Finish EM Ch.5: Magnetostatics Methods of Math. Physics, Thus. 10 March 2011, E.J. Zita • Lorentz Force • Ampere’s Law • Maxwell’s equations (d/dt=0) • Preview: Full Maxwell Equations (d/dt≠0) • Magnetic vector potential A || Electrostatic potential V • BC • Multipole expansion Lorentz Force F qv B IL B p.220 #10 (a) Lorentz Force F qv B IL B p.220 #12 Ampere’s Law boundary B dl 0 I , I J d a p.231 #13-16 You choose… Ampere’s Law boundary p.231 #14 B dl 0 I , I J d a Ampere’s Law boundary p.231 #15 B dl 0 I , I J d a Ampere’s Law boundary p.231 #16 B dl 0 I , I J d a Four laws of electromagnetism Electric Magnetic Gauss' Law Gauss' Law Charges → E fields No magnetic monopoles Ampere's Law Faraday's Law Currents → B fields 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 Maxwell’s equations Electric Magnetic Gauss' Law Gauss' Law Charges make E fields What if there were magnetic monopoles? Ampere's Law Faraday's Law Currents make B fields (so does changing E) Changing B make E fields Maxwell’s 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 Magnetic vector potential Electrostatic scalar potential V Electric dipole expansion of an arbitrary charge distribution r(r) Pn(cosq) are the Legendre Polynomials (p.148) Multipole expansion Magnetic field of a dipole B=A, where ˆ m rˆ m sinq φ Spring quarter in E&M • • • • • • • • Dynamics! dE/dt and dB/dt Bohm-Aharanov effect (A >B) Faraday’s law, motors, generators Magnetic monopole: Blas Cabrera’s 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

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