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emhw11 - Electromagnetic Theory I Problem Set 11 Due 14...

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Electromagnetic Theory I Problem Set 11 Due: 14 November 2011 41. In studying a quantum particle in the presence of an electromagnetic field we used the lagrangian L = 1 2 m ˙ r 2 + q ˙ r · A ( r , t ) ( r , t ) (1) When there is time dependence we showed near the beginning of the course that the relation between fields and potentials is B ( r , t ) = ∇ × A ( r , t ) , E ( r , t ) = −∇ φ ( r , t ) A ∂t ( r , t ) (2) a) Show that Lagrange’s equations of motion with this Lagrangian imply Newton’s equation with the Lorentz force on the right side. F = q E + q v × B (3) b) Derive the hamiltonian for this system. c) Repeat the discussion of parts a) and b) for the relativistic Lagrangian obtained by replacing the nonrelativistic kinetic energy term in L as follows 1 2 m ˙ r 2 mc 2 parenleftBigg 1 radicalbigg 1 ˙ r 2 c 2 parenrightBigg (4) 42. J, Problem 5.25. 43. J, Problem 5.32 44. Motional Emf from a Spinning Magnet (Revision of J, 6.4). Let the magnet be a uniformly magnetized sphere of radius R and magnetization M = M ˆ z , which is also a conductor with finite resistivity. Set the magnet spinning with angular speed
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