# Homework Assignment 6 Solution Set

Save this PDF as:
Size: px
Start display at page:

## Transcription

1 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 tht D depends only on the free chrge nd is therefore continuous cross the dielectric oundry with vcuum. Thus, the energy is W = D E = llspce Q = Q 8πɛ 0 ɛ(4πr dτ + ( ( + χ e ( + χ e = Q 8πɛ 0 + χ e +. Q ɛ 0 (4πr dτ Prolem (Griffiths 4. There is only one free chrge: q. So inside nd outside the dielectric we hve Inside the dielectric, then, we know nd So the ound chrges re E = D = P = ɛ 0 χ e E = σ = P ˆr r=r = q 4πr ˆr. q 4πɛ 0 ( + χ e r ˆr χ e q 4π( + χ e r ˆr. χ e q 4π( + χ e R ρ = P = 0 (except t r = 0. The totl ound chrge on the surfce is just q χe +χ e, nd it is compensted y the volume chrge tht is loclized immeditely round the implnted chrge.

2 Note lso tht the field outside the dielectric is the sme s if the dielectric were not present. The field is screened inside the dielectric ut is otherwise unffected once you re outside the dielectric gin. Prolem (Griffiths 4. - optionl The ehvior of E is known t the interfce from the oundry conditions tht rise due to the reltionships etween E, D nd the free nd ound chrges. We know tht the prllel component of E should e continous cross the oundry, ut the perpendiculr component hs discontinuity due to the surfce chrges there. However, we know tht D only depends on free chrges, so the perpendiculr component of D is continuous cross the interfce. Thus, tn θ tn θ = = E, E, E, E, D, ɛ D, ɛ = ɛ ɛ. In the region 0 < θ < π, tn θ is everywhere positive nd grows with incresing θ. Thus, electric field lines ehve opposite to the nlogous light wves in medi - they end towrd the norml when going from regions of lrge ɛ to smll ɛ. Therefore, convex lens would e diverging lens for electric field lines (i.e., they would defocus the field. Prolem 4 (Griffiths optionl This is oundry vlue prolem tht is very similr to severl others tht we hve done. This time, however, we dd nother oundry nd few more conditions sed on the properties of dielectrics. The generl solution is V i (r, θ = ( A li r li + B l i r li+ P li (cos θ l i where i denotes the region of interest (i =,, for inside the conductor, inside the dielectric, nd outside respectively with the following oundry conditions: V (r E 0 r cos θ ( V = 0 (for simplicity ( V (r = = V (r = = 0 ( V (r = = V (r = (4 ɛ V r r= = ɛ 0 V r r= (5

3 From ( we see tht ll A l vnish except for l =, for which A = E 0. Condition (4 then tells us tht nd Thus, B l l+ B l l+ A l l = 0 (for l 0 l = l. (B l B l l+ A l l = 0. (6 Now, condition (5 gives ( ɛ r (l A l l (l + B l l+ P l (cos θ = E 0 cos θ + (l + B l l+ P l (cos θ l l nd so, l + l+ (ɛ rb l B l ɛ r l l A l = 0 (for l 0. (7 Now, tke condition ( which gives nd comine it with results (6 & (7 ove to get A l = B l l+ (8 ( B l = B l + l+ l+ B l = B l ɛ r ( l + l+ + ll l+ (9 (0 B l = B l = A l = 0 (for l 0 ( So, ll tht s left is to find A nd B (we ll get B in the process. Go ck to conditions (4 nd (5 nd write them gin, this time with l = (sustituting for A vi (8. We hve E 0 + B = B E 0 + B = B ɛ r B = ( ( + E 0 ɛ r ( + + A = B = E 0 ɛ r ( + + ( ( (4 (5

4 THEREFORE... nd, so, E( r = V ( r V (r, θ = E 0 = ɛ r ( + + E 0 ɛ r ( + + (r r (( + r cos θˆr cos θ ( r sin θˆθ. Apprently the screening y the dielectric mkes it so tht the electric field in the dielectric hs some θ dependence s you move wy from r =. Thus, the ending of E 0 is weker thn if we hd just plced the conducting sphere there without ny dielectric. Prolem 5 (Griffiths 5. We cn just strt from the result of Exmple 5. y(t = C cos ωt + C sin ωt + E B t + C z(t = C cos ωt C sin ωt + C 4 nd pply the given initil conditions to solve for the unknown constnts. In ech cse we hve x(0 = y(0 = z(0 = 0 which tells us C = C C = C 4. dy dt t=0 = E B dz dt t=0 = 0 C = 0 C = 0 y(t = E B t z(t = 0. In this cse the mgnetic nd electric forces exctly cncel nd the trjectory is stright line in the ŷ direction. 4

5 dy dt t=0 = E B dz dt t=0 = 0 C = E ωb C = 0 y(t = E ωb sin ωt + E B t z(t = E cos ωt + E ωb ωb. Now the velocity in the ŷ direction is not enough to cuse the mgnetic force to cncel with the electric force, so there is some rolling like in Exmple 5.. The trjectory is sketched elow, with z s the verticle xis nd y s the horizontl xis, oth in units of E B with Q m such tht ω = πsec c dy dt t=0 = E B dz dt t=0 = E B C = 0 C = E ωb y(t = E ωb cos ωt + E B t + E ωb z(t = E sin ωt. ωb 5

6 This is nother cycloid, ut centered long the y xis. The sketch is elow in the sme units s the previous plot Prolem 6 (Griffiths 5.6 K(r = σ(r v(r = σωrˆθ J(r, θ = ρ v(r, θ = ρωr sin θ ˆφ Prolem 7 (Griffiths 5.8 The field due to finite stright segment of current is given in eq. 5.5 s B(s = µ 0I πs (sin θ sin θ where s is the perpendiculr distnce from the current to the point nd θ ndθ re the ngles to the ends of the current segment reltive to the perpendiculr s. The direction is determined y the right hnd rule. At the center of squre loop the contriution to the mgnetic field from ech of the four sides points in the sme direction (norml to the plne of the loop, so the totl field t the center of squre of side R is just B = µ ( ( 0I π ( sin sin π πr 4 4 µ0 I = πr. Generlizing to more sides just gives since sin θ sin( θ = sin θ. B = Nµ 0I 4πR sin π N 6

7 c As N we get µ 0 I N µ 0 I sin x lim sin θ = lim N πr π x 0 R x = µ 0I R. This grees with eq. 5.8, the field nywhere on the xis of circulr loop, when z = 0. Prolem 8 (Griffiths 5. From Ampere s lw we get quite esily B inside = 0 B outside = µ 0I πs ˆθ B inside = µ 0 πs I encl = µ 0 πs = Cµ 0s ˆθ, s 0 Cs πs ds ut how do we find C? We know tht the totl current is I, so This gives nd, of course, 0 Cs πs ds = I B inside = µ 0s C = π I. π I = µ 0Is π ˆθ B outside = µ 0I πs ˆθ. 7

### 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.

### 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

### Homework Assignment 9 Solution Set

Homework Assignment 9 Solution Set PHYCS 44 3 Mrch, 4 Problem (Griffiths 77) The mgnitude of the current in the loop is loop = ε induced = Φ B = A B = π = π µ n (µ n) = π µ nk According to Lense s Lw this

### Problem Solving 7: Faraday s Law Solution

MASSACHUSETTS NSTTUTE OF TECHNOLOGY Deprtment of Physics: 8.02 Prolem Solving 7: Frdy s Lw Solution Ojectives 1. To explore prticulr sitution tht cn led to chnging mgnetic flux through the open surfce

FINALTERM EXAMINATION 9 (Session - ) Clculus & Anlyticl Geometry-I Question No: ( Mrs: ) - Plese choose one f ( x) x According to Power-Rule of differentition, if d [ x n ] n x n n x n n x + ( n ) x n+

### 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

### 7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus

7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e

### 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

### Physics 202, Lecture 14

Physics 202, Lecture 14 Tody s Topics Sources of the Mgnetic Field (Ch. 28) Biot-Svrt Lw Ampere s Lw Mgnetism in Mtter Mxwell s Equtions Homework #7: due Tues 3/11 t 11 PM (4th problem optionl) Mgnetic

### 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

### Waveguide Guide: A and V. Ross L. Spencer

Wveguide Guide: A nd V Ross L. Spencer I relly think tht wveguide fields re esier to understnd using the potentils A nd V thn they re using the electric nd mgnetic fields. Since Griffiths doesn t do it

### 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

### 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

### 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,

### 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

### Physics 1402: Lecture 7 Today s Agenda

1 Physics 1402: Lecture 7 Tody s gend nnouncements: Lectures posted on: www.phys.uconn.edu/~rcote/ HW ssignments, solutions etc. Homework #2: On Msterphysics tody: due Fridy Go to msteringphysics.com Ls:

### I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3

2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is

### 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

### 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

### 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)

### 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

### 10 Vector Integral Calculus

Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve

Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite

### Topics Covered AP Calculus AB

Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.

### 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

### 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.

### 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

### On the diagram below the displacement is represented by the directed line segment OA.

Vectors Sclrs nd Vectors A vector is quntity tht hs mgnitude nd direction. One exmple of vector is velocity. The velocity of n oject is determined y the mgnitude(speed) nd direction of trvel. Other exmples

### Version 001 HW#6 - Electromagnetism arts (00224) 1

Version 001 HW#6 - Electromgnetism rts (00224) 1 This print-out should hve 11 questions. Multiple-choice questions my continue on the next column or pge find ll choices efore nswering. rightest Light ul

### 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

### Phys 7221, Fall 2006: Homework # 6

Phys 7221, Fll 2006: Homework # 6 Gbriel González October 29, 2006 Problem 3-7 In the lbortory system, the scttering ngle of the incident prticle is ϑ, nd tht of the initilly sttionry trget prticle, which

### Today in Physics 122: work, energy and potential in electrostatics

Tody in Physics 1: work, energy nd potentil in electrosttics Leftovers Perfect conductors Fields from chrges distriuted on perfect conductors Guss s lw for grvity Work nd energy Electrosttic potentil energy,

### 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

### 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

### Resistors. Consider a uniform cylinder of material with mediocre to poor to pathetic conductivity ( )

10/25/2005 Resistors.doc 1/7 Resistors Consider uniform cylinder of mteril with mediocre to poor to r. pthetic conductivity ( ) ˆ This cylinder is centered on the -xis, nd hs length. The surfce re of the

### Magnetic forces on a moving charge. EE Lecture 26. Lorentz Force Law and forces on currents. Laws of magnetostatics

Mgnetic forces on moving chrge o fr we ve studied electric forces between chrges t rest, nd the currents tht cn result in conducting medium 1. Mgnetic forces on chrge 2. Lws of mgnetosttics 3. Mgnetic

### 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

### Conducting Ellipsoid and Circular Disk

1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid,

### #6A&B Magnetic Field Mapping

#6A& Mgnetic Field Mpping Gol y performing this lb experiment, you will: 1. use mgnetic field mesurement technique bsed on Frdy s Lw (see the previous experiment),. study the mgnetic fields generted by

### set is not closed under matrix [ multiplication, ] and does not form a group.

Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed

### Section 7.1 Area of a Region Between Two Curves

Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region

### 4.4 Areas, Integrals and Antiderivatives

. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order

### 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

### 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

### 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

### 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

### Vectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3-dimensional vectors:

Vectors 1-23-2018 I ll look t vectors from n lgeric point of view nd geometric point of view. Algericlly, vector is n ordered list of (usully) rel numers. Here re some 2-dimensionl vectors: (2, 3), ( )

### Physics 9 Fall 2011 Homework 2 - Solutions Friday September 2, 2011

Physics 9 Fll 0 Homework - s Fridy September, 0 Mke sure your nme is on your homework, nd plese box your finl nswer. Becuse we will be giving prtil credit, be sure to ttempt ll the problems, even if you

### 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

### 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

### 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,

### 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

### Section 6: Area, Volume, and Average Value

Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

### Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +

Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd

### [ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves

Gols: 1. To find the re etween two curves Section 6.1 Are of Regions etween two Curves I. Are of Region Between Two Curves A. Grphicl Represention = _ B. Integrl Represention [ ( ) ( )] f x g x dx = C.

### Trigonometric Functions

Exercise. Degrees nd Rdins Chpter Trigonometric Functions EXERCISE. Degrees nd Rdins 4. Since 45 corresponds to rdin mesure of π/4 rd, we hve: 90 = 45 corresponds to π/4 or π/ rd. 5 = 7 45 corresponds

### 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.

### Andrew Ryba Math Intel Research Final Paper 6/7/09 (revision 6/17/09)

Andrew Ryb Mth ntel Reserch Finl Pper 6/7/09 (revision 6/17/09) Euler's formul tells us tht for every tringle, the squre of the distnce between its circumcenter nd incenter is R 2-2rR, where R is the circumrdius

### Physics 2135 Exam 3 April 21, 2015

Em Totl hysics 2135 Em 3 April 21, 2015 Key rinted Nme: 200 / 200 N/A Rec. Sec. Letter: Five multiple choice questions, 8 points ech. Choose the best or most nerly correct nswer. 1. C Two long stright

### PROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by

PROPERTES OF RES Centroid The concept of the centroid is prol lred fmilir to ou For plne shpe with n ovious geometric centre, (rectngle, circle) the centroid is t the centre f n re hs n is of smmetr, the

### THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.

THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem

### Section 4: Integration ECO4112F 2011

Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic

### Physics 202, Lecture 13. Today s Topics

Physics 202, Lecture 13 Tody s Topics Sources of the Mgnetic Field (Ch. 30) Clculting the B field due to currents Biot-Svrt Lw Emples: ring, stright wire Force between prllel wires Ampere s Lw: infinite

### 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

### Chapter 7 Steady Magnetic Field. september 2016 Microwave Laboratory Sogang University

Chpter 7 Stedy Mgnetic Field september 2016 Microwve Lbortory Sogng University Teching point Wht is the mgnetic field? Biot-Svrt s lw: Coulomb s lw of Mgnetic field Stedy current: current flow is independent

### How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=

### 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

### Chapter E - Problems

Chpter E - Prolems Blinn College - Physics 2426 - Terry Honn Prolem E.1 A wire with dimeter d feeds current to cpcitor. The chrge on the cpcitor vries with time s QHtL = Q 0 sin w t. Wht re the current

### Date Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( )

UNIT 5 TRIGONOMETRI RTIOS Dte Lesson Text TOPI Homework pr. 4 5.1 (48) Trigonometry Review WS 5.1 # 3 5, 9 11, (1, 13)doso pr. 6 5. (49) Relted ngles omplete lesson shell & WS 5. pr. 30 5.3 (50) 5.3 5.4

### a < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1

Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the

### 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

### The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).

The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples

### IMPORTANT. Read these directions carefully:

Physics 208: Electricity nd Mgnetism Finl Exm, Secs. 506 510. 7 My. 2004 Instructor: Dr. George R. Welch, 415 Engineering-Physics, 845-7737 Print your nme netly: Lst nme: First nme: Sign your nme: Plese

### PHYSICS ASSIGNMENT-9

MPS/PHY-XII-11/A9 PHYSICS ASSIGNMENT-9 *********************************************************************************************************** 1. A wire kept long the north-south direction is llowed

### KEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a

KEY CONCEPTS THINGS TO REMEMBER :. The re ounded y the curve y = f(), the -is nd the ordintes t = & = is given y, A = f () d = y d.. If the re is elow the is then A is negtive. The convention is to consider

### Phys. 506 Electricity and Magnetism Winter 2004 Prof. G. Raithel Problem Set 1 Total 30 Points. 1. Jackson Points

Phys. 56 Electricity nd Mgnetism Winter 4 Prof. G. Rithel Prolem Set Totl 3 Points. Jckson 8. Points : The electric field is the sme s in the -dimensionl electrosttic prolem of two concentric cylinders,

Mth 348 Fll 7 Lecture 3: Curves in Clculus Disclimer. As we hve textook, this lecture note is for guidnce nd supplement only. It should not e relied on when prepring for exms. In this lecture we set up

### The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O

1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

### Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.

1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples

### Shape and measurement

C H A P T E R 5 Shpe nd mesurement Wht is Pythgors theorem? How do we use Pythgors theorem? How do we find the perimeter of shpe? How do we find the re of shpe? How do we find the volume of shpe? How do

### l 2 p2 n 4n 2, the total surface area of the

Week 6 Lectures Sections 7.5, 7.6 Section 7.5: Surfce re of Revolution Surfce re of Cone: Let C be circle of rdius r. Let P n be n n-sided regulr polygon of perimeter p n with vertices on C. Form cone

### in a uniform magnetic flux density B = Boa z. (a) Show that the electron moves in a circular path. (b) Find the radius r o

6. THE TATC MAGNETC FELD 6- LOENTZ FOCE EQUATON Lorent force eqution F = Fe + Fm = q ( E + v B ) Exmple 6- An electron hs n initil velocity vo = vo y in uniform mgnetic flux density B = Bo. () how tht

### CHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx

CHAPTER 0 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS 0.. PARAMETRIC FUNCTIONS A) Recll tht for prmetric equtions,. B) If the equtions x f(t), nd y g(t) define y s twice-differentile function of x, then t

### Physics 712 Electricity and Magnetism Solutions to Final Exam, Spring 2016

Physics 7 Electricity nd Mgnetism Solutions to Finl Em, Spring 6 Plese note tht some possibly helpful formuls pper on the second pge The number of points on ech problem nd prt is mrked in squre brckets

### Section 17.2 Line Integrals

Section 7. Line Integrls Integrting Vector Fields nd Functions long urve In this section we consider the problem of integrting functions, both sclr nd vector (vector fields) long curve in the plne. We

### ( ) Same as above but m = f x = f x - symmetric to y-axis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.

AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find

### Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

### Triangles The following examples explore aspects of triangles:

Tringles The following exmples explore spects of tringles: xmple 1: ltitude of right ngled tringle + xmple : tringle ltitude of the symmetricl ltitude of n isosceles x x - 4 +x xmple 3: ltitude of the

### 9.4. The Vector Product. Introduction. Prerequisites. Learning Outcomes

The Vector Product 9.4 Introduction In this section we descrie how to find the vector product of two vectors. Like the sclr product its definition my seem strnge when first met ut the definition is chosen

### We divide the interval [a, b] into subintervals of equal length x = b a n

Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:

### 2.4 Linear Inequalities and Interval Notation

.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or

### 7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?

7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge

### Electromagnetism Answers to Problem Set 10 Spring 2006

Electromgnetism 76 Answers to Problem Set 1 Spring 6 1. Jckson Prob. 5.15: Shielded Bifilr Circuit: Two wires crrying oppositely directed currents re surrounded by cylindricl shell of inner rdius, outer

### ( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).

AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f

### ( ) as a fraction. Determine location of the highest

AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if

### 10. AREAS BETWEEN CURVES

. AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

### Improper Integrals, and Differential Equations

Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted

### The Wave Equation I. MA 436 Kurt Bryan

1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string