1 Physics 8.02 Equation Sheet Exam One Please Remove this Tear Sheet from Your Exam 23ˆ()44ooqqrrπε πε==Er r rrrr ˆpoints source q field pointrrr= from tor 31() ( )4Niiioiqπε==−−∑Er r rrrrrrvrv 201ˆ()4sourcedqπε=′−∫Er rrrrrrr qq=FErr encoclosedsurfaceQdε⋅=∫∫EArr points from inside to outsidedAr 0closedpath⋅=∫Edsrr bmoving from a to b b aaVVVΔ=−=−⋅∫Edsrr UqVΔ=Δ point charge;()04oqVVrπε=∞= many point charges1;()04NiioiqVVπε==∞=−∑rrrv 01() ; ( ) 04sourcedqVVπε=∞=′−∫rrrrrr ;()04ijall pairsoi jqqUUπε=∞=∑−rrrr V=−∇Err Er=−∂V∂rfor spherical and cylindrical symmetry xVEx∂=−∂ yVEy∂=−∂ zVEz∂=−∂ Circumferences, Areas, Volumes: 1) The area of a circle of radius r is πr2.Its circumference is 2πr. 2) The surface area of a sphere of radius r is 4πr2. Its volume is (4 / 3)πr3. 3) The area of the sides of a cylinder of radius r and height h is 2πrh . Its volume is πr2h . Some potentially useful numbers ke=14πε0= 9 × 109N m2C223 8.02 Exam One Spring 2010 FAMILY (last) NAME GIVEN (first) NAME Student ID Number Your Section: ____L01 MW 9 am ____L02 MW 11 am ____L03 MW 1 pm ____L04 MW 3 pm ____L05 TR 9 am ____L06 TR 11 am ____L07 TR 1 pm ____L08 TR 3 pm Your Table and Group (e.g. 10A): ______________ Score Grader Problem 1 (25 points) Problem 2 (25 points) Problem 3 (25 points) Problem 4 (25 points) TOTAL4 Problem 1 (25 points) In this problem you are asked to answer 5 questions, each worth 5 points. You do not have to show your work; in most cases you may simply circle the chosen answer. Question 1 (5 points) 1. Above we show the grass seeds representation of the field of four point charges, located at the positions indicated by the numbers. Which statement is true about the signs of these charges: a) All four charges have the same sign. b) Charges 1 and 2 have the same sign, and that sign is opposite the sign of 3 and 4. c) Charges 1 and 3 have the same sign, and that sign is opposite the sign of 2 and 4. d) Charges 1 and 4 have the same sign, and that sign is opposite the sign of 2 and 3. e) None of the above.5 Question 2 (5 points) The grass seeds figure below shows the electric field of three charges with charges +1, +1, and -1, The Gaussian surface in the figure is a sphere containing two of the charges. The total electric flux through the spherical Gaussian surface is a) Positive b) Negative c) Zero d) Impossible to determine without more information6 Question 3 (5 points) Two point-like charged objects with charges Q+ and Q− are placed on the bottom corners of a square of side a, as shown in the figure. You move an electron with charge e− from the upper right corner marked A to the upper left corner marked B. Which of the following statements is true? a) You do a negative amount of work on the electron equal to the amount of energy necessary to assemble the system of three charged objects with the electron at point B. b) You do a positive amount of work on the electron equal to the amount of energy necessary to assemble the system of three charged objects with the electron at point B. c) You do a positive amount of work on the electron and the potential energy of the system of three charged objects increases. d) You do a negative amount of work on the electron and the potential energy of the system of three charged objects decreases. e) You do a negative amount of work on the electron and the potential energy of the system of three charged objects increases. f) You do a positive amount of work on the electron and the potential energy of the system of three charged objects decreases.7 Question 4 (5 points) A graph of the electric potential ()Vz vs. z is shown in the figure below. Which of the following statements about the z -component of the electric field zE is true? a) 0for 3m 0and 0for0 3mzzEzEz<−<< < <<. b) 0for 3m 0and 0for0 3mzzEzEz<−<< > <<. c) 0for 3m 0and 0for0 3mzzEzEz>−<< < <<. d) 0for 3m 0and 0for0 3mzzEzEz>−<< > <<. e) None of the above because zE cannot be determined from information in the graph for the regions 3 m < 0 and 0 3 mzz−< <<.8 Question 5 (5 points) Careful measurements reveal an electric field 323ˆ1;()0;arrRrRrrR⎧⎛⎞−≤⎪⎜⎟=⎨⎝⎠⎪≥⎩rErr where a and R are constants. Which of the following best describes the charge distribution giving rise to this electric field? a) A negative point charge at the origin with charge 04qaπε= and a uniformly positive charged spherical shell of radius R with surface charge density 2/4qRσπ=− . b) A positive point charge at the origin with charge 04qaπε= and a uniformly negative charged spherical shell of radius R with surface charge density 2/4qRσπ=− . c) A positive point charge at the origin with charge 04qaπε= and a uniformly negative charged sphere of radius R with charge density 3/(4 /3)qRρπ=− . d) A negative point charge at the origin with charge 04qaπε−=− and a uniformly positive charged sphere of radius R with charge density 3/(4 /3)qRρπ= . e) Impossible to determine from the given information.9 Problem 2 (25 points) NOTE: YOU MUST SHOW WORK in order to get any credit for this problem. Make it clear to us that you understand what you are doing (use a few words!) . Four charged point-like objects, two of charge +q and two of charge –q, are arranged on the vertices of a square with sides of length 2a, as shown in the sketch. a) What is the electric field at point O, which is at the center of the square? Indicate the direction and the magnitude.10 b) What is the electric potential V at point O, the center of the square, taking the potential at infinity to be zero?11 c) Sketch on the figure below one path leading from infinity to the origin at O where the integral Od∞⋅∫Esrr is trivial to do by inspection. Does your answer here agree with your result in b)?12 Problem 3 (25 points) NOTE: YOU MUST SHOW WORK in order to get any credit for this problem. Make it clear to us that you understand what you are doing (use a few words!) Three infinite sheets of charge are located at xd=− , 0x=, and xd=, as shown in the sketch. The sheet at 0x = has a charge per unit area of 2σ, and the other two sheets have charge per unit area of σ− . a) What is
View Full Document
Unlocking...