MASSACHUSETTSINSTITUTEOFTECHNOLOGY …

1 MASSACHUSETTS INSTITUTE OF TECHNOLOGYP hysics DepartmentPhysics : Electromagnetism IIDecember 19, 2012 Prof. Alan GuthFINAL EXAMR eformatted to Remove Blank PagesTHE FORMULA SHEETS ARE AT THE END OF THE FINAL EXAM, FALL 2012p. 2 PROBLEM 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.

2 Write the total magnetic energy as1 Imag 22, and find an expression forImag.(c)(7 points)Calculate the angular momentum stored in the fields. Verify that themagnitude of the angular momentum coincides withImag . FINAL EXAM, FALL 2012p. 3 PROBLEM 2: CONSERVATION OF MOMENTUM IN THE PRESENCEOF MAGNETIC MONOPOLES(20 points)We have learned how to show, for the standard version of Maxwell s equations in-volving electric charge density eand current density Je, that momentum is detail, we learned how to show that for any volumeVbounded by a surfaceS,d 1 Pmech,i+ S3didxtc2V = Tijdaj,( )SwherePmech,iis thei th component of the total mechanical momentum inV,Siis thei thcomponent of the Poynting vector, andTijis the Maxwell stress tensor. Explicit formulasforSiandTijare given on p. 13 of the formula sheets. The goal of this problem is toextend the result to include the possibility that magnetic charges also exist, as describedby the extended version of Maxwell s equations shown in the formula sheets on p.

3 11. Tosimplify the algebra, however, we will consider the case whereONLY magnetic chargedensity mand magnetic current density Jmare present, but e= Je=0.(a)(5 points)If the magnetic charges are discrete, then the rate of change of the totalmechanical momentum in the region is just the sum of the forces acting on themagnetic monopoles:dPmech,i= (n)Fqn)i= (m 1 B v E ,dtc2nn ( )iwhere(n)Fiis thei th component of the force on then th monopole, and(n)qmis themagnetic charge of then th monopole. To demonstrate that Eq. ( ) holds in thepresence of monopoles, we need to write dPmech,i/dtin the form of a volume integralover a force densityfi,dPmech,i=dt fid3x,( )Vwhich we do by thinking of each infinitesimal volume element as a magnetic monopoleof magnetic charge d3 mx. Write an expression forfiin terms of m,Jm,E, that vshould not appear in your answer.(b)(5 points)Use Maxwell s equations to write the left-hand side of Eq. ( ), with yourresult from part (a), entirely in terms of Eand B.

4 (c)(10 points)Complete the proof, showing that the left hand side of Eq. ( ) canbe written as a surface integral. Does the Maxwell stress tensorTijrequire anymodification? (Hint:We recommend that you use index notation, but you mayuse vector notation if you prefer. The vector identities from the inside cover of thetextbook have been added to the formula sheet.) FINAL EXAM, FALL 2012p. 4 PROBLEM 3: THE ELECTRIC FIELD OF A CHARGED PARTICLEMOVING AT A CONSTANT VELOCITY(15 points)Suppose that a particle of chargeqis moving at speedv0along thex-axis, followingthe trajectory r(t)=v0t x. ( )From differentiating the Li enard-Wiechert potentials, we have learned that the electricfield of a point charge is given in general byqr E( r,t)=| r p| (c2 v2) +( r r4( ())pu p) "3( u ap),( )0 u r rp where the notation is defined in the formula sheets on p. 16. Suppose that an observer islocated on theyaxis at (0,y0,0), and measures the electric field at timet=0,exactlywhen the particle crosses the origin.

5 Note that all measurements are made in the samecoordinate system that was used in Eq. ( ) to describe the trajectory.(a)(5 points)Find the value of the retarded timetrappropriate to this measurement,and also find the values of r rpand u. (Remember that vector quantities must bedescribed as vectors, not num bers.)(b)(5 points)Use Eq. ( ) to find the value of Ethat the observer will measure. (Someof you may know the answer to this question, but to get full credit you must showhow to obtain it from Eq. ( ).)(c)(5 points)Find also the magnetic field Bthat the observer will measure at the 4: A ROTATING MAGNETIC DIPOLE(20 points)A rotatingmagneticdipole can be thought of as the su-perposition of twooscillatingmagnetic dipoles, one along thexaxis, and the other along theyaxis (see figure at the right),with the latter out of phase by 90 :m =m0[cos( t) x+sin( t) y].( )(a)(10 points)Suppose that an observer is located along thezaxis, at (0,0,z0).

6 Findthe electric and magnetic fields that she measures, as a function of time. (Rememberthat the electric and magnetic fields are vectors, not numbers.)(b)(5 points)Find the Poynting vector measured by the observer in part (a), as afunction of time, and also find the time-averaged value of the Poynting vector.(c)(5 points)Now suppose that the observer was located along thexaxis, at (x0,0,0).Find the electric and magnetic fields that she would measure at this point, as afunction of 5: SHORT QUESTIONS(25 points)(a)(5 points)Suppose I invented a functionVAlan( r,t), and defined the Alan gauge asthe gauge for whichV( r,t)=VAlan( r,t) for all rand allt. Is it always possible towrite the electromagnetic potentials in Alan gauge? If so, give a recipe for findingthe gauge function that takes one to Alan gauge, where the gauge transformationis written as A = A+ ,V =V .( ) tIf not, give an argument why not.(b)(5 points)Suppose I invented a vector function AAlan( r,t), and defined thevector-Alan gauge as the gauge for which A( r,t)=AAlan( r,t) for all rand allt.

7 Is it always possible to write the electromagnetic potentials in vector-Alan gauge?If so, give a recipe for finding the gauge function that takes one to Alan gauge,and if not, give an argument why.(c)(7 points)The potential energy function for an ideal (static) electric quadrupole isgiven by1r iV( r)=jQij,( )8 "0r3whereQijis the quadrupole moment tensor, which is traceless and symmetric. UsePoisson s equation to find the charge density ( r) for an ideal quadrupole. Youranswer should be expressed in terms ofQij, and some kind of derivative of a deltafunction. (Hint:Start by using the identity 1 ij 3 rir j4 i j=+ 33ij ( r)( )rr3from the formula sheet to expressr ir j/r3in terms of other quantities.)(d)(8 points)Accelerating charges normally radiate, but that is not always the a sphere with a total chargeQuniformly spread on its surface, with aradiusR(t) which pulsates:R(t)=R0+R1sin t ,( )where 0<R1<R0,soR(t) is always positive.

8 Show that this system does notradiate, even though the charges are accelerating. (Hint:take advantage of thesymmetry of the system to write down an exact solution to Maxwell s equations.)r MIT Electromagnetism IIFall 2012 For information about citing these materials or our Terms of Use, visit.