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MASSACHUSETTS INSTITUTE OF TECHNOLOGYPhysics DepartmentPhysics : Electromagnetism IIDecember 19, 2012Prof. Alan GuthFINAL EXAMReformatted to Remove Blank PagesTHE FORMULA SHEETS ARE AT THE END OF THE FINAL EXAM, FALL 2012p. 2PROBLEM 1: ANGULAR MOMENTUM AND A ROTATING SHELL OFCHARGE(20 points)This is an abbreviated version of Problem 3 of Problem Set total chargeQis uniformly distributed over the surface of a sphere of sphere rotates about thezaxis with angular velocity . The magnetic field of thisconstruction has been calculated, and is known to be Bin=b0 z ifr<R B=b R3( ) Bout=02cos r +sin ifr>R,2r3where 02Qb0=.( )4 3R(a)(7 points)Consider the case where = 0. Calculate the Faraday induced electricfield at the surface of the sphere as a function of . Calculate also the torque thisfield produces on the sphere.(b)(6 points)Now assume that is constant. Calculate the energy stored in the mag-netic field. Show that two-thirds is inside the sphere and one-third is outside thesphere.

B out = 0 2cosθrˆ+sinθθ ifr>R, 2r3 where µ0 2Q b0 =. (1.2) 4π3R (a) (7 points) Consider the case where ω˙ = 0.Calculate the Faraday induced electric field at the surface of the sphere asa function of θ.Calculatealsothe torque this fieldproducesonthesphere.