Maxwell’s Equations in Matter in vacuum in matter .E =  /  o .D =  free Poisson’s Equation .B = 0 .B = 0 No magnetic monopoles  x E = -∂B/∂t  x E = -∂B/∂t Faraday’s Law  x B =  o j +  o  o ∂E/∂t  x H = j free + ∂D/∂t Maxwell’s Displacement D =  o  E =  o (1+  )EConstitutive relation for D H = B/(  o  ) = (1-  B )B/  o Constitutive relation for H Solve with: model  for insulating, isotropic matter,  = 1,  free  = 0, j free = 0 model  for conducting, isotropic matter,  = 1,  free  = 0, j free =  (  )E

Bound and Free Charges Bound charges All valence electrons in insulators (materials with a ‘band gap’) Bound valence electrons in metals or semiconductors (band gap absent/small ) Free charges Conduction electrons in metals or semiconductors M ion k m electron k M ion Si ion Bound electron pair Resonance frequency  o ~ (k/M) 1/2 or ~ (k/m) 1/2 Ions: heavy, resonance in infra-red ~10 13 Hz Bound electrons: light, resonance in visible ~10 15 Hz Free electrons: no restoring force, no resonance

Bound and Free Charges Bound charges Resonance model for uncoupled electron pairs M ion k m electron k M ion

Bound and Free Charges Bound charges In and out of phase components of x(t) relative to E o cos(  t) M ion k m electron k M ion in phase out of phase

Bound and Free Charges Bound charges Connection to  and       Im{  } Re{  }

Bound and Free Charges Free charges Let   → 0 in  and  j pol = ∂P/∂t     Im{  }  Re{  } Drude ‘tail’

Maxwell’s Equations in Matter in vacuum in matter .E =  /  o .D =  free Poisson’s Equation .B = 0 .B = 0 No magnetic monopoles  x E = -∂B/∂t  x E = -∂B/∂t Faraday’s Law  x B =  o j +  o  o ∂E/∂t  x H = j free + ∂D/∂t Maxwell’s Displacement D =  o  E =  o (1+  )EConstitutive relation for D H = B/(  o  ) = (1-  B )B/  o Constitutive relation for H Solve with: model  for insulating, isotropic matter,  = 1,  = 0, j free = 0 model  for conducting, isotropic matter,  = 1,  = 0, j free =  (  )E

Maxwell’s Equations in Matter Solution of Maxwell’s equations in matter for  = 1,  free  = 0, j free = 0 Maxwell’s equations become  x E = -∂B/∂t  x H = ∂D/∂t H = B /  o D =  o  E  x B =  o  o  ∂E/∂t  x ∂B/∂t =  o  o  ∂ 2 E/∂t 2  x (-  x E) =  x ∂B/∂t =  o  o  ∂ 2 E/∂t 2 -  ( .E) +  2 E =  o  o  ∂ 2 E/∂t 2 .  E = . E = 0 since  free  = 0  2 E -  o  o  ∂ 2 E/∂t 2 = 0

Maxwell’s Equations in Matter  2 E -  o  o  ∂ 2 E/∂t 2 = 0 E(r, t) = E o e x Re{e i (k.r -  t) }  2 E = -k 2 E  o  o  ∂ 2 E/∂t 2 = -  o  o  2 E (-k 2  +  o  o  2 )E = 0  2 = k 2 /(  o  o  )  o  o   2 = k 2 k = ±  √(  o  o  ) k = ± √   /c Let  =  1  + i  2  be the real and imaginary parts of  and  = (n  + i  ) 2 We need √  = n  + i   = (n  + i  ) 2 = n 2  -  2 + i 2n   1  = n 2  -  2  2  = 2n  E(r, t) = E o e x Re{ e i (k.r -  t) } = E o e x Re{e i (kz -  t) } k || e z = E o e x Re{e i ((n + i  )  z/c -  t) } = E o e x Re{e i (n  z/c -  t) e -  z/c) } Attenuated wave with phase velocity v p = c/n

Maxwell’s Equations in Matter Solution of Maxwell’s equations in matter for  = 1,  free  = 0, j free =  (  )E Maxwell’s equations become  x E = -∂B/∂t  x H = j free + ∂D/∂t H = B /  o D =  o  E  x B =  o j free +  o  o  ∂E/∂t  x ∂B/∂t =  o  ∂E/∂t +  o  o  ∂ 2 E/∂t 2  x (-  x E) =  x ∂B/∂t =  o  ∂E/∂t +  o  o  ∂ 2 E/∂t 2 -  ( .E) +  2 E =  o  ∂E/∂t +  o  o  ∂ 2 E/∂t 2 .  E = . E = 0 since  free  = 0  2 E -  o  ∂E/∂t -  o  o  ∂ 2 E/∂t 2 = 0

Maxwell’s Equations in Matter  2 E -  o  ∂E/∂t -  o  o  ∂ 2 E/∂t 2 = 0 E(r, t) = E o e x Re{e i (k.r -  t) } k || e z  2 E = -k 2 E  o  ∂E/∂t =  o  i  E  o  o  ∂ 2 E/∂t 2 = -  o  o  2 E (-k 2  -  o  i  +  o  o  2 )E = 0  o  for a good conductor E(r, t) = E o e x Re{ e i (√(  o / 2 )z -  t) e -√(  o / 2 )z } NB wave travels in +z direction and is attenuated The skin depth  = √(2/  o ) is the thickness over which incident radiation is attenuated. For example, Cu metal DC conductivity is 5.7 x 10 7 (  m) -1 At 50 Hz  = 9 mm and at 10 kHz  = 0.7 mm