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Electromagnetism week 9 Physical Systems, Tuesday 6.Mar. 2007, EJZ

Electromagnetism week 9 Physical Systems, Tuesday 6.Mar. 2007, EJZ. Waves and wave equations Electromagnetism & Maxwell’s eqns Derive EM wave equation and speed of light Derive Max eqns in differential form Magnetic monopole  more symmetry Next quarter. Waves. Wave equation.

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Electromagnetism week 9 Physical Systems, Tuesday 6.Mar. 2007, EJZ

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  1. Electromagnetism week 9 Physical Systems, Tuesday 6.Mar. 2007, EJZ • Waves and wave equations • Electromagnetism & Maxwell’s eqns • Derive EM wave equation and speed of light • Derive Max eqns in differential form • Magnetic monopole  more symmetry • Next quarter

  2. Waves

  3. Wave equation 1. Differentiate dD/dt  d2D/dt2 2. Differentiate dD/dx  d2D/dx2 3. Find the speed from

  4. Gauss: E fields diverge from charges Lorentz force: E fields can move charges Causes and effects of E F = q E

  5. Ampere: B fields curl around currents Lorentz force: B fields can bend moving charges Causes and effects of B F = q v x B = IL x B

  6. Changing fields create new fields! Faraday: Changing magnetic flux induces circulating electric field Guess what a changing E field induces?

  7. Changing E field creates B field! Current piles charge onto capacitor Magnetic field doesn’t stop Changing electric flux • “displacement current” • magnetic circulation

  8. Partial Maxwell’s equations

  9. Maxwell eqns  electromagnetic waves Consider waves traveling in the x direction with frequency f= w/2p and wavelength l= 2p/k E(x,t)=E0 sin (kx-wt) and B(x,t)=B0 sin (kx-wt) Do these solve Faraday and Ampere’s laws?

  10. Faraday + Ampere

  11. Sub in: E=E0 sin (kx-wt) and B=B0 sin (kx-wt)

  12. Speed of Maxwellian waves? Faraday: wB0 = k E0 Ampere: m0e0wE0=kB0 Eliminate B0/E0 and solve for v=w/k e0 = 8.85 x 10-12C2 N/m2 m0= 4 p x 10-7 Tm/A

  13. Maxwell equations  Light E(x,t)=E0 sin (kx-wt) and B(x,t)=B0 sin (kx-wt) solve Faraday’s and Ampere’s laws. Electromagnetic waves in vacuum have speed c and energy/volume = 1/2 e0 E2 = B2 /(2m0 )

  14. Full Maxwell equations inintegral form

  15. Integral to differential form Gauss’ Law: apply Divergence Thm: and the Definition of charge density: to find the Differential form:

  16. Integral to differential form Ampere’s Law: apply Curl Thm: and the Definition of current density: to find the Differential form:

  17. Integral to differential form Faraday’s Law: apply Curl Thm: to find the Differential form:

  18. Finish integral to differential form…

  19. Finish integral to differential form…

  20. Maxwell eqns in differential form Notice the asymmetries – how can we make these symmetric by adding a magnetic monopole?

  21. If there were magnetic monopoles… where J = rv

  22. Next quarter: ElectroDYNAMICS, quantitatively, including Ohm’s law, Faraday’s law and induction, Maxwell equations Conservation laws, Energy and momentum Electromagnetic waves Potentials and fields Electrodynamics and relativity, field tensors Magnetism is a relativistic consequence of the Lorentz invariance of charge!

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