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Time-dependent fields

Time-dependent fields. .D = r  x E = 0. .B = 0  x H = J. Static fields decoupled. D = e E. B = m H. Electric fields have zero curl Caused by Static Charges Magnetic fields have zero divergence Caused by Static Currents Asymmetry in E and B field properties. .D = r

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Time-dependent fields

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  1. Time-dependent fields

  2. .D = r  x E = 0 .B = 0  x H = J Static fields decoupled D = eE B = mH Electric fields have zero curl Caused by Static Charges Magnetic fields have zero divergence Caused by Static Currents Asymmetry in E and B field properties

  3. .D = r  x E = - ∂B/∂t .B = 0  x H = J + ∂D/∂t Dynamic fields coupled D = eE B = mH Maxwell’s equations for Electromagnetism Field equations more symmetric (fields resemble each other)

  4. .D = r  x E = - ∂B/∂t .B = 0  x H = J + ∂D/∂t Kirchhoff’s Law V=LdI/dt Deriving Circuit Theory !

  5. The 4 Maxwell equations .D = r Gauss’ Law for electrostatics (Flux prop. to enclosed charge)

  6. The 4 Maxwell equations .B = 0 Gauss’ Law for magnetostatics (There is no magnetic charge)

  7.  x E = - ∂B/∂t The 4 Maxwell equations Faraday’s law of induction (Changing magnetic flux creates voltage)

  8. The 4 Maxwell equations  x H = J + ∂D/∂t Ampere’s law (Changing electric flux creates magnetic field)

  9. .D = r  x E = - ∂B/∂t .B = 0  x H = J + ∂D/∂t D = eE B = mH Constitutive equations (Maxwell’s eqns don’t give e,m, s, r Need quantum mechanics/solid state/statistical physics for this!) We treat them as external inputs The 4 Maxwell’s equations Not just E,B but D,H and also inputs r, J = sE

  10. .D = r  x E = - ∂B/∂t .B = 0  x H = J + ∂D/∂t The 4 Maxwell’s equations Most important consequence: Electromagnetic Waves (Chapter 7) Here we’ll learn the two new equations (Faraday’s Law and Ampere’s Law)

  11.  x E = - ∂B/∂t Faraday’s Law Since  x E ≠ 0 , can’t have E = -U Changing magnetic flux creates voltage

  12. Faraday’s Law  x E .dA = - ∂B.dA/∂t Integrate both sides

  13. Faraday’s Law E .dl = - ∂FB/∂t Stokes Theorem

  14. Faraday’s Law E .dl = - ∂FB/∂t Changing magnetic flux creates voltage

  15. Definition: FB/I = L (Solenoid) Faraday’s Law Vemf = - ∂FB/∂t Vemf = - LdI/dt

  16.  x E = - ∂B/∂t Faraday’s Law Changing magnetic flux creates voltage (and thus current)

  17. How to change magnetic flux? http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html

  18. Lenz’s Law opposing induced flux Increasing flux Induced current opposes any change in flux (Nature Prefers Inertia)

  19. Lenz’s Law opposing induced flux Decreasing flux Induced current opposes any change in flux (Nature Prefers Inertia)

  20. Force argument for Lenz’s Law Lorentz Force -e(v x B) downwards on el Wire motion Current flows upward in slider Current flows upward in slider Current flows upward in slider Decreasing B Flux towards you  induced B also towards you Increasing B Flux towards you  induced B away from you

  21. No matter which way you see it, the current in the slider flows the same way !!

  22. http://micro.magnet.fsu.edu/electromag/java/faraday/ /faraday2 /lenzlaw/ /transformer /detector

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