ENGI 4430 Gauss & Stokes Theorems; Potentials Page 10.01
|
|
- Nickolas Summers
- 5 years ago
- Views:
Transcription
1 ENGI 443 Gauss & tokes heorems; Potentials Page.. Gauss Divergence heorem Let be a piecewise-smooth closed surface enclosing a volume in vector field. hen the net flux of F out of is F d F d, N 3 and let F be a where F N is the component of F normal to the surface. But the divergence of F is a flux density, or an outflow per unit volume at a point. Integrating div F over the entire enclosed volume must match the net flux out through the boundary of the volume. Gauss divergence theorem then follows: F d F d Example. (Example 9.8 repeated) Find the total flux of the vector field F z k ˆ through the simple closed surface x y z a b c Use Gauss Divergence heorem: F is differentiable everywhere in F d div F d 3, so Gauss divergence theorem is valid. ˆ zk z div F x y z 4 abc div F d d the volume of the ellipsoid! 3 herefore F d 4abc 3 In fact, the flux of that surface. F z k ˆ through any simple closed surface is just the volume enclosed by
2 ENGI 443 Gauss & tokes heorems; Potentials Page. Example. Archimedes Principle Gauss divergence theorem may be used to derive Archimedes principle for the buoyant force on a body totally immersed in a fluid of constant density (independent of depth). Examine an elementary section of the surface of the immersed body, at a depth z below the surface of the fluid: he pressure at any depth z is the weight of fluid per unit area from the column of fluid above that area. z is the height of the column (note z < ). the mass is Az he volume of the column is Az the weight is Az g and the pressure is the weight per unit area, so pressure = p = gz he normal vector N to is directed outward, but the hydrostatic force on the surface (due to the pressure p) acts inward. he element of hydrostatic force on is pressure area direction g z N ˆ g z N ˆ he element of buoyant force on of ˆk (vertically upwards): is the component of the hydrostatic force in the direction g z Nˆ k ˆ Define F gzk ˆ and d N ˆ d. umming over all such elements, the total buoyant force on the immersed object is
3 ENGI 443 Gauss & tokes heorems; Potentials Page.3 Example. Archimedes Principle (continued) g z ˆ k d g z kˆ Nˆ d F d F d (by the Gauss Divergence heorem) x But g z ˆ k g z g z g y z z gz z provided that z g, which is a reasonable assumption for incompressible fluids over depths shallow enough for the acceleration g due to gravity to be nearly constant. total buoyant force g d g = weight of fluid displaced by the immersed body herefore the total buoyant force on an object fully immersed in a fluid equals the weight of the fluid displaced by the immersed object (Archimedes principle).
4 ENGI 443 Gauss & tokes heorems; Potentials Page.4 Gauss Law A point charge q at the origin O generates an electric field q q E r r ˆ 3 4 r 4 r If is a smooth simple closed surface not enclosing the charge, then the total flux through is E d E d (Gauss divergence theorem) But Example 6.4 showed that r r 3. r E d. herefore here is no net outflow of electric flux through any closed surface not enclosing the source of the electrostatic field. If does enclose the charge, then one cannot use Gauss divergence theorem, because E is undefined at the origin. Remedy: Construct a surface identical to except for a small hole cut where a narrow tube connects it to another surface, a sphere of radius a centre O and entirely inside. Let * (which is a simple closed surface), then O is outside *! Applying Gauss divergence theorem to *, E d * * E d E d E d E d
5 ENGI 443 Gauss & tokes heorems; Potentials Page.5 Gauss Law (continued) As the tube approaches zero thickness, EN on opposite walls of the tube tend to become equal in magnitude and opposite in direction, cancelling each other perfectly in the limit of zero thickness all over the tube. E d and therefore E d E d But is a sphere, centre O, radius a. Using parameters, on the sphere, r sin cos a sin sin cos r r Finding, as before leads to Nasin r. But the outward normal to actually points towards O. N a sin r on the sphere q and E r everywhere on 3. 4 a Also r arˆ r r a q sin sin 3 sin q q E N r a r r r 4 a 4 a 4 Recall that d Nd N N d d N d d q E d E N d d sin d d 4
6 ENGI 443 Gauss & tokes heorems; Potentials Page.6 Gauss Law (continued) q cos q q 4 4 q E d E d (which is independent of the radius of the sphere ) a O, But, as he surface looks more and more like the surface as the tube collapses to a line and the sphere collapses into a point at the origin. Gauss law then follows. Gauss law for the net flux through any smooth simple closed surface, in the presence of a point charge q at the origin, then follows: E d q if encloses otherwise O
7 ENGI 443 Gauss & tokes heorems; Potentials Page.7 Example.3 Poisson s Equation he exact location of the enclosed charge is immaterial, provided it is somewhere inside the volume enclosed by the surface. he charge therefore does not need to be a concentrated point charge, but can be spread out within the enclosed volume. Let the charge density be (x, y, z), then the total charge enclosed by is q d Gauss law E d q Apply Gauss divergence theorem to the left hand side, substitute for q on the right hand side and assume that the permittivity is constant throughout the volume: E d d E d his identity will hold for all volumes only if the integrand is zero everywhere. Poisson s equation then follows: E E and his reduces to Laplace s equation when.
8 ENGI 443 Gauss & tokes heorems; Potentials Page.8 tokes heorem Let F be a vector field acting parallel to the xy-plane. Represent its Cartesian components by F f ˆ ˆ i f j f f. hen ˆ i x f ˆ ˆ f f F j f F k y x y ˆ k z Green s theorem can then be expressed in the form F dr F kˆ da C D Now let us twist the simple closed curve C and its enclosed surface out of the xy-plane, so that the unit normal vector ˆk is replaced by a more general normal vector N. If the surface (that is bounded in 3 by the simple closed curve C) can be represented by z f x, y, then a normal vector at any point on is z z N x y Coherent orientation: C is oriented coherently with respect to if, as one travels along C with N pointing from one s feet to one s head, is always on one s left side. he resulting generalization of Green s theorem is tokes theorem: F dr F N d curl F d C his can be extended further, to a non-flat surface with a non-constant normal vector N.
9 ENGI 443 Gauss & tokes heorems; Potentials Page.9 Example.4 Find the circulation of xy F xyz xz e around C : the unit square in the xz-plane. Because of the right-hand rule, the positive orientation around the square is OGHJ (the y axis is directed into the page). In the xz plane y = F xz Computing the line integral around the four sides of the square: d OG : r r t dt t and d r F F dt G F dr dt t O In a similar way (in Problem et 9 Question 5), it can be shown that H J O F dr, F dr and F dr G H J F dr C OR use tokes theorem:
10 ENGI 443 Gauss & tokes heorems; Potentials Page. Example.4 (continued) Note that we must find F first before setting y = : F y F y ˆ i x yz x xy x e x ˆ xy F j xz x y y e y z xz ˆ xy k e z On D F x x z xz z x da ˆ j da F da z x da da F da F dr D C Note that this vector field F is not conservative, because F. Domain A region of 3 is a domain if and only if ) For all points P o in, there exists a sphere, centre P o, all of whose interior points are inside ; and ) For all points P o and P in, there exists a piecewise smooth curve C, entirely in, from o P to P. A domain is simply connected if it has no holes.
11 ENGI 443 Gauss & tokes heorems; Potentials Page. Example.5 Are these regions simply-connected domains? he interior of a sphere. he interior of a torus. he first octant. YE NO YE On a simply-connected domain the following statements are either all true or all false: F is conservative. F F P P C F dr - independent of the path between the two points. end F dr C C start Example.6 Find a potential function x, y, z for the vector field x y z F. First, check that a potential function exists at all: ˆ i x x curl ˆ F F j y y ˆ k z z herefore F is conservative on 3 and a potential function exists.
12 ENGI 443 Gauss & tokes heorems; Potentials Page. Example.6 (continued) F x y z x x x g y, z g y g y, z y h z y y x y h z dh z hz z c z dz x y z c We have a free choice for the value of the arbitrary constant c. Choose c =, then x, y, z x y z r Often we can spot the potential function, then check that it has the correct partial derivatives.
13 ENGI 443 Gauss & tokes heorems; Potentials Page.3 Example.7 y y Find a potential function for F e ˆi xe z ˆj yz k ˆ that has the value at the origin. We seek a function x, y, z such that y y e, xe z and yz. x y z y Immediately we can spot that xe z y Checking this candidate solution, we find y y xe z y e x and y xe z y yz z suggests xe z y. y y,, x y z xe z y c, where c is an arbitrary constant.,, c c x, y, z xe z y herefore y
14 ENGI 443 Gauss & tokes heorems; Potentials Page.4 Maxwell s Equations (not examinable in this course) We have seen how Gauss and tokes theorems have led to Poisson s equation, relating the electric intensity vector E to the electric charge density : E Where the permittivity is constant, the corresponding equation for the electric flux density D is one of Maxwell s equations: D. Another of Maxwell s equations follows from the absence of isolated magnetic charges (no magnetic monopoles): H B, where H is the magnetic intensity and B is the magnetic flux density. Faraday s law, connecting electric intensity with the rate of change of magnetic flux density, is E dr t B d. Applying tokes theorem to the left side produces C E B t Ampère s circuital law, I H dl, leads to H J Jd, where C the current density is J E v, is the conductivity, is the volume charge D density; and the displacement charge density is J d t he fourth Maxwell equation is H D J t he four Maxwell s equations together allow the derivation of the equations of propagating electromagnetic waves. [End of Chapter ]
ENGI Surface Integrals Page 2.01
ENGI 543. urface Integrals Page.1. urface Integrals This chapter introduces the theorems of Green, Gauss and tokes. Two different methods of integrating a function of two variables over a curved surface
More informationThe Divergence Theorem Stokes Theorem Applications of Vector Calculus. Calculus. Vector Calculus (III)
Calculus Vector Calculus (III) Outline 1 The Divergence Theorem 2 Stokes Theorem 3 Applications of Vector Calculus The Divergence Theorem (I) Recall that at the end of section 12.5, we had rewritten Green
More informationENGI 4430 Line Integrals; Green s Theorem Page 8.01
ENGI 443 Line Integrals; Green s Theorem Page 8. 8. Line Integrals Two applications of line integrals are treated here: the evaluation of work done on a particle as it travels along a curve in the presence
More informationENGI 4430 Line Integrals; Green s Theorem Page 8.01
ENGI 4430 Line Integrals; Green s Theorem Page 8.01 8. Line Integrals Two applications of line integrals are treated here: the evaluation of work done on a particle as it travels along a curve in the presence
More informationIntegral 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
More informationVector field and Inductance. P.Ravindran, PHY041: Electricity & Magnetism 19 February 2013: Vector Field, Inductance.
Vector field and Inductance Earth s Magnetic Field Earth s field looks similar to what we d expect if 11.5 there were a giant bar magnet imbedded inside it, but the dipole axis of this magnet is offset
More informationENGI 4430 Surface Integrals Page and 0 2 r
ENGI 4430 Surface Integrals Page 9.01 9. Surface Integrals - Projection Method Surfaces in 3 In 3 a surface can be represented by a vector parametric equation r x u, v ˆi y u, v ˆj z u, v k ˆ where u,
More informationElectromagnetic Field Theory Chapter 9: Time-varying EM Fields
Electromagnetic Field Theory Chapter 9: Time-varying EM Fields Faraday s law of induction We have learned that a constant current induces magnetic field and a constant charge (or a voltage) makes an electric
More informationA Brief Revision of Vector Calculus and Maxwell s Equations
A Brief Revision of Vector Calculus and Maxwell s Equations Debapratim Ghosh Electronic Systems Group Department of Electrical Engineering Indian Institute of Technology Bombay e-mail: dghosh@ee.iitb.ac.in
More informationENGI Gradient, Divergence, Curl Page 5.01
ENGI 94 5. - Gradient, Divergence, Curl Page 5. 5. The Gradient Operator A brief review is provided here for the gradient operator in both Cartesian and orthogonal non-cartesian coordinate systems. Sections
More informationUnit-1 Electrostatics-1
1. Describe about Co-ordinate Systems. Co-ordinate Systems Unit-1 Electrostatics-1 In order to describe the spatial variations of the quantities, we require using appropriate coordinate system. A point
More informationJim 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
More informationIntroduction to Vector Calculus (29) SOLVED EXAMPLES. (d) B. C A. (f) a unit vector perpendicular to both B. = ˆ 2k = = 8 = = 8
Introduction to Vector Calculus (9) SOLVED EXAMPLES Q. If vector A i ˆ ˆj k, ˆ B i ˆ ˆj, C i ˆ 3j ˆ kˆ (a) A B (e) A B C (g) Solution: (b) A B (c) A. B C (d) B. C A then find (f) a unit vector perpendicular
More information6. Vector Integral Calculus in Space
6. Vector Integral alculus in pace 6A. Vector Fields in pace 6A-1 Describegeometricallythefollowingvectorfields: a) xi +yj +zk ρ b) xi zk 6A-2 Write down the vector field where each vector runs from (x,y,z)
More informationMATH 0350 PRACTICE FINAL FALL 2017 SAMUEL S. WATSON. a c. b c.
MATH 35 PRACTICE FINAL FALL 17 SAMUEL S. WATSON Problem 1 Verify that if a and b are nonzero vectors, the vector c = a b + b a bisects the angle between a and b. The cosine of the angle between a and c
More informationMATH 263 ASSIGNMENT 9 SOLUTIONS. F dv =
MAH AIGNMEN 9 OLUION ) Let F = (x yz)î + (y + xz)ĵ + (z + xy)ˆk and let be the portion of the cylinder x + y = that lies inside the sphere x + y + z = 4 be the portion of the sphere x + y + z = 4 that
More information송석호 ( 물리학과 )
http://optics.hanyang.ac.kr/~shsong 송석호 ( 물리학과 ) Field and Wave Electromagnetics, David K. Cheng Reviews on (Week 1). Vector Analysis 3. tatic Electric Fields (Week ) 4. olution of Electrostatic Problems
More informationDivergence Theorem December 2013
Divergence Theorem 17.3 11 December 2013 Fundamental Theorem, Four Ways. b F (x) dx = F (b) F (a) a [a, b] F (x) on boundary of If C path from P to Q, ( φ) ds = φ(q) φ(p) C φ on boundary of C Green s Theorem:
More information53. Flux Integrals. Here, R is the region over which the double integral is evaluated.
53. Flux Integrals Let be an orientable surface within 3. An orientable surface, roughly speaking, is one with two distinct sides. At any point on an orientable surface, there exists two normal vectors,
More informationDifferential Operators and the Divergence Theorem
1 of 6 1/15/2007 6:31 PM Differential Operators and the Divergence Theorem One of the most important and useful mathematical constructs is the "del operator", usually denoted by the symbol Ñ (which is
More informationMath 11 Fall 2016 Final Practice Problem Solutions
Math 11 Fall 216 Final Practice Problem olutions Here are some problems on the material we covered since the second midterm. This collection of problems is not intended to mimic the final in length, content,
More informationDivergence Theorem Fundamental Theorem, Four Ways. 3D Fundamental Theorem. Divergence Theorem
Divergence Theorem 17.3 11 December 213 Fundamental Theorem, Four Ways. b F (x) dx = F (b) F (a) a [a, b] F (x) on boundary of If C path from P to Q, ( φ) ds = φ(q) φ(p) C φ on boundary of C Green s Theorem:
More informationMATH 332: Vector Analysis Summer 2005 Homework
MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,
More informationChapter Three: Propagation of light waves
Chapter Three Propagation of Light Waves CHAPTER OUTLINE 3.1 Maxwell s Equations 3.2 Physical Significance of Maxwell s Equations 3.3 Properties of Electromagnetic Waves 3.4 Constitutive Relations 3.5
More informationProblem 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
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 information(a) 0 (b) 1/4 (c) 1/3 (d) 1/2 (e) 2/3 (f) 3/4 (g) 1 (h) 4/3
Math 114 Practice Problems for Test 3 omments: 0. urface integrals, tokes Theorem and Gauss Theorem used to be in the Math40 syllabus until last year, so we will look at some of the questions from those
More informationName: Date: 12/06/2018. M20550 Calculus III Tutorial Worksheet 11
1. ompute the surface integral M255 alculus III Tutorial Worksheet 11 x + y + z) d, where is a surface given by ru, v) u + v, u v, 1 + 2u + v and u 2, v 1. olution: First, we know x + y + z) d [ ] u +
More informationStokes s Theorem 17.2
Stokes s Theorem 17.2 6 December 213 Stokes s Theorem is the generalization of Green s Theorem to surfaces not just flat surfaces (regions in R 2 ). Relate a double integral over a surface with a line
More informationPRACTICE PROBLEMS. Please let me know if you find any mistakes in the text so that i can fix them. 1. Mixed partial derivatives.
PRACTICE PROBLEMS Please let me know if you find any mistakes in the text so that i can fix them. 1.1. Let Show that f is C 1 and yet How is that possible? 1. Mixed partial derivatives f(x, y) = {xy x
More informationMaxwell s Equations in Differential Form, and Uniform Plane Waves in Free Space
C H A P T E R 3 Maxwell s Equations in Differential Form, and Uniform Plane Waves in Free Space In Chapter 2, we introduced Maxwell s equations in integral form. We learned that the quantities involved
More informationPhysics 114 Exam 1 Fall 2016
Physics 114 Exam 1 Fall 2016 Name: For grading purposes (do not write here): Question 1. 1. 2. 2. 3. 3. Problem Answer each of the following questions and each of the problems. Points for each question
More informationENGI Multiple Integration Page 8-01
ENGI 345 8. Multiple Integration Page 8-01 8. Multiple Integration This chapter provides only a very brief introduction to the major topic of multiple integration. Uses of multiple integration include
More information(You may need to make a sin / cos-type trigonometric substitution.) Solution.
MTHE 7 Problem Set Solutions. As a reminder, a torus with radii a and b is the surface of revolution of the circle (x b) + z = a in the xz-plane about the z-axis (a and b are positive real numbers, with
More informationUNIT 1. INTRODUCTION
UNIT 1. INTRODUCTION Objective: The aim of this chapter is to gain knowledge on Basics of electromagnetic fields Scalar and vector quantities, vector calculus Various co-ordinate systems namely Cartesian,
More informationTransmission Lines and E. M. Waves Prof. R. K. Shevgaonkar Department of Electrical Engineering Indian Institute of Technology, Bombay
Transmission Lines and E. M. Waves Prof. R. K. Shevgaonkar Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture 18 Basic Laws of Electromagnetics We saw in the earlier lecture
More informationOne side of each sheet is blank and may be used as scratch paper.
Math 244 Spring 2017 (Practice) Final 5/11/2017 Time Limit: 2 hours Name: No calculators or notes are allowed. One side of each sheet is blank and may be used as scratch paper. heck your answers whenever
More informationLecture 8. Engineering applications
Lecture 8 Engineering applications In Lecture 6 we saw one classic example of the application of vector calculus to Maxwell s equation. In this lecture we explore a few more examples from fluid mechanics
More information송석호 ( 물리학과 )
http://optics.hanyang.ac.kr/~shsong 송석호 ( 물리학과 ) Introduction to Electrodynamics, David J. Griffiths Review: 1. Vector analysis 2. Electrostatics 3. Special techniques 4. Electric fields in mater 5. Magnetostatics
More informationBefore seeing some applications of vector calculus to Physics, we note that vector calculus is easy, because... There s only one Theorem!
16.10 Summary and Applications Before seeing some applications of vector calculus to Physics, we note that vector calculus is easy, because... There s only one Theorem! Green s, Stokes, and the Divergence
More informationDIVERGENCE AND CURL THEOREMS
This document is stored in Documents/4C/Gausstokes.tex. with LaTex. Compile it November 29, 2014 Hans P. Paar DIVERGENCE AND CURL THEOREM 1 Introduction We discuss the theorems of Gauss and tokes also
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 informationMathematical Notes for E&M Gradient, Divergence, and Curl
Mathematical Notes for E&M Gradient, Divergence, and Curl In these notes I explain the differential operators gradient, divergence, and curl (also known as rotor), the relations between them, the integral
More informationMath 233. Practice Problems Chapter 15. i j k
Math 233. Practice Problems hapter 15 1. ompute the curl and divergence of the vector field F given by F (4 cos(x 2 ) 2y)i + (4 sin(y 2 ) + 6x)j + (6x 2 y 6x + 4e 3z )k olution: The curl of F is computed
More informationF S (2) (3) D 2. 1 Stefan Sint, see also
Note I.6 1 These are Conor Houghton s notes from 6 to whom many thanks! I have just edited a few minor things and corrected some typos. The pictures are in separate files and have also been drawn by Conor.
More informationSome Important Concepts and Theorems of Vector Calculus
ome Important oncepts and Theorems of Vector alculus 1. The Directional Derivative. If f(x) is a scalar-valued function of, say x, y, and z, andu is a direction (i.e. a unit vector), then the Directional
More informationMATHS 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
More informationMath 11 Fall 2007 Practice Problem Solutions
Math 11 Fall 27 Practice Problem olutions Here are some problems on the material we covered since the second midterm. This collection of problems is not intended to mimic the final in length, content,
More informationCalculus 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.
More informationENGI Gradient, Divergence, Curl Page 5.01
ENGI 940 5.0 - Gradient, Divergence, Curl Page 5.0 5. e Gradient Operator A brief review is provided ere for te gradient operator in bot Cartesian and ortogonal non-cartesian coordinate systems. Sections
More informationMagnetostatics. Lecture 23: Electromagnetic Theory. Professor D. K. Ghosh, Physics Department, I.I.T., Bombay
Magnetostatics Lecture 23: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay Magnetostatics Up until now, we have been discussing electrostatics, which deals with physics
More informationS12.1 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
More informationECE 3209 Electromagnetic Fields Final Exam Example. University of Virginia Solutions
ECE 3209 Electromagnetic Fields Final Exam Example University of Virginia Solutions (print name above) This exam is closed book and closed notes. Please perform all work on the exam sheets in a neat and
More informationBasics of Electromagnetics Maxwell s Equations (Part - I)
Basics of Electromagnetics Maxwell s Equations (Part - I) Soln. 1. C A. dl = C. d S [GATE 1994: 1 Mark] A. dl = A. da using Stoke s Theorem = S A. ds 2. The electric field strength at distant point, P,
More information18.02 Multivariable Calculus Fall 2007
MIT OpenourseWare http://ocw.mit.edu 18.02 Multivariable alculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.02 Lecture 30. Tue, Nov
More informationF ds, where F and S are as given.
Math 21a Integral Theorems Review pring, 29 1 For these problems, find F dr, where F and are as given. a) F x, y, z and is parameterized by rt) t, t, t t 1) b) F x, y, z and is parameterized by rt) t,
More informationV Relation to Physics
V15.2-3 Relation to Physics The three theorems we have studied: the divergence theorem and tokes theorem in space, and Green s theorem in the plane (which is really just a special case of tokes theorem)
More informationMath Review for Exam 3
1. ompute oln: (8x + 36xy)ds = Math 235 - Review for Exam 3 (8x + 36xy)ds, where c(t) = (t, t 2, t 3 ) on the interval t 1. 1 (8t + 36t 3 ) 1 + 4t 2 + 9t 4 dt = 2 3 (1 + 4t2 + 9t 4 ) 3 2 1 = 2 3 ((14)
More informationENERGY IN ELECTROSTATICS
ENERGY IN ELECTROSTATICS We now turn to the question of energy in electrostatics. The first question to consider is whether or not the force is conservative. You will recall from last semester that a conservative
More informationIntroduction to Electromagnetic Theory
Introduction to Electromagnetic Theory Lecture topics Laws of magnetism and electricity Meaning of Maxwell s equations Solution of Maxwell s equations Electromagnetic radiation: wave model James Clerk
More informationMaxwell s Equations in Differential Form
Maxwell s Equations in Differential Form CHAPTER 3 In Chapter 2 we introduced Maxwell s equations in integral form. We learned that the quantities involved in the formulation of these equations are the
More informationARNOLD PIZER rochester problib from CVS Summer 2003
ARNOLD PIZER rochester problib from VS Summer 003 WeBWorK assignment Vectoralculus due 5/3/08 at :00 AM.( pt) setvectoralculus/ur V.pg onsider the transformation T : x 8 53 u 45 45 53v y 53 u 8 53 v A.
More informationEE2007: Engineering Mathematics II Vector Calculus
EE2007: Engineering Mathematics II Vector Calculus Ling KV School of EEE, NTU ekvling@ntu.edu.sg Rm: S2-B2b-22 Ver 1.1: Ling KV, October 22, 2006 Ver 1.0: Ling KV, Jul 2005 EE2007/Ling KV/Aug 2006 EE2007:
More informationES.182A Topic 45 Notes Jeremy Orloff
E.8A Topic 45 Notes Jeremy Orloff 45 More surface integrals; divergence theorem Note: Much of these notes are taken directly from the upplementary Notes V0 by Arthur Mattuck. 45. Closed urfaces A closed
More informationr r 1 r r 1 2 = q 1 p = qd and it points from the negative charge to the positive charge.
MP204, Important Equations page 1 Below is a list of important equations that we meet in our study of Electromagnetism in the MP204 module. For your exam, you are expected to understand all of these, and
More informationVector Calculus. Vector Fields. Reading Trim Vector Fields. Assignment web page assignment #9. Chapter 14 will examine a vector field.
Vector alculus Vector Fields Reading Trim 14.1 Vector Fields Assignment web page assignment #9 hapter 14 will eamine a vector field. For eample, if we eamine the temperature conditions in a room, for ever
More informationMath Review Night: Work and the Dot Product
Math Review Night: Work and the Dot Product Dot Product A scalar quantity Magnitude: A B = A B cosθ The dot product can be positive, zero, or negative Two types of projections: the dot product is the parallel
More informationMATH2000 Flux integrals and Gauss divergence theorem (solutions)
DEPARTMENT O MATHEMATIC MATH lux integrals and Gauss divergence theorem (solutions ( The hemisphere can be represented as We have by direct calculation in terms of spherical coordinates. = {(r, θ, φ r,
More informationINTRODUCTION MAGNETIC FIELD OF A MOVING POINT CHARGE. Introduction. Magnetic field due to a moving point charge. Units.
Chapter 9 THE MAGNETC FELD ntroduction Magnetic field due to a moving point charge Units Biot-Savart Law Gauss s Law for magnetism Ampère s Law Maxwell s equations for statics Summary NTRODUCTON Last lecture
More information5.1 Fluid momentum equation Hydrostatics Archimedes theorem The vorticity equation... 42
Chapter 5 Euler s equation Contents 5.1 Fluid momentum equation........................ 39 5. Hydrostatics................................ 40 5.3 Archimedes theorem........................... 41 5.4 The
More informationfree space (vacuum) permittivity [ F/m]
Electrostatic Fields Electrostatic fields are static (time-invariant) electric fields produced by static (stationary) charge distributions. The mathematical definition of the electrostatic field is derived
More informationFLUX OF VECTOR FIELD INTRODUCTION
Chapter 3 GAUSS LAW ntroduction Flux of vector field Solid angle Gauss s Law Symmetry Spherical symmetry Cylindrical symmetry Plane symmetry Superposition of symmetric geometries Motion of point charges
More informationVector Calculus, Maths II
Section A Vector Calculus, Maths II REVISION (VECTORS) 1. Position vector of a point P(x, y, z) is given as + y and its magnitude by 2. The scalar components of a vector are its direction ratios, and represent
More informationChapter 21: Gauss s Law
Chapter 21: Gauss s Law Electric field lines Electric field lines provide a convenient and insightful way to represent electric fields. A field line is a curve whose direction at each point is the direction
More informationLINE AND SURFACE INTEGRALS: A SUMMARY OF CALCULUS 3 UNIT 4
LINE AN URFAE INTEGRAL: A UMMARY OF ALULU 3 UNIT 4 The final unit of material in multivariable calculus introduces many unfamiliar and non-intuitive concepts in a short amount of time. This document attempts
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 informationRead this cover page completely before you start.
I affirm that I have worked this exam independently, without texts, outside help, integral tables, calculator, solutions, or software. (Please sign legibly.) Read this cover page completely before you
More informationMAXWELL S DIFFERENTIAL LAWS IN FREE SPACE
MAXWELL S DIFFERENTIAL LAWS IN FREE SPACE.0 INTRODUCTION Maxwell s integral laws encompass the laws of electrical circuits. The transition from fields to circuits is made by associating the relevant volumes,
More information( ) ( ) ( ) ( ) Calculus III - Problem Drill 24: Stokes and Divergence Theorem
alculus III - Problem Drill 4: tokes and Divergence Theorem Question No. 1 of 1 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as needed () Pick the 1. Use
More informationMath 5BI: Problem Set 9 Integral Theorems of Vector Calculus
Math 5BI: Problem et 9 Integral Theorems of Vector Calculus June 2, 2010 A. ivergence and Curl The gradient operator = i + y j + z k operates not only on scalar-valued functions f, yielding the gradient
More informationMagnetostatic fields! steady magnetic fields produced by steady (DC) currents or stationary magnetic materials.
ECE 3313 Electromagnetics I! Static (time-invariant) fields Electrostatic or magnetostatic fields are not coupled together. (one can exist without the other.) Electrostatic fields! steady electric fields
More informationMath 11 Fall 2018 Practice Final Exam
Math 11 Fall 218 Practice Final Exam Disclaimer: This practice exam should give you an idea of the sort of questions we may ask on the actual exam. Since the practice exam (like the real exam) is not long
More informationLecture 12 Notes, Electromagnetic Theory I Dr. Christopher S. Baird University of Massachusetts Lowell
Lecture 12 Notes, Electromagnetic Theory I Dr. Christopher S. Baird University of Massachusetts Lowell 1. Review of Magnetostatics in Magnetic Materials - Currents give rise to curling magnetic fields:
More information18.1. Math 1920 November 29, ) Solution: In this function P = x 2 y and Q = 0, therefore Q. Converting to polar coordinates, this gives I =
Homework 1 elected olutions Math 19 November 9, 18 18.1 5) olution: In this function P = x y and Q =, therefore Q x P = x. We obtain the following integral: ( Q I = x ydx = x P ) da = x da. onverting to
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 informationDivergence Theorem and Its Application in Characterizing
Divergence Theorem and Its Application in Characterizing Fluid Flow Let v be the velocity of flow of a fluid element and ρ(x, y, z, t) be the mass density of fluid at a point (x, y, z) at time t. Thus,
More informationHOMEWORK 8 SOLUTIONS
HOMEWOK 8 OLUTION. Let and φ = xdy dz + ydz dx + zdx dy. let be the disk at height given by: : x + y, z =, let X be the region in 3 bounded by the cone and the disk. We orient X via dx dy dz, then by definition
More informationProblem Solving 9: Displacement Current, Poynting Vector and Energy Flow
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Problem Solving 9: Displacement Current, Poynting Vector and Energy Flow Section Table and Group Names Hand in one copy per group at the end
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 information18.02 Multivariable Calculus Fall 2007
MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. V9. Surface Integrals Surface
More informationENGI Partial Differentiation Page y f x
ENGI 3424 4 Partial Differentiation Page 4-01 4. Partial Differentiation For functions of one variable, be found unambiguously by differentiation: y f x, the rate of change of the dependent variable can
More informationLECTURE 15 CONDUCTORS, ELECTRIC FLUX & GAUSS S LAW. Instructor: Kazumi Tolich
LECTURE 15 CONDUCTORS, ELECTRIC FLUX & GAUSS S LAW Instructor: Kazumi Tolich Lecture 15 2! Reading chapter 19-6 to 19-7.! Properties of conductors! Charge by Induction! Electric flux! Gauss's law! Calculating
More informationStudy Guide for Exam #2
Physical Mechanics METR103 November, 000 Study Guide for Exam # The information even below is meant to serve as a guide to help you to prepare for the second hour exam. The absence of a topic or point
More informationLINE AND SURFACE INTEGRALS: A SUMMARY OF CALCULUS 3 UNIT 4
LINE AN URFAE INTEGRAL: A UMMARY OF ALULU 3 UNIT 4 The final unit of material in multivariable calculus introduces many unfamiliar and non-intuitive concepts in a short amount of time. This document attempts
More information52. The Del Operator: Divergence and Curl
52. The Del Operator: Divergence and Curl Let F(x, y, z) = M(x, y, z), N(x, y, z), P(x, y, z) be a vector field in R 3. The del operator is represented by the symbol, and is written = x, y, z, or = x,
More informationMath 263 Final. (b) The cross product is. i j k c. =< c 1, 1, 1 >
Math 63 Final Problem 1: [ points, 5 points to each part] Given the points P : (1, 1, 1), Q : (1,, ), R : (,, c 1), where c is a parameter, find (a) the vector equation of the line through P and Q. (b)
More informationPoynting Vector and Energy Flow W14D1
Poynting Vector and Energy Flow W14D1 1 Announcements Week 14 Prepset due online Friday 8:30 am PS 11 due Week 14 Friday at 9 pm in boxes outside 26-152 Sunday Tutoring 1-5 pm in 26-152 2 Outline Poynting
More informationMaxwell s Equations. In the previous chapters we saw the four fundamental equations governging electrostatics and magnetostatics. They are.
Maxwell s Equations Introduction In the previous chapters we saw the four fundamental equations governging electrostatics and magnetostatics. They are D = ρ () E = 0 (2) B = 0 (3) H = J (4) In the integral
More informationMaxwell s Equations:
Course Instructor Dr. Raymond C. Rumpf Office: A-337 Phone: (915) 747-6958 E-Mail: rcrumpf@utep.edu Maxwell s Equations: Physical Interpretation EE-3321 Electromagnetic Field Theory Outline Maxwell s Equations
More informationMaxwell's Equations and Electromagnetic Waves
BACK Maxwell's Equations and Electromagnetic Waves Michael Fowler, Physics Department, UVa 5/9/9 The Equations Maxwell s four equations describe the electric and magnetic fields arising from varying distributions
More information