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Electromagnetism Ch.7 Methods of Math. Physics, Friday 8 April 2011, EJZ

Electromagnetism Ch.7 Methods of Math. Physics, Friday 8 April 2011, EJZ. Inductors and inductance Waves and wave equations Electromagnetism & Maxwell’s eqns Derive EM wave equation and speed of light. Faraday’s Law. Break time…. Waves. Wave equation.

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Electromagnetism Ch.7 Methods of Math. Physics, Friday 8 April 2011, EJZ

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  1. Electromagnetism Ch.7 Methods of Math. Physics, Friday 8 April 2011, EJZ • Inductors and inductance • Waves and wave equations • Electromagnetism & Maxwell’s eqns • Derive EM wave equation and speed of light

  2. Faraday’s Law

  3. Break time…

  4. Waves

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

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

  7. 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

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

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

  10. Partial Maxwell’s equations

  11. 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?

  12. Faraday + Ampere

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

  14. Speed of Maxwellian waves? Faraday: wB0 = k E0 Ampere: m0e0wE0=kB0 Eliminate B0/E0 and solve for v=w/k e0= 8.85 x 10-12 C2 N/m2 m0 = 4 px10-7 Tm/A

  15. 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 )

  16. Full Maxwell equations inintegral form

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

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

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

  20. Finish integral to differential form…

  21. Finish integral to differential form…

  22. Maxwell eqns in differential form

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