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:

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Presentation transcript:

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 p.220 #10 (a)

Lorentz Force p.220 #12

Ampere’s Law p.231 #13-16 You choose…

Ampere’s Law p.231 #14

Ampere’s Law p.231 #15

Ampere’s 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 Maxwell’s equations

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

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) 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