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Time variation

Time variation. Combining electrostatics and magnetostatics: (1) . E = r / e o where r = r f + r b (2) . B = 0 “no magnetic monopoles” (3)  x E = 0 “conservative” (4)  x B = m o j where j = j f + j M Under time-variation: (1) and (2) are unchanged,

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Time variation

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  1. Time variation Combining electrostatics and magnetostatics: (1) .E = r/eo wherer=rf+ rb (2) .B = 0 “no magnetic monopoles” (3)  x E = 0 “conservative” (4)  x B = moj where j= jf + jM Under time-variation: (1) and (2) are unchanged, (3) becomes Faraday’s Law (4) acquires an extra term, plus 3rd component of j

  2. B dS dℓ Faraday’s Law of Induction emfx induced in a circuit equals the rate of change of magnetic flux through the circuit

  3. Problem! Displacement current Ampere’s Law Continuity equation Steady current implies constant charge density so Ampere’s law consistent with the Continuity equation for steady currents Ampere’s law inconsistent with the continuity equation (conservation of charge) when charge density time dependent

  4. Extending Ampere’s Law add term to LHS such that taking Div makes LHS also identically equal to zero: The extra term is in the bracket extended Ampere’s Law

  5. eoE eo E/t - + - + I C S R Illustration of displacement current Discharging capacitor

  6. Displacement current magnitude Suppose E varies harmonically in time w ~ 10.19 rads-1 for eoE/t to be comparable to sE

  7. k M= sin(ay) k j i jM= curl M = a cos(ay) i Types of current j Total current • Polarisation current density from oscillation of charges in • electric dipoles • Magnetisation current density variation in magnitude of • magnetic dipoles in space/time

  8. 1st form of Maxwell’s Equations all field terms on LHS and all source terms on RHS The sources (r and j) are multiple (free, bound, mag, pol) special status of free source suggests 2nd Form

  9. Extending Ampere’s Law to H D/t is displacement current postulated by Maxwell (1862) to exist in the gap of a capacitor being charged In vacuum D = eoE and displacement current exists throughout space

  10. 2nd form of Maxwell’s Equations Applies only to well behaved LIH media Focus on sources means equations (2) and (3) unchanged! Recall Gauss’ Law for D In this version of (1), rrfandeoe Recall H version of (4) In this version of (4), jjf , alsomom andeoe

  11. 2nd and 3rd forms LHS: 2nd form, free sources only, other sources hidden in permittivity and permeability constants RHS: 3rd form (Minkowsky) free sources only, mixed fields, no constants

  12. Electromagnetic Wave Equation First form

  13. w w = ck = 2pc/l k Electromagnetic Waves in Vacuum Dispersion relation speed of light in vacuum

  14. k || k B || j E || i Relationship between E and B

  15. l r r k r|| Consecutive wave fronts Plane waves in a nutshell

  16. EM Waves in insulating LIH medium w Slope=±c/n w = ck/n = 2pc/nl k Dispersion relation Less than speed of light in vacuum e,m complex in general, real (as has been assumed) if hn<<Eg

  17. vB xB +p/2 t xB w 0 vB -p/2 wo -p Bound Charges Bound charge displacement xB Or velocity vB versus time Phase relative to driving field vs frequency

  18. Free Charges For a free charge, spring constant and wo tend to zero

  19. Dielectric susceptibility

  20. EM waves in conducting LIH medium

  21. EM waves in conducting LIH medium EM wave is attenuated within ~ skin depth in conducting media NB Insulating materials become ‘conducting’ when radiation frequency tuned above Eg

  22. Energy in Electromagnetic Waves Energy density in matter for static fields Average obtained over one cycle of light wave

  23. ct b a Energy in Electromagnetic Waves Average energy over one cycle of light wave Distance travelled by light over one cycle = 2pc/w = ct Average energy in volume ab ct

  24. Energy in Electromagnetic Waves

  25. Poynting Vector N = E x H is the Poynting vector Equal to the instantaneous energy flow associated with an EM wave In vacuum N || wave vector k Example If the E amplitude of a plane wave is 0.1 Vm-1 Energy crossing unit area per second is

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