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Physics PHYS 354 Electricity and Magnetism II Problem Set #7 1. Coulomb and Lorentz Gauges a) Show that Maxwell’s (time-dependent form) equations imply 0 A , B A and E 4 t where is a scalar potential. b) Show that the gauge transformations , A A and t where is a scalar function, do not affect the gauge-independent values of E or B . c) d) Suppose that A and do not satisfy the Coulomb gauge condition. Show that the Coulomb gauge can be reached by selecting a suitable , which is the solution to the differential equation 2 A . Suppose that A and do not satisfy the Lorentz gauge condition, which is 1 A 2 0. c t Show that the Lorentz gauge can be reached by selecting a suitable , which is the solution to the differential equation 2 1 1 2 A 2 . c2 t 2 c t 2. Relativistic generalization of Ohm's Law In the rest frame of a medium one has J E where J is the current density, E is the electric field, and is the conductivity (measured in the rest frame). This is called Ohm's Law. a) Why is this equation not relativistic? b) Try setting 1 J KF u (Eq. 1) c where u is the 4-velocity of the medium. Find K such that in the rest frame u c,0 the above equation reduces to Ohm's Law. c) The above equation implies u J 0 (Eq. 2) Why? Show by going to the rest frame that Eq. 2 is not generally true. d) Fix up Eq. 1 by adding an extra term J M u J u F u 2 c Show that M 1 . e) If the medium has a velocity c with respect to an inertial frame, show that the current in that frame is given by J E B E (Eq. 3) where E , B , and are measured in this inertial frame, and show J0 1 J E (Eq. 4). 2 Hint: show Eq. 4 first, then use it to get Eq. 3. 2 f) (Harder) Assume the medium is uncharged in its rest frame 0 . Using the Lorentz transformation and Eq. 3 show that 3. c Moving Wire An infinitely long thin wire along the z axis moves with a speed v along the z axis and carries a charge per unit length of q, and so a current qv. a) b) c) d) 4. E and J E B . Calculate the charge density and current as seen by an observer in S' moving with a speed v along the z axis. Calculate the electric and magnetic fields seen by this observer. Transform these fields to find the fields in the original frame. Check that these agree with a direct calculation in the original frame S. Maxwell's Equations in a Dielectric Medium We want to construct Maxwell's equations in a uniformly moving dielectric medium. Assume i) ii) that G is constructed from D and H just as F is constructed from E and B and transforms like a tensor, that, therefore, Maxwell's equations are written , G 0 J x iii) and that in the rest frame of the medium 1 D E and H B . In general, the 4-velocity of the medium is u . The relation between G and F must be G KF L u u F u u F c2 where K and L are constants depending on and . Find K and L. 3 5. Relativistic Equations of Motion a) The relativistic equations of motion of a charged particle in a given electromagnetic field are m d 2 x dx qF 2 d d where d is the proper time interval d dt . Show that the force in a particular reference frame therefore reduces to the Lorentz force, dp F q E v B dt where p mv is the relativistic momentum. b) A particle travels with varying speed v t in a circle of fixed radius R, with equation r x 2 y 2 R and z 0 . The components of dv dt are then dv dt along the tangent and v 2 R towards the center. On the circular path, E is along the tangent and has a magnitude E and B is in the z direction and has a magnitude B. Write down the components of the equation of motion in the radial and tangential directions. You may find that it helps to write d d v v dv . dt dt dt c) Show that the equations in (b) can both be satisfied if E t R dBt dt where E and B are both functions of time alone. Show that this equation is a form a Faraday's law (from general physics) assuming that the average magnetic flux through the circular loop is twice that on the boundary. 4