a stretched horizontal membrane are often used to illustrate gravitation. Investigate the system further.
More informationGraduate Diploma in Engineering Circuits and waves
9210-112 Graduate Diploma in Engineering Circuits and waves You should have the following for this examination one answer book non-programmable calculator pen, pencil, ruler No additional data is attached
More informationELECTRO MAGNETIC FIELDS
SET - 1 1. a) State and explain Gauss law in differential form and also list the limitations of Guess law. b) A square sheet defined by -2 x 2m, -2 y 2m lies in the = -2m plane. The charge density on the
More informationUniversity of Alabama Department of Physics and Astronomy. PH 125 / LeClair Spring A Short Math Guide. Cartesian (x, y) Polar (r, θ)
University of Alabama Department of Physics and Astronomy PH 125 / LeClair Spring 2009 A Short Math Guide 1 Definition of coordinates Relationship between 2D cartesian (, y) and polar (r, θ) coordinates.
More informationProperties of Coordinate Systems
Properties of Coordinate Systems Cartesian Coordinates Position vector: r yy For Two Neighboring Points P and P : Displacement between two neighboring points: dsdr d dy y d Distance between two neighboring
More informationLaplace equation in polar coordinates
Laplace equation in polar coordinates The Laplace equation is given by 2 F 2 + 2 F 2 = 0 We have x = r cos θ, y = r sin θ, and also r 2 = x 2 + y 2, tan θ = y/x We have for the partials with respect to
More informationo Two-wire transmission line (end view is shown, the radius of the conductors = a, the distance between the centers of the two conductors = d)
Homework 2 Due Monday, 14 June 1. There is a small number of simple conductor/dielectric configurations for which we can relatively easily find the capacitance. Students of electromagnetics should be sure
More informationWritten Examination. Antennas and Propagation (AA ) June 22, 2018.
Written Examination Antennas and Propagation (AA. 7-8 June, 8. Problem ( points A circular loop of radius a = cm is positioned at a height h over a perfectly electric conductive ground plane as in figure,
More informationp. 1/ Section 1.4: Cylindrical and Spherical Coordinates
p. 1/ Section 1.4: Cylindrical and Spherical Coordinates p. / Cylindrical Coordinate (r,θ,w) where θ is measured counterclockwise as viewed from the positive w-axis. p. / Cylindrical Coordinate (r,θ,w)
More informationTutorial 3 - Solutions Electromagnetic Waves
Tutorial 3 - Solutions Electromagnetic Waves You can find formulas you require for vector calculus at the end of this tutorial. 1. Find the Divergence and Curl of the following functions - (a) k r ˆr f
More informationPhysics 202, Exam 1 Review
Physics 202, Exam 1 Review Logistics Topics: Electrostatics (Chapters 21-24.6) Point charges: electric force, field, potential energy, and potential Distributions: electric field, electric potential. Interaction
More informationDepartment of Physics IIT Kanpur, Semester II,
Department of Phsics IIT Kanpur, Semester II, 7-8 PHYA: Phsics II Solutions # Instructors: AKJ & SC Solution.: (a) At the top of the hill, the gradient of the height function should be ero, that is, h(,
More information2 nd ORDER O.D.E.s SUBSTITUTIONS
nd ORDER O.D.E.s SUBSTITUTIONS Question 1 (***+) d y y 8y + 16y = d d d, y 0, Find the general solution of the above differential equation by using the transformation equation t = y. Give the answer in
More informationn i exp E g 2kT lnn i E g 2kT
HOMEWORK #10 12.19 For intrinsic semiconductors, the intrinsic carrier concentration n i depends on temperature as follows: n i exp E g 2kT (28.35a) or taking natural logarithms, lnn i E g 2kT (12.35b)
More informationIn this chapter we study elliptical PDEs. That is, PDEs of the form. 2 u = lots,
Chapter 8 Elliptic PDEs In this chapter we study elliptical PDEs. That is, PDEs of the form 2 u = lots, where lots means lower-order terms (u x, u y,..., u, f). Here are some ways to think about the physical
More informationGreen s function for the wave equation
Green s function for the wave equation Non-relativistic case January 2018 1 The wave equations In the Lorentz Gauge, the wave equations for the potentials are (Notes 1 eqns 44 and 43): 1 2 2 2 2 0 (1)
More informationSolutions to PS 2 Physics 201
Solutions to PS Physics 1 1. ke dq E = i (1) r = i = i k eλ = i k eλ = i k eλ k e λ xdx () (x x) (x x )dx (x x ) + x dx () (x x ) x ln + x x + x x (4) x + x ln + x (5) x + x To find the field for x, we
More information1. (3) Write Gauss Law in differential form. Explain the physical meaning.
Electrodynamics I Midterm Exam - Part A - Closed Book KSU 204/0/23 Name Electro Dynamic Instructions: Use SI units. Where appropriate, define all variables or symbols you use, in words. Try to tell about
More informationNotes 19 Gradient and Laplacian
ECE 3318 Applied Electricity and Magnetism Spring 218 Prof. David R. Jackson Dept. of ECE Notes 19 Gradient and Laplacian 1 Gradient Φ ( x, y, z) =scalar function Φ Φ Φ grad Φ xˆ + yˆ + zˆ x y z We can
More informationContents. 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)
More informationKeble College - Hilary 2015 CP3&4: Mathematical methods I&II Tutorial 4 - Vector calculus and multiple integrals II
Keble ollege - Hilary 2015 P3&4: Mathematical methods I&II Tutorial 4 - Vector calculus and multiple integrals II Tomi Johnson 1 Prepare full solutions to the problems with a self assessment of your progress
More informationReview of Electrostatics. Define the gradient operation on a field F = F(x, y, z) by;
Review of Electrostatics 1 Gradient Define the gradient operation on a field F = F(x, y, z) by; F = ˆx F x + ŷ F y + ẑ F z This operation forms a vector as may be shown by its transformation properties
More informationHomework #5 Solutions
Homework #5 Solutions Problem 4.4 a) Laplace's equation in spherical coordinates is If V 1 = C 1 /R 2 V = (1/R 2 )d/dr(r 2 (dv/dr)) + 1/(R 2 sinθ)d/dθ(sinθdv/dθ) + 1/(R 2 sin 2 θ)d 2 V/dϕ 2 = 0 dv 1 /dr
More informationMath 575-Lecture 19. In this lecture, we continue to investigate the solutions of the Stokes equations.
Math 575-Lecture 9 In this lecture, we continue to investigate the solutions of the Stokes equations. Energy balance Rewrite the equation to σ = f. We begin the energy estimate by dotting u in the Stokes
More informationElectrodynamics I Midterm - Part A - Closed Book KSU 2005/10/17 Electro Dynamic
Electrodynamics I Midterm - Part A - Closed Book KSU 5//7 Name Electro Dynamic. () Write Gauss Law in differential form. E( r) =ρ( r)/ɛ, or D = ρ, E= electricfield,ρ=volume charge density, ɛ =permittivity
More information1. (3) Write Gauss Law in differential form. Explain the physical meaning.
Electrodynamics I Midterm Exam - Part A - Closed Book KSU 204/0/23 Name Instructions: Use SI units. Where appropriate, define all variables or symbols you use, in words. Try to tell about the physics involved,
More informationReview of Electrostatics
Review of Electrostatics 1 Gradient Define the gradient operation on a field F = F(x, y, z) by; F = ˆx F x + ŷ F y + ẑ F z This operation forms a vector as may be shown by its transformation properties
More informationProblem Set #3: 2.11, 2.15, 2.21, 2.26, 2.40, 2.42, 2.43, 2.46 (Due Thursday Feb. 27th)
Chapter Electrostatics Problem Set #3:.,.5,.,.6,.40,.4,.43,.46 (Due Thursday Feb. 7th). Coulomb s Law Coulomb showed experimentally that for two point charges the force is - proportional to each of the
More informationTwo common difficul:es with HW 2: Problem 1c: v = r n ˆr
Two common difficul:es with HW : Problem 1c: For what values of n does the divergence of v = r n ˆr diverge at the origin? In this context, diverge means becomes arbitrarily large ( goes to infinity ).
More informationPhysics 7B Final Exam: Monday December 14th, 2015 Instructors: Prof. R.J. Birgeneau/Dr. A. Frano
Physics 7B Final Exam: Monday December 14th, 15 Instructors: Prof. R.J. Birgeneau/Dr. A. Frano Total points: 1 (7 problems) Show all your work and take particular care to explain what you are doing. Partial
More informationSpotlight on Laplace s Equation
16 Spotlight on Laplace s Equation Reference: Sections 1.1,1.2, and 1.5. Laplace s equation is the undriven, linear, second-order PDE 2 u = (1) We defined diffusivity on page 587. where 2 is the Laplacian
More informationElectromagnetism Physics 15b
Electromagnetism Physics 15b Lecture #5 Curl Conductors Purcell 2.13 3.3 What We Did Last Time Defined divergence: Defined the Laplacian: From Gauss s Law: Laplace s equation: F da divf = lim S V 0 V Guass
More informationQuantum Mechanics in 3-Dimensions
Quantum Mechanics in 3-Dimensions Pavithran S Iyer, 2nd yr BSc Physics, Chennai Mathematical Institute Email: pavithra@cmi.ac.in August 28 th, 2009 1 Schrodinger equation in Spherical Coordinates 1.1 Transforming
More informationDOING PHYSICS WITH MATLAB. ELECTRIC FIELD AND ELECTRIC POTENTIAL: POISSON S and LAPLACES S EQUATIONS
DOING PHYSICS WITH MATLAB ELECTRIC FIELD AND ELECTRIC POTENTIAL: POISSON S and LAPLACES S EQUATIONS Ian Cooper School of Physics, University of Sydney ian.cooper@sydney.edu.au DOWNLOAD DIRECTORY FOR MATLAB
More informationPhysics 202, Exam 1 Review
Physics 202, Exam 1 Review Logistics Topics: Electrostatics + Capacitors (Chapters 21-24) Point charges: electric force, field, potential energy, and potential Distributions: electric field, electric potential.
More informationEEE321 Electromagnetic Fileds and Waves. Prof. Dr. Hasan Hüseyin BALIK. (1 st Week)
EEE321 Electromagnetic Fileds and Waves Prof. Dr. Hasan Hüseyin BALIK (1 st Week) Outline Course Information and Policies Course Syllabus Vector Operators Coordinate Systems Course Information (see web
More informationAN EXPRESSION FOR THE RADAR CROSS SECTION COMPUTATION OF AN ELECTRICALLY LARGE PERFECT CONDUCTING CYLINDER LOCATED OVER A DIELECTRIC HALF-SPACE
Progress In Electromagnetics Research, PIER 77, 267 272, 27 AN EXPRESSION FOR THE RADAR CROSS SECTION COMPUTATION OF AN ELECTRICALLY LARGE PERFECT CONDUCTING CYLINDER LOCATED OVER A DIELECTRIC HALF-SPACE
More informationReview. Spring Semester /21/14. Physics for Scientists & Engineers 2 1
Review Spring Semester 2014 Physics for Scientists & Engineers 2 1 Notes! Homework set 13 extended to Tuesday, 4/22! Remember to fill out SIRS form: https://sirsonline.msu.edu Physics for Scientists &
More information5) Two large metal plates are held a distance h apart, one at a potential zero, the other
Promlems 1) Find charge distribution on a grounded conducting sphere with radious R centered at the origin due to a charge q at a position (r,θ,φ) outside of the sphere. Plot the charge distribution as
More informationElectromagnetic Field Theory (EMT)
Electromagnetic Field Theory (EMT) Lecture # 9 1) Coulomb s Law and Field Intensity 2) Electric Fields Due to Continuous Charge Distributions Line Charge Surface Charge Volume Charge Coulomb's Law Coulomb's
More informationCreated by T. Madas VECTOR OPERATORS. Created by T. Madas
VECTOR OPERATORS GRADIENT gradϕ ϕ Question 1 A surface S is given by the Cartesian equation x 2 2 + y = 25. a) Draw a sketch of S, and describe it geometrically. b) Determine an equation of the tangent
More informationPhysics 505 Fall Homework Assignment #9 Solutions
Physics 55 Fall 25 Textbook problems: Ch. 5: 5.2, 5.22, 5.26 Ch. 6: 6.1 Homework Assignment #9 olutions 5.2 a) tarting from the force equation (5.12) and the fact that a magnetization M inside a volume
More informationUNIT-I INTRODUCTION TO COORDINATE SYSTEMS AND VECTOR ALGEBRA
SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) Subject with Code : EMF(16EE214) Sem: II-B.Tech & II-Sem Course & Branch: B.Tech - EEE Year
More informationMATH20411 PDEs and Vector Calculus B
MATH2411 PDEs and Vector Calculus B Dr Stefan Güttel Acknowledgement The lecture notes and other course materials are based on notes provided by Dr Catherine Powell. SECTION 1: Introctory Material MATH2411
More informationMultiple Choice. Compute the Jacobian, (u, v), of the coordinate transformation x = u2 v 4, y = uv. (a) 2u 2 + 4v 4 (b) xu yv (c) 3u 2 + 7v 6
.(5pts) y = uv. ompute the Jacobian, Multiple hoice (x, y) (u, v), of the coordinate transformation x = u v 4, (a) u + 4v 4 (b) xu yv (c) u + 7v 6 (d) u (e) u v uv 4 Solution. u v 4v u = u + 4v 4..(5pts)
More informationTUTORIAL 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
More informationElectrodynamics and Microwaves 3. Gradient, Curl and Divergence
1 Module 3 Gradient, Divergence and Curl 1. Introduction 2. The operators & 2 3. Gradient 4. Divergence 5. Curl 6. Mathematical expressions for gradient, divergence and curl in different coordinate systems.
More information1.1 a.) Suppose we have a conductor and place some charge density within it. For a surface S inside the conductor enclose the charge density!
1.1 a. uppose we have a conductor and place some charge density within it. # Q = d 3 x x V ( For a surface inside the conductor enclose the charge density E d a = 1 d 3 x $ %( x$ # V This will cause an
More informationE & M Qualifier. January 11, To insure that the your work is graded correctly you MUST:
E & M Qualifier 1 January 11, 2017 To insure that the your work is graded correctly you MUST: 1. use only the blank answer paper provided, 2. use only the reference material supplied (Schaum s Guides),
More informationALGEBRA 2 X. Final Exam. Review Packet
ALGEBRA X Final Exam Review Packet Multiple Choice Match: 1) x + y = r a) equation of a line ) x = 5y 4y+ b) equation of a hyperbola ) 4) x y + = 1 64 9 c) equation of a parabola x y = 1 4 49 d) equation
More informationPhys. 505 Electricity and Magnetism Fall 2003 Prof. G. Raithel Problem Set 1
Phys. 505 Electricity and Magnetism Fall 2003 Prof. G. Raithel Problem Set Problem.3 a): By symmetry, the solution must be of the form ρ(x) = ρ(r) = Qδ(r R)f, with a constant f to be specified by the condition
More informationMultiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015
Multiple Integrals and Vector Calculus (Oxford Physics) Ramin Golestanian Synopsis and Problem Sets; Hilary 215 The outline of the material, which will be covered in 14 lectures, is as follows: 1. Introduction
More information1 st ORDER O.D.E. EXAM QUESTIONS
1 st ORDER O.D.E. EXAM QUESTIONS Question 1 (**) 4y + = 6x 5, x > 0. dx x Determine the solution of the above differential equation subject to the boundary condition is y = 1 at x = 1. Give the answer
More information(a) Consider a sphere of charge with radius a and charge density ρ(r) that varies with radius as. ρ(r) = Ar n for r a
Physics 7B Midterm 2 - Fall 207 Professor R. Birgeneau Total Points: 00 ( Problems) This exam is out of 00 points. Show all your work and take particular care to explain your steps. Partial credit will
More informationKINGS COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING QUESTION BANK
KINGS COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING QUESTION BANK SUB.NAME : ELECTROMAGNETIC FIELDS SUBJECT CODE : EC 2253 YEAR / SEMESTER : II / IV UNIT- I - STATIC ELECTRIC
More information13 - ELECTROSTATICS Page 1 ( Answers at the end of all questions )
3 - ELECTROSTATICS Page ) Two point charges 8 and - are located at x = 0 and x = L respectively. The location of a point on the x axis at which the net electric field due to these two point charges is
More informationELECTROMAGNETIC CONIC SECTIONS
ELECTROMAGNETIC CONIC SECTIONS Tevian Dray Department of Mathematics, Oregon State University, Corvallis, OR 97331 tevian@math.orst.edu Corinne A. Manogue Department of Physics, Oregon State University,
More informationPhys102 General Physics II. Chapter 24: Gauss s Law
Phys102 General Physics II Gauss Law Chapter 24: Gauss s Law Flux Electric Flux Gauss Law Coulombs Law from Gauss Law Isolated conductor and Electric field outside conductor Application of Gauss Law Charged
More informationProjectile Motion and 2-D Dynamics
Projectile Motion and 2-D Dynamics Vector Notation Vectors vs. Scalars In Physics 11, you learned the difference between vectors and scalars. A vector is a quantity that includes both direction and magnitude
More informationIntegrals in cylindrical, spherical coordinates (Sect. 15.7)
Integrals in clindrical, spherical coordinates (Sect. 15.7 Integration in spherical coordinates. Review: Clindrical coordinates. Spherical coordinates in space. Triple integral in spherical coordinates.
More informationElectrodynamics Qualifier Examination
Electrodynamics Qualifier Examination August 15, 2007 General Instructions: In all cases, be sure to state your system of units. Show all your work, write only on one side of the designated paper, and
More information