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Maxwell ’ s Equations

.D = r  x E = -∂B/∂t. .B = 0  x H = J + ∂D/∂t. J = s E. D = e E. B = m H. Maxwell ’ s Equations. Add in ‘ constitutive ’ material relations. .B = 0  x H = J + ∂D/∂t. .E = r/e  x E = -∂B/∂t. Free space, r =J =0. .E = r/e  x E = -∂B/∂t

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Maxwell ’ s Equations

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  1. .D = r  x E = -∂B/∂t .B = 0  x H = J + ∂D/∂t J = sE D = eE B = mH Maxwell’s Equations Add in ‘constitutive’ material relations

  2. .B = 0  x H = J + ∂D/∂t .E = r/e  x E = -∂B/∂t Free space, r =J =0

  3. .E = r/e  x E = -∂B/∂t Do  x ( x E) .B = 0  x H = J + ∂D/∂t 2E = me∂2E/∂t2 Solution: Plane waves Wave equation One set of Solutions: Plane waves E(r,t) = E0ejwt-jbr b = mew = w/v

  4. Maxwell in a metal: finite conductivity (r=0,J=sE) .B = 0  x H = s E + ∂D/∂t .E = r/e  x E = -∂B/∂t • Dielectric relaxation time t (Time for charges to adjust in DC field) • Skin Depth d (Penetration for low frequency) • Plasma frequency wp (Time for charges to rearrange in AC field)

  5. Maxwell in a metal: finite conductivity (r=0,J=sE) .B = 0  x H = s E + ∂D/∂t .E = r/e  x E = -∂B/∂t • Dielectric relaxation time t (DC conductivity) • Skin Depth d (Distance over which AC fields decay) • Plasma frequency wp (Response time in alternating fields)

  6. Maxwell in metal (r=0,J=sE) .D = 0  x E = - ∂B/∂t .B = 0  x H = J + ∂D/∂t ∂/∂t  jw   -jb • x ( x E) = -∂( x B)/∂t 2E = m∂J/∂t + me∂2E/∂t2 Use plane waves and phasors b2 = -jwsm + mew2 = meeffw2 whereeeff = e –js/w Complex b gives decaying wave

  7. Loss: finite conductivity(r=0,J=sE) .D = 0  x E = - ∂B/∂t .B = 0  x H = J + ∂D/∂t eeff = e(1 –js/ew) (Note dimensions between s and e ! What “time” is this?)

  8. Dielectric Relaxation Time EOC: ∂r/∂t = -. J = -s . E = -s . D/e • = -(s/e)r • = -r/t r = r0e-t/t • = e/s Higher conductivity, better metal  smaller t Cu: 1/s = 1.7mW-cm, er = 6 , t = 5 x 10-19s After time t, charges/fields escape towards surface How close to surface ???

  9. Skin Depth in Metals 2E = jwsmE - mew2E b2 = -jwsm + mew2 meeffw2 d Note that b is complex! • = bR + jbI E = E0ej(br-wt) = E0ej(bRr-wt)e-bIr Decays! P ~ |E|2 = |E0|2e-2bIr = P0e-r/d where • = 1/2bI is the skin depth (distance over which power decays)

  10. Skin Depth in Metals 2E = jwsmE - mew2E b2 = -jwsm + mew2 Low frequency, drop second term • = bR - jbI ≈ (-jwsm)1/2 =(wsm)1/2 [1-j]/√2 • I ≈ (wsm/2)1/2 From previous slide, d = 1/2bI = (2/wsm)1/2

  11. Skin Depth in Metals d = 2/(wsm) = 1/pfsm  Skin Depth Cu: 1/s = 1.7mW-cm, f = 409 GHz, d = 0.1033 mm d High conductivity, high frequency signals eliminated from conductor Dimensions: d = c (t/pf)  Length

  12. Skin depth for various frequencies and materials Courtesy: Wikipedia

  13. Why don’t fields penetrate? EM fields oscillate and create eddy currents Inside bulk, eddy currents oppose conduction current On surface over depth d, eddy currents helpconduction current (Courtesy: Wikipedia)

  14. Is this true for fast signals? • If AC field varies too fast, eddy currents cannot keep up, neither can charges move fast enough to screen it in the bulk. • This means fast frequencies should have different screening properties than slow frequencies • Meaning even the bare e is frequency dependent !! For a conductor, eeff(w) = e(w) - js/w

  15. The freq-dependent e -Im(e) Re(e) w e = e1(w) + je2(w) w0 wp Plasma frequency Resonant frequency

  16. Plasmonics: A Field unto itself! Changing wp dynamics with size Nanogold: red in color!

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