1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2013 Exam One Equation Sheet !F12= keq1q2r122ˆr12, force by 1 on 2 F = q!E !E(!r) = keqr2ˆr = keqr3!r !E(!r) = keqi!r !"ri3(!r !"ri)i=1N" !E(!r) = kedqr2source!ˆr = ked"q (!r #"!r )!r #"!r3source! !E ! d!Aclosed surface"""=qenc#o , d!A =ˆnoutdA !VAB= VB" VA= "!E #d!sAB$ V (r) ! V (") = keqr V (!r) ! V (") = keqi!r !"rii=1N# V (!r) ! V (") = ked#q!r !#!rsource$ !U = q!V q!V + !K = 0 Ustored= keqiqjrijall pairs!; U (") = 0 Electric Dipole !p = qi"rii=1N! V (!r) ! V (") = ke!p #ˆrr2 !! =!p "!E Constants ke=14!"0= 9 # 109N $ m2$ C%2 Circumferences, Areas, Volumes 1) The area of a circle of radius r is !r2. Its circumference is 2!r. 2) The surface area of a sphere of radius r is 4!r2. Its volume is (4 / 3)!r3. 3) The area of the sides of a cylinder of radius r and height h is 2!rh. Its volume is !r2h.2 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2013 Exam 1 Practice Problems Part I: Short Questions and Concept Questions Problem 1: Spark Plug Pictured at right is a typical spark plug (for scale, the thread diameter is about 10 mm). About what voltage does your car ignition system need to generate to make a spark, if the breakdown field in a gas/air mixture is about 10 times higher than in air? For those of you unfamiliar with spark plugs, the spark is generated in the gap at the left of the top picture (top of the bottom picture). The white on the right is a ceramic which acts as an insulator between the high voltage center and the grounded outer (threaded) part. Problem 2 Consider the three charges Q+ , 2Q+, and Q!, and a mathematical spherical surface (it does not physically exist) as shown in the figure below. What is the net electric flux on the spherical surface between the charges? Briefly explain your answer.3 Problem 3: A pyramid has a square base of side a, and four faces which are equilateral triangles. A charge Q is placed on the center of the base of the pyramid. What is the net flux of electric field emerging from one of the triangular faces of the pyramid? 1. 0 2. Q8!0 3. Qa22!0 4. Q2!0 5. Undetermined: we must know whether Q is infinitesimally above or below the plane? Problem 4: The work done by an external agent in moving a positively charged object that starts from rest at infinity and ends at rest at the point P midway between two charges of magnitude +Q and !2Q, 1. is positive. 2. is negative. 3. is zero. 4. cannot be determined – not enough info is given.4 Problem 5 Four charges of equal magnitude (two positive and two negative) are placed on the corners of a square as pictured at left (positive charges at positions A & B, negative at positions C & D). You decide that you want to swap charges B & C so that the charges of like sign will be on the same diagonal rather than on the same side of the square. In order to do this, you must do… 1. … positive work 2. … negative work 3. … no net work 4. … an indeterminate amount of work, as it depends on exactly how you choose to move the charges. Problem 6 Two opposite charges are placed on a line as shown in the figure below. The charge on the right is three times the magnitude of the charge on the left. Besides infinite, where else can electric field possibly be zero? 1. between the two charges. 2. to the right of the charge on the right. 3. to the left of the charge on the left. 4. the electric field is nowhere zero.5 Part II Analytic Problems Problem 1 Non-uniformly charged sphere A sphere of radius R has a charge density !=!0(r / R) where!0 is a constant and r is the distance from the center of the sphere. a) What is the total charge inside the sphere? b) Find the electric field everywhere (both inside and outside the sphere). Problem 2 Electric field and force A positively charged wire is bent into a semicircle of radius R, as shown in the figure below. The total charge on the semicircle is Q. However, the charge per unit length along the semicircle is non-uniform and given by 0cos! ! "=. a) What is the relationship between 0!, R and Q? b) If a particle with a charge q is placed at the origin, what is the total force on the particle? Show all your work including setting up and integrating any necessary integrals.6 Problem 3: Charged slab and sheets An infinite slab of charge carrying a uniform volume charge density ! has its boundaries located at x = -2 meters and x = +2 meters. It is infinite in the y direction and in the z direction (out of the page). Two similarly infinite charge sheets (zero thickness) are located at x = -6 meters and x = +6 meters, with uniform surface charge densities !1 and !2 respectively. In the accessible regions you’ve measured the electric field to be: !E(x) =!0 ; x < !6 m(10 N "C!1)ˆi ; ! 6 m < x < ! 2 m(!10 N "C!1)ˆi ; 2 m < x < 6 m!0 ; x > 6 m#$%%&%% (a) What is the charge density ! of the slab? (b) Find the two surface charge densities !1 and !2 of the left and right charged sheets. Problem 4 Concentric charged cylinders A very long uniformly charged solid cylinder (length L and radius a) carrying a positive charge +q is surrounded by a thin uniformly charged cylindrical shell (length L and radius b) with negative charge !q, as shown in the figure. You may ignore edge effects. Find a vector expression for the electric field in each of the regions (i) r < a, (ii) a r b< <, and (iii) r > b.Problem 5 Electric field and potential of charged objects Two charges +q and –q lie along the y-axis and are separated by a distance 2d as shown in the figure. (a) Calculate the total electric field E! at position A, a distance a from the y-axis. Indicate its direction on the sketch (draw an arrow). (b) Calculate the total electric field E! at position B, a distance b from the x-axis. Indicate its direction on the sketch (draw an arrow). (c) Find the electric potential V at position A with the electric potential zero at infinity. (d) Find the electric potential V at position B. (e) A positively charged dust particle with mass m and charge q+ is released from rest at point
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