Finish EM Ch.5: Magnetostatics Methods of Math. Physics, Thus. 10 March 2011, E.J. Zita Lorentz Force Amperes Law Maxwells equations (d/dt=0) Preview: Full Maxwell Equations (d/dt0) Magnetic vector potential A || Electrostatic potential V BC Multipole expansion Lorentz Force p.220 #10 (a) Lorentz Force p.220 #12 Amperes Law p.231 #13-16 You choose Amperes Law p.231 #14 Amperes Law p.231 #15 Amperes Law p.231 #16 Four laws of electromagnetism Electrodynamics Changing E(t) make B(x) Changing B(t) make E(x) Wave equations for E and B Electromagnetic waves Motors and generators Dynamic Sun Full Maxwells equations Maxwells Eqns with magnetic monopole Lorentz Force: Continuity equation: Vector Fields: Helmholtz Theorem For some vector field F, if the divergence = D = F, and the curl = C = F then (a) what do you know about C ? and (b) Can you find F? (a) C = 0, because ( F) 0 (b) We can find F iff we have boundary conditions, and require the field to vanish at infinity. Helmholtz: A vector field is uniquely determined by its div and curl (with BC) Vector Fields: Potentials.1 For some vector field F = - V, find F : (hint: look at identities inside front cover) F = 0 F = - V Curl-free fields can be written as the gradient of a scalar potential (physically, these are conservative fields, e.g. gravity or electrostatic). Theorem 1 examples The second part of each question illustrates Theorem 2, which follows Vector Fields: Potentials.2 For some vector field F = A, find F : F = 0 F = A Divergence-free fields can be written as the curl of a vector potential (physically, these have closed field lines, e.g. magnetic). Optional Proof of Thm.2 Practice with vector field theorems Magnetic vector potential Electrostatic scalar potential V Electric dipole expansion of an arbitrary charge distribution (r) (p.148) P n (cos ) are the Legendre Polynomials Multipole expansion Magnetic field of a dipole B= A, where Spring quarter in E&M Dynamics! dE/dt and dB/dt Bohm-Aharanov effect (A >B) Faradays law, motors, generators Magnetic monopole: Blas Cabreras measurement Conservation laws, EM energy EM waves Relativistic phenomena EM field tensor Spring quarter in MMP Tuesday: Boas and DiffEq Hamiltonians / Lagrangians Thursday: Quantum Mechanics Friday: Electromagnetism and Research