Unformatted text preview: Princeton University--Physics 501 Problem Set 6 Due: Nov. 22, 2011 1) J6.14 (this problem is rather lengthy and will be weighted accordingly). An ideal circular parallel plate capacitor of radius a and plate separation d a is connected to a current source by axial leads. The current in the wire is I(t) = I0 cos t. a) Calculate the electric and magnetic fields between the plates to second order in powers of the frequency (or wave number), neglecting the effects of fringing fields. b) Calculate the volume integrals of we and wm that enter the definition of the reactance X, to second order in . Show that in terms of the input current, Ii = -iQ, where Q is the total charge on one plate, these energies are 1 |Ii | d we d x = 4 0 2 a2
3 2 and 0 |Ii | d wm d x = 4 8
3 2 1+ 2 a2 12c2 Note: The reactance is given by X= 4 |Ii |
2 V (wm - we ) d3 x c) Show that the equivalent series circuit has C o a2 /a, L 0 d/8, and that the resonant frequency is res 2 2c/a. Compare with the first root of J0 (x). 2) Some experimental searches for magnetic monopoles rely on the discrete (although not necessarily abrupt) change in current flowing in a superconducting loop when it is traversed by a Dirac monopole. a) Calculate the minimum possible change for a circular loop of radius a traversed by a Dirac monopole. Hints: Use the integral form of Faraday's law with JM = 0. Write JM as a 3-D function involving position and time. b) Calculate I(t) in the loop assuming that the monopole starts at z = - and passes through the center of the loop at t = 0 (the z axis is perpendicular to the loop and passes through its center). You may state your answers to (a) and (b) in terms of the L, the loop's inductance. ...
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