Conducting Ellipsoid and Circular Disk

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Conducting Ellipsoid and Circular Disk"

Transcription

1 1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid, cn be written σ ellipsoid = x + y b + z = 1, (1) c, () 4πbc x + y + z where is the totl chrge. Show tht if the chrge distribution of the ellipsoid is projected onto ny of its symmetry plnes, the result is independent of the extent of the ellipse perpendiculr to the plne of projection (i.e., σ xy, the projection of σ ellipsoid on the x-y plne, is independent of prmeter c). Thus, the projection of the chrge distribution of conducting oblte or prolte spheroid onto its equtoril plne is the sme s the projected chrge distribution of conducting sphere. By considering thin conducting circulr disk of rdius s specil cse of n ellipsoid, show tht its surfce chrge density (summed over both sides) cn be written V 0 π r, (3) where the electric potentil V 0 of the disk is relted by V 0 = π/. Show lso tht if the chrge distribution on the conducting ellipsoid is projected onto ny of the coordinte xes, the result is uniform (i.e., the chrge distribution projected onto the x-xis is σ x = /). In prticulr, we expect tht the chrge distribution long conducting needle will be uniform, since the needle cn be considered s the limit of conducting ellipsoid, two of whose three xes hve shrunk to zero. Solution The chrge distribution (3) on thin, conducting disk cn be deduced in vriety of wys [1]. We record here highly geometric derivtion following Thomson []. The strting point is the elementry result tht the electric field is zero in the interior of sphericl shell of ny thickness tht hs uniform volume chrge density between the inner nd outer surfces of the shell. A well-known geometric rgument (due to Newton) for this is illustrted in Fig. 1. The electric field t point r 0 in the interior of the shell due to lmin of thickness δ nd re A 1 centered on point r 1 tht lies within nrrow cone whose vertex is point 0 is 1

2 Figure 1: For ny point r 0 in the interior of uniformly chrged shell of chrge, the xis of nrrow bicone intercepts the inner surfce of the shell t points r 1 nd r. The corresponding res on the inner surfce of the shell intercepted by the bicone re A 1 nd A. In the limit of smll res, A 1 /R 01 = A /R 0. given by E 1 = ρ dvol 1 ˆR R01 01, (4) where ρ is the volume chrge density, dvol 1 = A 1 δ, R 01 = r 0 r 1, nd the center of the sphere is tken to be t the origin. Likewise, the electric field from lmin of re A centered on point r defined by the intercept with the shell of the sme nrrow cone extended in the opposite direction (forming bicone) is given by E = ρ dvol R 0 ˆR 0, (5) In the limit of bicones of smll hlf ngle, the two prts of the bicone s truncted by the shell re similr, so tht A 1 = A dvol 1, = dvol, (6) R01 R0 R01 R0 nd, of course, ˆR 0 = ˆR 01. Hence E 1 + E = 0. Since this construction cn be pplied to ll points in the mteril of the sphericl shell, nd for ll pirs of surfce elements subtended by (nrrow) bicones, the totl electric field in the interior of the shell is zero. We now reconsider the bove rgument fter rbitrry scle trnsformtions hve been pplied to the rectngulr coordinte xes, x k 1 x, y k y, z k 3 z. (7) A sphericl shell of rdius s is thereby trnsformed into n ellipsoid, x s + y s + z s = 1 x s /k 1 + y s /k + z s /k 3 = 1. (8)

3 As prmeter s is vried, one obtins set of similr ellipsoids, centered on the origin. A smll volume element obeys the trnsformtion dvol = dxdydz k 1 k k 3 dxdydz = k 1 k k 3 dvol. (9) The three points 0, 1, nd in Fig. 1 lie long line, so tht R 01 = r 0 r 1 = CR 0 = C(r 0 r ), (10) where C is (negtive) constnt. This reltion is invrint under the scle trnsformtion (7), so tht together with eq. (9) the reltion dvol 1 R 01 = dvol R 0, (11) is lso invrint. Hence, if the ellipsoidl shell, which is the trnsform of the sphericl shell of Fig. 1, contins uniform volume chrge density, the reltion E 1 + E = 0 remins true t the vertex of ny bicone in the interior of the shell, which implies tht the totl electric field is zero there. This proof is bsed on the premise tht the ellipsoidl shell is bounded by two similr ellipsoids, nd tht the volume chrge density in the shell is uniform. If we let the outer ellipsoid of the shell pproch the inner one, lwys remining similr to the ltter, we rech configurtion tht is equivlent to thin, conducting ellipsoid, since in both cses the electric field is zero in the interior. Hence, the surfce chrge distribution on thin, conducting ellipsoid must the sme s the projection onto its surfce of uniform chrge distribution between tht surfce nd similr, but slightly lrger ellipsoidl surfce. The chrge σ per unit re on the surfce of thin, conducting ellipsoid is therefore proportionl to the thickness, which we write s δd, of the ellispodl shell formed by tht surfce nd similr, but slightly lrger ellipsoid: σ = ρ δd, (1) where constnt ρ is to be determined from knowledge of the totl chrge on the conducting ellipsoid. The thickness δd of thin ellipsoidl shell t some point on its inner surfce is the distnce between the plne tht is tngent to the inner surfce t the specified point, nd the plne tht is tngent to the outer surfce t the point similr to the specifiedl point. These plnes re prllel since the ellipsoids re prllel. In prticulr if the semimjor xes of the inner ellipsoid re clled, b, nd c, then those of the outer ellipsoid cn be written + δ, b + δb nd c + δc. Let the (perpendiculr) distnce from the plne tngent to the specified point on the inner ellipsoid to its center be clled d, nd the corresponding distnce from the outer tngent plne be d + δd, so tht δd is the desired thickness of the shell t the specified point. Then, the condition of similrity is tht δ = δb b = δc c = δd d. (13) Since the volume of n ellipsoid with semimjor xes, b, nd c is 4πbc/3, the volume of the ellispoidl shell is 4π(+δ)(b+δb)(c+δc)/3 4πbc/3 = 4πbc(δd/d), using eq. (13). 3

4 As the constnt ρ hs n interprettion s the uniform chrge density within the mteril of the ellipsoidl shell, we find tht the totl chrge on the conducting ellipsoid is relted by = ρ Vol shell = 4πbc ρ δd, (14) d nd hence, σ = ρ δd = d 4πbc. (15) It remins to find n expression for the distnce d to the tngent plne. If we write the eqution for the ellipsoid in the form f(x, y, z) = x + y b + z 1 = 0, (16) c then the grdient of f is perpendiculr to the tngent plne. Thus, the vector d from the center of the ellipsoid to the tngent plne is proportionl to f. Tht is, The unit vector ˆd is therefore ( x d f =, y b, z ). (17) c ˆd = ( x, y b, z c ). (18) x + y + z The mgnitude d of the vector d is relted to the vector r = (x, y, z) of the specified point on the ellipse by d = r ˆd x + y + z = b c 1 =. (19) x + y + z x + y + z be At length, we hve found the chrge density on the surfce of conducting ellipsoid to where is the totl chrge. σ ellipsoid =, (0) 4πbc x + y + z To deduce the projection of the chrge distribution of the conducting ellipse onto one of its symmetry plnes, sy the x-y plne, note tht projected re element dx dy corresponds to re da on the surfce of the ellipsoid tht is relted by dx dy = da ẑ = da d z, (1) where the unit vector ˆd tht is norml to the surfce of the ellipsoid t the point (x, y) is given by eq. (19). Thus, da = dx dy ˆd z = c dx dy x z + y 4 b + z 4 c. () 4 4

5 The chrge projected onto the x-y plne from both z > 0 nd z < 0 is d xy = σ ellipsoid da = c dx dy πbz = dx dy, (3) πb 1 x y b combining eqs. (1), (0) nd (). The projected chrge density (due to both hlves of the ellipsoid), σ xy = d xy /dx dy, is independent of the prmeter c tht specifies the size of the ellipsoid in z. For exmple, the chrge distributions of sphere, disk, nd both n oblte nd prolte spheroid, ll of the sme equtoril dimeter, re the sme when projected onto the equtoril plne. If the project the chrge d xy of eq. (3) onto the x xis, the result is d x = dx π b b x b b x dy = dx, (4) b b x y which is uniform in x! In prticulr, the uniform chrge distribution on conducting sphere projects to uniform chrge distribution on ny dimeter; nd the chrge distribution is uniform long conducting needle tht is the limit of conducting ellipsoid. The cse of thin, conducting ellipticl disk in the x-y plne cn be obtined from eq. (0) by letting c go to zero. For this we note tht eq. (16) for generl ellipsoid permits us to write c x 4 + y b 4 + z c 4 = ( x c + y 4 b 4 ) + 1 x y b 1 x y b. (5) The chrge density on ech side of conducting ellipticl disk is therefore σ ellipticl disk =. (6) 4πb 1 x + y b The chrge density on ech side of conducting circulr disk of rdius follows immeditely s 4π r, (7) where r = x + y. Such disk hs potentil V 0, which cn be found by clculting the potentil t the center of the disk ccording to V 0 = V (r = 0, z = 0) = 0 σ(r) πr dr = r 0 dr r = π. (8) Hence, conducting disk of rdius t potentil V 0 hs chrge density V 0 π r. (9) 3 References [1] W.R. Smythe, Sttic nd Dynmic Electricity, 3rd ed. (McGrw-Hill, New York, 1968). [] W. Thomson (Lord Kelvin), Ppers on Electricity nd Mgnetism, nd ed. (Mcmilln, London, 1884), pp. 7 nd

Physics 3323, Fall 2016 Problem Set 7 due Oct 14, 2016

Physics 3323, Fall 2016 Problem Set 7 due Oct 14, 2016 Physics 333, Fll 16 Problem Set 7 due Oct 14, 16 Reding: Griffiths 4.1 through 4.4.1 1. Electric dipole An electric dipole with p = p ẑ is locted t the origin nd is sitting in n otherwise uniform electric

More information

Problems for HW X. C. Gwinn. November 30, 2009

Problems for HW X. C. Gwinn. November 30, 2009 Problems for HW X C. Gwinn November 30, 2009 These problems will not be grded. 1 HWX Problem 1 Suppose thn n object is composed of liner dielectric mteril, with constnt reltive permittivity ɛ r. The object

More information

ragsdale (zdr82) HW2 ditmire (58335) 1

ragsdale (zdr82) HW2 ditmire (58335) 1 rgsdle (zdr82) HW2 ditmire (58335) This print-out should hve 22 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. 00 0.0 points A chrge of 8. µc

More information

Exam 1 Solutions (1) C, D, A, B (2) C, A, D, B (3) C, B, D, A (4) A, C, D, B (5) D, C, A, B

Exam 1 Solutions (1) C, D, A, B (2) C, A, D, B (3) C, B, D, A (4) A, C, D, B (5) D, C, A, B PHY 249, Fll 216 Exm 1 Solutions nswer 1 is correct for ll problems. 1. Two uniformly chrged spheres, nd B, re plced t lrge distnce from ech other, with their centers on the x xis. The chrge on sphere

More information

Homework Assignment 3 Solution Set

Homework Assignment 3 Solution Set Homework Assignment 3 Solution Set PHYCS 44 6 Ferury, 4 Prolem 1 (Griffiths.5(c The potentil due to ny continuous chrge distriution is the sum of the contriutions from ech infinitesiml chrge in the distriution.

More information

JUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 8 (First moments of a volume) A.J.Hobson

JUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 8 (First moments of a volume) A.J.Hobson JUST THE MATHS UNIT NUMBER 3.8 INTEGRATIN APPLICATINS 8 (First moments of volume) b A.J.Hobson 3.8. Introduction 3.8. First moment of volume of revolution bout plne through the origin, perpendiculr to

More information

APPLICATIONS OF THE DEFINITE INTEGRAL

APPLICATIONS OF THE DEFINITE INTEGRAL APPLICATIONS OF THE DEFINITE INTEGRAL. Volume: Slicing, disks nd wshers.. Volumes by Slicing. Suppose solid object hs boundries extending from x =, to x = b, nd tht its cross-section in plne pssing through

More information

Physics 2135 Exam 1 February 14, 2017

Physics 2135 Exam 1 February 14, 2017 Exm Totl / 200 Physics 215 Exm 1 Ferury 14, 2017 Printed Nme: Rec. Sec. Letter: Five multiple choice questions, 8 points ech. Choose the est or most nerly correct nswer. 1. Two chrges 1 nd 2 re seprted

More information

The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Factor of an Insulated Two-Wire Transmission Line The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the

More information

CAPACITORS AND DIELECTRICS

CAPACITORS AND DIELECTRICS Importnt Definitions nd Units Cpcitnce: CAPACITORS AND DIELECTRICS The property of system of electricl conductors nd insultors which enbles it to store electric chrge when potentil difference exists between

More information

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero

More information

Lecture 13 - Linking E, ϕ, and ρ

Lecture 13 - Linking E, ϕ, and ρ Lecture 13 - Linking E, ϕ, nd ρ A Puzzle... Inner-Surfce Chrge Density A positive point chrge q is locted off-center inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on

More information

Physics 24 Exam 1 February 18, 2014

Physics 24 Exam 1 February 18, 2014 Exm Totl / 200 Physics 24 Exm 1 Februry 18, 2014 Printed Nme: Rec. Sec. Letter: Five multiple choice questions, 8 points ech. Choose the best or most nerly correct nswer. 1. The totl electric flux pssing

More information

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS 33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in

More information

KINEMATICS OF RIGID BODIES

KINEMATICS OF RIGID BODIES KINEMTICS OF RIGID ODIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. Description

More information

Phys 4321 Final Exam December 14, 2009

Phys 4321 Final Exam December 14, 2009 Phys 4321 Finl Exm December 14, 2009 You my NOT use the text book or notes to complete this exm. You nd my not receive ny id from nyone other tht the instructor. You will hve 3 hours to finish. DO YOUR

More information

12 TRANSFORMING BIVARIATE DENSITY FUNCTIONS

12 TRANSFORMING BIVARIATE DENSITY FUNCTIONS 1 TRANSFORMING BIVARIATE DENSITY FUNCTIONS Hving seen how to trnsform the probbility density functions ssocited with single rndom vrible, the next logicl step is to see how to trnsform bivrite probbility

More information

2. VECTORS AND MATRICES IN 3 DIMENSIONS

2. VECTORS AND MATRICES IN 3 DIMENSIONS 2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

More information

Summary: Method of Separation of Variables

Summary: Method of Separation of Variables Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section

More information

This final is a three hour open book, open notes exam. Do all four problems.

This final is a three hour open book, open notes exam. Do all four problems. Physics 55 Fll 27 Finl Exm Solutions This finl is three hour open book, open notes exm. Do ll four problems. [25 pts] 1. A point electric dipole with dipole moment p is locted in vcuum pointing wy from

More information

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007 A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

More information

Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution

Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: olumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge

More information

Section 14.3 Arc Length and Curvature

Section 14.3 Arc Length and Curvature Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in

More information

Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.

Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space. Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)

More information

Prof. Anchordoqui. Problems set # 4 Physics 169 March 3, 2015

Prof. Anchordoqui. Problems set # 4 Physics 169 March 3, 2015 Prof. Anchordoui Problems set # 4 Physics 169 Mrch 3, 15 1. (i) Eight eul chrges re locted t corners of cube of side s, s shown in Fig. 1. Find electric potentil t one corner, tking zero potentil to be

More information

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

More information

TImath.com Algebra 2. Constructing an Ellipse

TImath.com Algebra 2. Constructing an Ellipse TImth.com Algebr Constructing n Ellipse ID: 9980 Time required 60 minutes Activity Overview This ctivity introduces ellipses from geometric perspective. Two different methods for constructing n ellipse

More information

Problem Set 4: Mostly Magnetic

Problem Set 4: Mostly Magnetic University of Albm Deprtment of Physics nd Astronomy PH 102 / LeClir Summer 2012 nstructions: Problem Set 4: Mostly Mgnetic 1. Answer ll questions below. Show your work for full credit. 2. All problems

More information

Definite integral. Mathematics FRDIS MENDELU

Definite integral. Mathematics FRDIS MENDELU Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the

More information

Physics Jonathan Dowling. Lecture 9 FIRST MIDTERM REVIEW

Physics Jonathan Dowling. Lecture 9 FIRST MIDTERM REVIEW Physics 10 Jonthn Dowling Physics 10 ecture 9 FIRST MIDTERM REVIEW A few concepts: electric force, field nd potentil Electric force: Wht is the force on chrge produced by other chrges? Wht is the force

More information

Definite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30

Definite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30 Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function

More information

200 points 5 Problems on 4 Pages and 20 Multiple Choice/Short Answer Questions on 5 pages 1 hour, 48 minutes

200 points 5 Problems on 4 Pages and 20 Multiple Choice/Short Answer Questions on 5 pages 1 hour, 48 minutes PHYSICS 132 Smple Finl 200 points 5 Problems on 4 Pges nd 20 Multiple Choice/Short Answer Questions on 5 pges 1 hour, 48 minutes Student Nme: Recittion Instructor (circle one): nme1 nme2 nme3 nme4 Write

More information

7.6 The Use of Definite Integrals in Physics and Engineering

7.6 The Use of Definite Integrals in Physics and Engineering Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems

More information

Math 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008

Math 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008 Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 2-5pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of

More information

Line Integrals. Chapter Definition

Line Integrals. Chapter Definition hpter 2 Line Integrls 2.1 Definition When we re integrting function of one vrible, we integrte long n intervl on one of the xes. We now generlize this ide by integrting long ny curve in the xy-plne. It

More information

p(t) dt + i 1 re it ireit dt =

p(t) dt + i 1 re it ireit dt = Note: This mteril is contined in Kreyszig, Chpter 13. Complex integrtion We will define integrls of complex functions long curves in C. (This is bit similr to [relvlued] line integrls P dx + Q dy in R2.)

More information

Final Exam Solutions, MAC 3474 Calculus 3 Honors, Fall 2018

Final Exam Solutions, MAC 3474 Calculus 3 Honors, Fall 2018 Finl xm olutions, MA 3474 lculus 3 Honors, Fll 28. Find the re of the prt of the sddle surfce z xy/ tht lies inside the cylinder x 2 + y 2 2 in the first positive) octnt; is positive constnt. olution:

More information

Candidates must show on each answer book the type of calculator used.

Candidates must show on each answer book the type of calculator used. UNIVERSITY OF EAST ANGLIA School of Mthemtics My/June UG Exmintion 2007 2008 ELECTRICITY AND MAGNETISM Time llowed: 3 hours Attempt FIVE questions. Cndidtes must show on ech nswer book the type of clcultor

More information

Homework Assignment 5 Solution Set

Homework Assignment 5 Solution Set Homework Assignment 5 Solution Set PHYCS 44 3 Februry, 4 Problem Griffiths 3.8 The first imge chrge gurntees potentil of zero on the surfce. The secon imge chrge won t chnge the contribution to the potentil

More information

Math 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions

Math 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions Mth 1102: Clculus I (Mth/Sci mjors) MWF 3pm, Fulton Hll 230 Homework 2 solutions Plese write netly, nd show ll work. Cution: An nswer with no work is wrong! Do the following problems from Chpter III: 6,

More information

Partial Derivatives. Limits. For a single variable function f (x), the limit lim

Partial Derivatives. Limits. For a single variable function f (x), the limit lim Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles

More information

Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution

Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: Volumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge

More information

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

More information

Mathematics. Area under Curve.

Mathematics. Area under Curve. Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding

More information

Math 0230 Calculus 2 Lectures

Math 0230 Calculus 2 Lectures Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two

More information

BME 207 Introduction to Biomechanics Spring 2018

BME 207 Introduction to Biomechanics Spring 2018 April 6, 28 UNIVERSITY O RHODE ISAND Deprtment of Electricl, Computer nd Biomedicl Engineering BME 27 Introduction to Biomechnics Spring 28 Homework 8 Prolem 14.6 in the textook. In ddition to prts -e,

More information

Reference. Vector Analysis Chapter 2

Reference. Vector Analysis Chapter 2 Reference Vector nlsis Chpter Sttic Electric Fields (3 Weeks) Chpter 3.3 Coulomb s Lw Chpter 3.4 Guss s Lw nd pplictions Chpter 3.5 Electric Potentil Chpter 3.6 Mteril Medi in Sttic Electric Field Chpter

More information

Math 8 Winter 2015 Applications of Integration

Math 8 Winter 2015 Applications of Integration Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

More information

Phys 6321 Final Exam - Solutions May 3, 2013

Phys 6321 Final Exam - Solutions May 3, 2013 Phys 6321 Finl Exm - Solutions My 3, 2013 You my NOT use ny book or notes other thn tht supplied with this test. You will hve 3 hours to finish. DO YOUR OWN WORK. Express your nswers clerly nd concisely

More information

Homework Assignment 6 Solution Set

Homework Assignment 6 Solution Set Homework Assignment 6 Solution Set PHYCS 440 Mrch, 004 Prolem (Griffiths 4.6 One wy to find the energy is to find the E nd D fields everywhere nd then integrte the energy density for those fields. We know

More information

Problem 1. Solution: a) The coordinate of a point on the disc is given by r r cos,sin,0. The potential at P is then given by. r z 2 rcos 2 rsin 2

Problem 1. Solution: a) The coordinate of a point on the disc is given by r r cos,sin,0. The potential at P is then given by. r z 2 rcos 2 rsin 2 Prolem Consider disc of chrge density r r nd rdius R tht lies within the xy-plne. The origin of the coordinte systems is locted t the center of the ring. ) Give the potentil t the point P,,z in terms of,r,

More information

I. Equations of a Circle a. At the origin center= r= b. Standard from: center= r=

I. Equations of a Circle a. At the origin center= r= b. Standard from: center= r= 11.: Circle & Ellipse I cn Write the eqution of circle given specific informtion Grph circle in coordinte plne. Grph n ellipse nd determine ll criticl informtion. Write the eqution of n ellipse from rel

More information

Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim

Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)

More information

JUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 6 (First moments of an arc) A.J.Hobson

JUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 6 (First moments of an arc) A.J.Hobson JUST THE MATHS UNIT NUMBER 13.6 INTEGRATION APPLICATIONS 6 (First moments of n rc) by A.J.Hobson 13.6.1 Introduction 13.6. First moment of n rc bout the y-xis 13.6.3 First moment of n rc bout the x-xis

More information

Physics Graduate Prelim exam

Physics Graduate Prelim exam Physics Grdute Prelim exm Fll 2008 Instructions: This exm hs 3 sections: Mechnics, EM nd Quntum. There re 3 problems in ech section You re required to solve 2 from ech section. Show ll work. This exm is

More information

THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES

THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES 1. TRANSLATION Figure shows rigid body trnslting in three-dimensionl spce. Any two points in the body, such s A nd B, will move long prllel stright lines if

More information

Sample Exam 5 - Skip Problems 1-3

Sample Exam 5 - Skip Problems 1-3 Smple Exm 5 - Skip Problems 1-3 Physics 121 Common Exm 2: Fll 2010 Nme (Print): 4 igit I: Section: Honors Code Pledge: As n NJIT student I, pledge to comply with the provisions of the NJIT Acdemic Honor

More information

JUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 12 (Second moments of an area (B)) A.J.Hobson

JUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 12 (Second moments of an area (B)) A.J.Hobson JUST THE MATHS UNIT NUMBE 13.1 INTEGATION APPLICATIONS 1 (Second moments of n re (B)) b A.J.Hobson 13.1.1 The prllel xis theorem 13.1. The perpendiculr xis theorem 13.1.3 The rdius of grtion of n re 13.1.4

More information

13.4 Work done by Constant Forces

13.4 Work done by Constant Forces 13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push

More information

Note 16. Stokes theorem Differential Geometry, 2005

Note 16. Stokes theorem Differential Geometry, 2005 Note 16. Stokes theorem ifferentil Geometry, 2005 Stokes theorem is the centrl result in the theory of integrtion on mnifolds. It gives the reltion between exterior differentition (see Note 14) nd integrtion

More information

Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions

Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner

More information

JUST THE MATHS SLIDES NUMBER INTEGRATION APPLICATIONS 12 (Second moments of an area (B)) A.J.Hobson

JUST THE MATHS SLIDES NUMBER INTEGRATION APPLICATIONS 12 (Second moments of an area (B)) A.J.Hobson JUST THE MATHS SLIDES NUMBER 13.12 INTEGRATION APPLICATIONS 12 (Second moments of n re (B)) b A.J.Hobson 13.12.1 The prllel xis theorem 13.12.2 The perpendiculr xis theorem 13.12.3 The rdius of grtion

More information

(b) Let S 1 : f(x, y, z) = (x a) 2 + (y b) 2 + (z c) 2 = 1, this is a level set in 3D, hence

(b) Let S 1 : f(x, y, z) = (x a) 2 + (y b) 2 + (z c) 2 = 1, this is a level set in 3D, hence Problem ( points) Find the vector eqution of the line tht joins points on the two lines L : r ( + t) i t j ( + t) k L : r t i + (t ) j ( + t) k nd is perpendiculr to both those lines. Find the set of ll

More information

Math 120 Answers for Homework 13

Math 120 Answers for Homework 13 Mth 12 Answers for Homework 13 1. In this problem we will use the fct tht if m f(x M on n intervl [, b] (nd if f is integrble on [, b] then (* m(b f dx M(b. ( The function f(x = 1 + x 3 is n incresing

More information

HW Solutions # MIT - Prof. Kowalski. Friction, circular dynamics, and Work-Kinetic Energy.

HW Solutions # MIT - Prof. Kowalski. Friction, circular dynamics, and Work-Kinetic Energy. HW Solutions # 5-8.01 MIT - Prof. Kowlski Friction, circulr dynmics, nd Work-Kinetic Energy. 1) 5.80 If the block were to remin t rest reltive to the truck, the friction force would need to cuse n ccelertion

More information

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.) MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

More information

CONIC SECTIONS. Chapter 11

CONIC SECTIONS. Chapter 11 CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round

More information

Physics 121 Sample Common Exam 1 NOTE: ANSWERS ARE ON PAGE 8. Instructions:

Physics 121 Sample Common Exam 1 NOTE: ANSWERS ARE ON PAGE 8. Instructions: Physics 121 Smple Common Exm 1 NOTE: ANSWERS ARE ON PAGE 8 Nme (Print): 4 Digit ID: Section: Instructions: Answer ll questions. uestions 1 through 16 re multiple choice questions worth 5 points ech. You

More information

Indefinite Integral. Chapter Integration - reverse of differentiation

Indefinite Integral. Chapter Integration - reverse of differentiation Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the

More information

Polynomials and Division Theory

Polynomials and Division Theory Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the

More information

Contour and surface integrals

Contour and surface integrals Contour nd surfce integrls Contour integrls of the sclr type These re integrls of the type I = f@rd l C where C is contour nd l is n infinitesiml element of the contour length. If the contour cn be described

More information

US01CMTH02 UNIT Curvature

US01CMTH02 UNIT Curvature Stu mteril of BSc(Semester - I) US1CMTH (Rdius of Curvture nd Rectifiction) Prepred by Nilesh Y Ptel Hed,Mthemtics Deprtment,VPnd RPTPScience College US1CMTH UNIT- 1 Curvture Let f : I R be sufficiently

More information

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014 SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.

More information

Not for reproduction

Not for reproduction AREA OF A SURFACE OF REVOLUTION cut h FIGURE FIGURE πr r r l h FIGURE A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundry of solid of revolution of the type

More information

Theoretische Physik 2: Elektrodynamik (Prof. A.-S. Smith) Home assignment 4

Theoretische Physik 2: Elektrodynamik (Prof. A.-S. Smith) Home assignment 4 WiSe 1 8.1.1 Prof. Dr. A.-S. Smith Dipl.-Phys. Ellen Fischermeier Dipl.-Phys. Mtthis Sb m Lehrstuhl für Theoretische Physik I Deprtment für Physik Friedrich-Alexnder-Universität Erlngen-Nürnberg Theoretische

More information

Chapter 6 Electrostatic Boundary Value Problems. Dr. Talal Skaik

Chapter 6 Electrostatic Boundary Value Problems. Dr. Talal Skaik Chpter 6 Electrosttic Boundry lue Problems Dr. Tll Skik 1 1 Introduction In previous chpters, E ws determined by coulombs lw or Guss lw when chrge distribution is known, or potentil is known throughout

More information

SAINT IGNATIUS COLLEGE

SAINT IGNATIUS COLLEGE SAINT IGNATIUS COLLEGE Directions to Students Tril Higher School Certificte 0 MATHEMATICS Reding Time : 5 minutes Totl Mrks 00 Working Time : hours Write using blue or blck pen. (sketches in pencil). This

More information

Problem Set 3 Solutions

Problem Set 3 Solutions Msschusetts Institute of Technology Deprtment of Physics Physics 8.07 Fll 2005 Problem Set 3 Solutions Problem 1: Cylindricl Cpcitor Griffiths Problems 2.39: Let the totl chrge per unit length on the inner

More information

Time : 3 hours 03 - Mathematics - March 2007 Marks : 100 Pg - 1 S E CT I O N - A

Time : 3 hours 03 - Mathematics - March 2007 Marks : 100 Pg - 1 S E CT I O N - A Time : hours 0 - Mthemtics - Mrch 007 Mrks : 100 Pg - 1 Instructions : 1. Answer ll questions.. Write your nswers ccording to the instructions given below with the questions.. Begin ech section on new

More information

DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS

DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS 3 DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS This chpter summrizes few properties of Cli ord Algebr nd describe its usefulness in e ecting vector rottions. 3.1 De nition of Associtive

More information

Line Integrals. Partitioning the Curve. Estimating the Mass

Line Integrals. Partitioning the Curve. Estimating the Mass Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to

More information

5.2 Volumes: Disks and Washers

5.2 Volumes: Disks and Washers 4 pplictions of definite integrls 5. Volumes: Disks nd Wshers In the previous section, we computed volumes of solids for which we could determine the re of cross-section or slice. In this section, we restrict

More information

ES.182A Topic 32 Notes Jeremy Orloff

ES.182A Topic 32 Notes Jeremy Orloff ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In

More information

Thomas Whitham Sixth Form

Thomas Whitham Sixth Form Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos

More information

The Properties of Stars

The Properties of Stars 10/11/010 The Properties of Strs sses Using Newton s Lw of Grvity to Determine the ss of Celestil ody ny two prticles in the universe ttrct ech other with force tht is directly proportionl to the product

More information

1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.

1.2. Linear Variable Coefficient Equations. y + b ! = a y + b  Remark: The case b = 0 and a non-constant can be solved with the same idea as above. 1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt

More information

Math 100 Review Sheet

Math 100 Review Sheet Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s

More information

k ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola.

k ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola. Stndrd Eqution of Prol with vertex ( h, k ) nd directrix y = k p is ( x h) p ( y k ) = 4. Verticl xis of symmetry Stndrd Eqution of Prol with vertex ( h, k ) nd directrix x = h p is ( y k ) p( x h) = 4.

More information

Geometric and Mechanical Applications of Integrals

Geometric and Mechanical Applications of Integrals 5 Geometric nd Mechnicl Applictions of Integrls 5.1 Computing Are 5.1.1 Using Crtesin Coordintes Suppose curve is given by n eqution y = f(x), x b, where f : [, b] R is continuous function such tht f(x)

More information

KINEMATICS OF RIGID BODIES

KINEMATICS OF RIGID BODIES KINEMTICS OF RIGI OIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. escription

More information

Probability Distributions for Gradient Directions in Uncertain 3D Scalar Fields

Probability Distributions for Gradient Directions in Uncertain 3D Scalar Fields Technicl Report 7.8. Technische Universität München Probbility Distributions for Grdient Directions in Uncertin 3D Sclr Fields Tobis Pfffelmoser, Mihel Mihi, nd Rüdiger Westermnn Computer Grphics nd Visuliztion

More information

Heat flux and total heat

Heat flux and total heat Het flux nd totl het John McCun Mrch 14, 2017 1 Introduction Yesterdy (if I remember correctly) Ms. Prsd sked me question bout the condition of insulted boundry for the 1D het eqution, nd (bsed on glnce

More information

F is on a moving charged particle. F = 0, if B v. (sin " = 0)

F is on a moving charged particle. F = 0, if B v. (sin  = 0) F is on moving chrged prticle. Chpter 29 Mgnetic Fields Ech mgnet hs two poles, north pole nd south pole, regrdless the size nd shpe of the mgnet. Like poles repel ech other, unlike poles ttrct ech other.

More information

50. Use symmetry to evaluate xx D is the region bounded by the square with vertices 5, Prove Property 11. y y CAS

50. Use symmetry to evaluate xx D is the region bounded by the square with vertices 5, Prove Property 11. y y CAS 68 CHAPTE MULTIPLE INTEGALS 46. e da, 49. Evlute tn 3 4 da, where,. [Hint: Eploit the fct tht is the disk with center the origin nd rdius is smmetric with respect to both es.] 5. Use smmetr to evlute 3

More information

Lecture Outline. Dispersion Relation Electromagnetic Wave Polarization 8/7/2018. EE 4347 Applied Electromagnetics. Topic 3c

Lecture Outline. Dispersion Relation Electromagnetic Wave Polarization 8/7/2018. EE 4347 Applied Electromagnetics. Topic 3c Course Instructor Dr. Rymond C. Rumpf Office: A 337 Phone: (915) 747 6958 E Mil: rcrumpf@utep.edu EE 4347 Applied Electromgnetics Topic 3c Wve Dispersion & Polriztion Wve Dispersion These notes & Polriztion

More information

Hung problem # 3 April 10, 2011 () [4 pts.] The electric field points rdilly inwrd [1 pt.]. Since the chrge distribution is cylindriclly symmetric, we pick cylinder of rdius r for our Gussin surfce S.

More information

Method of Localisation and Controlled Ejection of Swarms of Likely Charged Particles

Method of Localisation and Controlled Ejection of Swarms of Likely Charged Particles Method of Loclistion nd Controlled Ejection of Swrms of Likely Chrged Prticles I. N. Tukev July 3, 17 Astrct This work considers Coulom forces cting on chrged point prticle locted etween the two coxil,

More information

Things to Memorize: A Partial List. January 27, 2017

Things to Memorize: A Partial List. January 27, 2017 Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved

More information

( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that

( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we

More information

MATH 13 FINAL STUDY GUIDE, WINTER 2012

MATH 13 FINAL STUDY GUIDE, WINTER 2012 MATH 13 FINAL TUY GUI, WINTR 2012 This is ment to be quick reference guide for the topics you might wnt to know for the finl. It probbly isn t comprehensive, but should cover most of wht we studied in

More information