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Chapter 34

Chapter 34. The Laws of Electromagnetism Maxwell’s Equations Displacement Current Electromagnetic Radiation. The Electromagnetic Spectrum. infra -red. ultra -violet. Radio waves. g -rays. m -wave. x-rays. The Equations of Electromagnetism (at this point …).

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Chapter 34

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  1. Chapter 34 The Laws of Electromagnetism Maxwell’s Equations Displacement Current Electromagnetic Radiation

  2. The Electromagnetic Spectrum infra -red ultra -violet Radio waves g-rays m-wave x-rays

  3. The Equations of Electromagnetism (at this point …) Gauss’ Law for Electrostatics Gauss’ Law for Magnetism Faraday’s Law of Induction Ampere’s Law

  4. The Equations of Electromagnetism ..monopole.. Gauss’s Laws 1 ? 2 ...there’s no magnetic monopole....!!

  5. The Equations of Electromagnetism .. if you change a magnetic field you induce an electric field......... Faraday’s Law 3 Ampere’s Law 4 .......is the reverse true..?

  6. E B ...lets take a look at charge flowing into a capacitor... ...when we derived Ampere’s Law we assumed constant current...

  7. E B ...lets take a look at charge flowing into a capacitor... ...when we derived Ampere’s Law we assumed constant current... E B .. if the loop encloses one plate of the capacitor..there is a problem … I = 0 Side view:(Surface is now like a bag:)

  8. E B Maxwell solved this problem by realizing that.... Inside the capacitor there must be an induced magnetic field... How?.

  9. E B x x x x x x x x x x x x Maxwell solved this problem by realizing that.... Inside the capacitor there must be an induced magnetic field... How?. Inside the capacitor there is a changing E  B A changing electric field induces a magnetic field E

  10. E B x x x x x x x x x x x x Maxwell solved this problem by realizing that.... Inside the capacitor there must be an induced magnetic field... How?. Inside the capacitor there is a changing E  B A changing electric field induces a magnetic field E where Id is called the displacement current

  11. E B x x x x x x x x x x x x Maxwell solved this problem by realizing that.... Inside the capacitor there must be an induced magnetic field... How?. Inside the capacitor there is a changing E  B A changing electric field induces a magnetic field E where Id is called the displacement current Therefore, Maxwell’s revision of Ampere’s Law becomes....

  12. Derivation of Displacement Current For a capacitor, and . Now, the electric flux is given by EA, so: , where this current , not being associated with charges, is called the “Displacement current”, Id. Hence: and:

  13. Derivation of Displacement Current For a capacitor, and . Now, the electric flux is given by EA, so: , where this current, not being associated with charges, is called the “Displacement Current”, Id. Hence: and:

  14. Maxwell’s Equations of Electromagnetism Gauss’ Law for Electrostatics Gauss’ Law for Magnetism Faraday’s Law of Induction Ampere’s Law

  15. Maxwell’s Equations of Electromagnetismin Vacuum (no charges, no masses) Consider these equations in a vacuum..... ......no mass, no charges. no currents.....

  16. Maxwell’s Equations of Electromagnetismin Vacuum (no charges, no masses)

  17. Faraday’s law: dB/dt electric field Maxwell’s modification of Ampere’s law dE/dt magnetic field ˆ j ˆ z E(x, t) = EP sin (kx-t) B(x, t) = BP sin (kx-t) Electromagnetic Waves These two equations can be solved simultaneously. The result is:

  18. Electromagnetic Waves B E

  19. Electromagnetic Waves B E Special case..PLANE WAVES... satisfy the wave equation Maxwell’s Solution

  20. E(x, t) = EP sin (kx-t) B(x, t) = BP sin (kx-t) ˆ j ˆ z Plane Electromagnetic Waves Ey Bz c x

  21. F(x) x  F(x) v x Static wave F(x) = FP sin (kx + ) k = 2   k = wavenumber  = wavelength Moving wave F(x, t) = FP sin (kx - t)  = 2  f  = angular frequency f = frequency v =  / k

  22. F v Moving wave F(x, t) = FP sin (kx - t) x What happens at x = 0 as a function of time? F(0, t) = FP sin (-t) For x = 0 and t = 0  F(0, 0) = FP sin (0) For x = 0 and t = t  F (0, t) = FP sin (0 – t) = FP sin (– t) This is equivalent to: kx = - t  x = - (/k) t F(x=0) at time t is the same as F[x=-(/k)t] at time 0 The wave moves to the right with speed /k

  23. E(x, t) = EP sin (kx-t) B(x, t) = BP sin (kx-t) ˆ j ˆ z Plane Electromagnetic Waves Ey Bz Notes: Waves are in Phase, but fields oriented at 900. k=2p/l. Speed of wave is c=w/k (= fl) At all times E=cB. c x

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