1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2010 Exam Two Equation Sheet Force Law: ()ext extq=+×FEvBrr rr Source Equations: 231ˆ44oodq dqdrrπε πε==Errrr 23ˆ44ooId Iddrrμμππ××==sr srBrrrr ˆrrr=rpoints from source to field point Current Density and Current: open surfaceId=⋅∫∫Jarr Gauss’s Law: insideclosedsurface/odQε⋅=∫∫EArr Gauss’s Law for Magnetism: 0closedsurfaced⋅=∫∫BArr Electric Potential Difference: bbaaVVV dΔ= − ≡− ⋅∫Esrr V=−∇Err Potential Energy: UqVΔ=Δ Capacitance: VQCΔ= 221122QUCVC==Δ Capacitors in Parallel: 12eqCCC=++⋅⋅⋅ Capacitors in Series: 12111eqCCC=++⋅⋅⋅ Ohm’s Law: VIRΔ= where is the conductivityccσσ=JErr where is the resistivityrrρρ=EJrr Resistors in Parallel: 12111eqRRR= + +⋅⋅⋅ Resistors in Series: 12eqRRR=++⋅⋅⋅ Power Loss in Resistor: 22Joule/PIVIRVR=Δ = =Δ Magnetic Dipole: = IAμnr) Torque on Magnetic Dipole: =×τμBrrr Constants: 92-201/4 9 10 N m Cekπε==× ⋅ ⋅ 7-10/4 10 T m Aμπ−=⋅⋅23 8.02 Exam Two 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 TTh 9 am ____L06 TTh 11 am ____L07 TTh 1 pm ____L08 TTh 3 pm Your 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) Five Concept Questions. Please circle your answers. Question 1: (5 points) Circle the correct answer. Consider a simple parallel-plate capacitor whose plates are given equal and opposite charges and are separated by a distance d. The capacitor is connected to a battery. Suppose the plates are pushed together until they are separated by a distance D = d/2. How does the final electrostatic energy stored in the capacitor compare to the initial energy? a) Final is half the initial. b) Final is one fourth the initial. c) Final is twice than initial. d) Final is four times the initial. e) They are the same.5 Question 2: (5 points) Circle the correct answer. A conducting wire is attached to an initially charged spherical conducting shell of radius 2a . The other end of the wire is attached to the outer surface of a neutral conducting spherical shell of radius a that is located a very large distance away (at infinity). When electrostatic equilibrium is reached, the charge on the shell of radius 2a is equal to a) one fourth the charge on the shell of radius a . b) half the charge on the shell of radius a . c) twice the charge on the shell of radius a . d) four time the charge on the shell of radius a . e) None of the above.6 Question 3: (5 points) Circle the correct answer. What is the correct order for the total power dissipated in the following circuits, from least to greatest? Assume all bulbs and all batteries are identical. Ignore any internal resistance of the batteries. a) A < B = C < D < E b) D < C < B = E < A c) D < B < E < A < C d) A = B < D < C < E e) B < A < C = D < E7 Question 4: (5 points) Circle the correct answer. Consider a triangular loop of wire with sidesa and b . The loop carries a current I in the direction shown, and is placed in a uniform magnetic field that has magnitude B and points in the same direction as the current in side OM of the loop. At the moment shown in the figure the torque on the current loop a) points in the ˆ− i -direction and has magnitude /2IabB . b) points in the ˆ+ i-direction and has magnitude /2IabB . c) points in the ˆ−j-direction and has magnitude /2IabB . d) points in the ˆ+j-direction and has magnitude /2IabB . e) points in the ˆ− i-direction and has magnitude IabB . f) points in the ˆ+ i -direction and has magnitude IabB . g) points in the ˆ−j-direction and has magnitude IabB . h) points in the ˆ+j-direction and has magnitude IabB . i) None of the above.8 Question 5: (5 points) Circle the correct answer. A particle with charge q and velocity vr enters through the hole in screen 1 and passes through a region with non-zero electric and magnetic fields (see sketch). If 0q < and the magnitude of the electric field E is greater than the product of the magnitude of the initial velocity v and the magnitude of the magnetic fieldB, that is EvB> , then the force on the particle a) is zero and the particle will move in a straight line and pass through the hole on screen 2. b) is constant and the particle will follow a parabolic trajectory hitting the screen 2 above the hole. c) is constant and the particle will follow a parabolic trajectory hitting screen 2 below the hole. d) is constant in magnitude but changes direction and the particle will follow a circular trajectory hitting the screen 2 above the hole. e) is constant in magnitude but changes direction and the particle will follow a circular trajectory hitting the screen 2 below the hole. f) changes magnitude and direction and the particle will follow a curved trajectory hitting the screen 2 above the hole. g) changes magnitude and direction and the particle will follow a curved trajectory hitting the screen 2 below the hole.910 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!) . Consider a spherical vacuum capacitor consisting of inner and outer thin conducting spherical shells with charge Q+ on the inner shell of radius a and charge Q− on the outer shell of radius b. You may neglect the thickness of each shell. a) What are the magnitude and direction of the electric field everywhere in space as a function of r , the distance from the center of the spherical conductors?11 b) What is the capacitance of this capacitor?12 c) Now consider the case that the dimension of the outer shell is doubled from b to 2b . Assuming that the charge on the shells is not changed, how does the stored potential energy change? That is, find an expression for after beforeUU UΔ≡− in terms of a , b , and 0ε.1314 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!) .
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