MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2012 Exam One Equation Sheet Please Remove this Tear Sheet from Your Exam ! qq F12 = ke 1 2 2 r̂12 , force by 1 on 2 r ! 12 F = qE ! ! q q! E(r) = ke 2 r̂ = ke 3 r r r N ! ! q ! " E(r) = ke " ! i" 3 (r ! ri ) i=1 r ! r i ! ! ! ! dq dq"(r # r " ) E(r) = ke ! 2 r̂ = ke ! ! ! 3 r # r" source r source ! ! q E ! d A = enc #o closed surface " "" ! dA points from inside to outside B ! ! !V AB = VB " V A = " $ E # d s A V (r) ! V (") = ke q r N q ! V (r) ! V (") = ke # ! i " i=1 r ! ri ! V (r) ! V (") = ke $ source dq# ! ! r ! r# !U = q!V q!V + !K = 0 U stored = ke ! all pairs qi q j rij ; U (") = 0 Electric Dipole ! N " p = ! qi ri i=1 ! p # r̂ ! V (r) ! V (") = ke 2 r Gradient ! ! E = !"V "V "V "V Ey = ! Ex = ! Ez = ! "y "x "z "V Er = ! for spherical and cylindrical "r symmetry 1 #V E! = " for cylindrical symmetry r #! Constants ke = 1 = 9 # 109 N $ m 2 $ C%2 4!" 0 Circumferences, Areas, Volumes 1) The area of a circle of radius r is ! r 2 . Its circumference is 2! r . 2) The surface area of a sphere of radius r is 4! r 2 . Its volume is (4 / 3)! r 3 . 3) The area of the sides of a cylinder of radius r and height h is 2! rh . Its volume is ! r 2 h . 1 2 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2012 Exam One Your Name ______________________________________________ Your Table and Group (e.g. 10A): ______________ Your Section: 32-082 ____L01 Figueroa MW 10-12 am ____L05 Gore TR 11-1 pm ____L02 Tegmark MW 12 -2 pm ____L06 Rajagopal TR 1-3 pm ____L03 Soljacic MW 2 -4 pm ____L07 Belcher TR 3-5 pm ____L04 Dourmashkin TR 9 -11 am Your Section: 26-152 ____L08 Stevens MW 10-12 pm ____L09 Chen MW 2-4 pm Score Grader Problem 1 (25 points) Problem 2 (25 points) Problem 3 (25 points) Problem 4 (25 points) TOTAL 3 Problem 1 (25 points) In this problem you are asked to answer 5 questions, each worth 5 points. Question 1 (5 points) Two small objects each with a charge of +Q exert a force of ! magnitude F ! F on each other. We replace one of the objects with another whose charge is +4Q , and we move the +Q and +4Q charges to be 3 times as far apart as they were: ! What is the magnitude of the force F1 ! F1 on the +4Q charge above? a) F1 = F / 9 b) F1 = F / 3 c) F1 = 4F / 9 d) F1 = 4F / 3 e) None of the above. The correct answer is ________________________. 4 Question 2 (5 points) Three infinite sheets of charge lying in the yz -plane are shown in the sketch. The sheet on the right at x = d is positively charged with charge per unit area of +! , the sheet in the middle at x = 0 is positively charged with charge per unit area of +4! , and the sheet on the left at x = !d is negatively charged with charge per unit area of !" . What is the electric field at the point P in the region x > d ? a) zero. ! ! b) E = î . "0 ! " c) E = ! î . #0 ! 2! d) E = î . "0 ! 2" e) E = ! î . #0 ! 4! f) E = î . "0 ! 4" g) E = ! î . #0 h) None of the above. The correct answer is ________________________. 5 Question 3 (5 points) We have two electric dipoles. Each dipole consists of two equal and opposite point charges at the ends of an insulating rod of length d . The dipoles sit along the x -axis a distance r apart, oriented at right angles to each other as shown below. Their separation r >> d . You may take d to be small but not zero. The dipole ON RIGHT a) will feel a force to the left and a torque trying to make it rotate clockwise. b) will feel a force to the left and a torque trying to make it rotate counterclockwise. c) will feel a force to the left and no torque. d) will feel a force to the right and a torque trying to make it rotate clockwise. e) will feel a force to the right and a torque trying to make it rotate counterclockwise. f) will feel a force to the right and no torque. g) will feel no force and a torque trying to make it rotate clockwise. h) will feel no force and a torque trying to make it rotate counterclockwise. i) will feel no force and no torque. The correct answer is ________________________. 6 Question 4 (5 points) A thin rod extends along the x -axis from x = !l / 2 to x = l / 2 . The rod carries a uniformly distributed positive charge +Q . Consider a point P lying in the z = 0 plane with coordinates (x, y,0) . Which of the following expressions describes the electric potential difference V (P) ! V (") , between infinity and the point P ? a) keQ x # = l / 2 ((x ! x # ) î + yĵ) dx # . V (P) ! V (") = l x # =$! l / 2 ((x ! x # )2 + y 2 )3/ 2 b) V (P) ! V (") = keQ x # = l / 2 ((x ! x # ) î + yĵ) dx # . l x # =$! l / 2 ((x ! x # )2 + y 2 )1/ 2 c) V (P) ! V (") = keQ x # = l / 2 (x ! x # ) dx # . $ l x # = ! l / 2 ((x ! x # )2 + y 2 )3/ 2 d) keQ x # = l / 2 (x ! x # ) dx # . V (P) ! V (") = $ l x # = ! l / 2 ((x ! x # )2 + y 2 )1/ 2 e) keQ x # = l / 2 dx # . V (P) ! V (") = $ l x # = ! l / 2 ((x ! x # )2 + y 2 )3/ 2 f) keQ x # = l / 2 dx # . V (P) ! V (") = $ l x # = ! l / 2 ((x ! x # )2 + y 2 )1/ 2 g) None of the above. The correct answer is ________________________. 7 Question 5 (5 points) ! The area vector dA at each point on a closed surface (i.e., a surface that surrounds a volume) are always chosen to point out of the enclosed volume. A closed imaginary surface is called a Gaussian surface. The Gaussian surface below is a cylinder. A positive charge is located on the cylinder axis above the Gaussian cylinder, as shown on the figure to the right. Which statement is correct about the flux "" ! ! E ! d A through surface B and surface B through the entire closed surface A + B + C ? a) The flux through surface B is positive and through the entire closed surface it is positive. b) The flux through B is positive and through the entire surface it is negative. c) The flux through B is positive and through the entire surface it is zero. d) The flux through B is negative and through the entire surface it is zero. e) The flux through B is negative and through the entire surface it is negative. f) The flux through B is negative and through the entire surface it is positive. The correct answer is ________________________. 8 Problem 2 (25 points) NOTE: YOU MUST SHOW WORK in order to get any credit for this problem. Three charges equal to +Q , !2Q , and +Q are placed on the vertices of an equilateral triangle with sides a, as shown in the sketch. The point P is at the origin. Note that sin(60! ) = 3 / 2 and cos(60! ) = 1 / 2 . ! a) What is the x-component of the electric field E at point P ? ! b) What is the y-component of the electric field E at point P ? 9 c) What is the electric potential V at point P, assuming that V (!) = 0 ? d) Suppose you now move a point charge with charge +5Q from infinity to point P in the above problem. How much work does it take you to do this? 10 e) What is the dipole moment vector of the original set of three charges? f) Suppose the original three charges were glued to a wooden equilateral triangle which was free to rotate ! about its center, and then that triangle was placed in a uniform electric field E = Eo ŷ , where Eo > 0 . Will the triangle rotate counterclockwise or clockwise? 11 Problem 3 (25 points) NOTE: YOU MUST SHOW WORK in order to get any credit for this problem. A very thin disk of radius R is positively charged with uniform charge per unit area +! . Consider a point P lying a distance x > 0 from the center of the disk along the axis of symmetry of the disk. Integration formulas you may find useful: x2 " dr ! r! = 2 +C 2 3/ 2 (x + r ! ) (x + r ! 2 )1/ 2 " (x 2 " (x r !dr ! 1 = # +C 2 + r ! 2 )3/ 2 (x 2 + r ! 2 )1/ 2 " (x dr ! = ln r ! + (x 2 + r ! 2 )1/ 2 + C 2 2 1/ 2 + r! ) 2 r !dr ! = (x 2 + r ! 2 )1/ 2 + C + r ! 2 )1/ 2 a) What is the electric potential difference V (P) ! V (") ? Hint: you may find the integration area element da! = 2" r !dr ! useful. 12 b) What is the direction and magnitude of the electric field at the point P ? 13 c) Find an expression for the direction and magnitude of the electric field when x << R . d) How does your result in part c) compare to the electric field of a uniformly charged infinite plane with charge per area +! ? Explain your reasoning. 14 15 Problem 4 (25 points) NOTE: YOU MUST SHOW WORK in order to get any credit for this problem. A very long solid cylinder of radius R1 carries a non-uniform volume charge density ! = Cr ; 0 < r " R1 , where C is a positive constant with units [C ! m "4 ] . It is surrounded by a very thin negatively charged cylindrical shell of radius R2 with uniform charge per area ! < 0 as shown in the figure. The electric field is zero for r > R2 . You may assume the cylinders are infinite. a) Find an expression for the charge per area ! on the outer shell in terms of C , R1 , and R2 as needed. 16 b) Find a vector expression for the electric field in the regions (i) 0 < r ! R1 and (ii) R1 < r < R2 . For each region clearly show your choice of Gaussian surface. Express your answers in terms of r , C , R1 , and R2 as needed. 17 c) A positively charged object of charge Q > 0 and mass M is released from rest at r = R1 . What is the speed of the object when it reaches r = R2 ? Express your answers in terms of r , C , Q , M , R1 , and R2 as needed. 18 19