Maxwell’s Equations in Matter

1. Maxwell’s Equations in Matter in vacuum in matter .E = r /eo .D = rfreePoisson’s Equation .B = 0 .B = 0 No magnetic monopoles  x E = -∂B/∂t  x E = -∂B/∂t Faraday’s Law  x B = moj + moeo∂E/∂t  x H = jfree+ ∂D/∂t Maxwell’s Displacement D = eoe E = eo(1+ c)E Constitutive relation for D H = B/(mom)= (1- cB)B/moConstitutive relation for H Solve with: model e for insulating, isotropic matter, m= 1,rfree= 0,jfree= 0 model e for conducting, isotropic matter, m= 1,rfree= 0,jfree= s(w)E

2. Mion k melectron kMion Si ion Bound electron pair 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 Resonance frequency wo ~ (k/M)1/2 or ~ (k/m)1/2 Ions: heavy, resonance in infra-red ~1013Hz Bound electrons: light, resonance in visible ~1015Hz Free electrons: no restoring force, no resonance

3. Mion k melectron kMion Bound and Free Charges Bound charges Resonance model for uncoupled electron pairs

4. Mion k melectron kMion Bound and Free Charges Bound charges In and out of phase components of x(t) relative to Eo cos(wt) in phase out of phase

5. Bound and Free Charges Bound charges Connection to c and e e(w) Im{e(w)} w = wo w/wo Re{e(w)}

6. s(w) Re{e(w)} wo = 0 Drude ‘tail’ w Im{s(w)} Bound and Free Charges Free charges Let wo→ 0 in c and e jpol = ∂P/∂t

7. Maxwell’s Equations in Matter in vacuum in matter .E = r /eo .D = rfreePoisson’s Equation .B = 0 .B = 0 No magnetic monopoles  x E = -∂B/∂t  x E = -∂B/∂t Faraday’s Law  x B = moj + moeo∂E/∂t  x H = jfree+ ∂D/∂t Maxwell’s Displacement D = eoe E = eo(1+ c)E Constitutive relation for D H = B/(mom)= (1- cB)B/moConstitutive relation for H Solve with: model e for insulating, isotropic matter, m= 1,r = 0,jfree= 0 model e for conducting, isotropic matter, m= 1,r = 0,jfree= s(w)E

8. Maxwell’s Equations in Matter Solution of Maxwell’s equations in matter for m= 1, rfree= 0,jfree= 0 Maxwell’s equations become  x E = -∂B/∂t  x H = ∂D/∂t H = B /moD = eoeE  x B = moeoe∂E/∂t  x ∂B/∂t = moeoe∂2E/∂t2  x (- x E) =  x ∂B/∂t = moeoe∂2E/∂t2 -(.E) + 2E = moeoe∂2E/∂t2 . e E= e . E = 0 since rfree= 0 2E - moeoe∂2E/∂t2 = 0

9. Maxwell’s Equations in Matter • 2E - moeoe∂2E/∂t2 = 0 E(r, t) = EoexRe{ei(k.r-wt)} • 2E = -k2E moeoe∂2E/∂t2 = - moeoe w2E • (-k2+moeoe w2)E = 0 • w2 = k2/(moeoe)moeoew2 = k2 k = ± w√(moeoe) k = ± √ew/c • Let e = e1+ie2be the real and imaginary parts of e and e = (n+ik)2 • We need √e = n+ik e = (n+ik)2 = n2- k2 +i 2nke1= n2- k2 e2= 2nk • E(r, t) = Eoex Re{ ei(k.r- wt) } = Eoex Re{ei(kz- wt)} k || ez • = Eoex Re{ei((n + ik)wz/c - wt)}= Eoex Re{ei(nwz/c -wt)e- kwz/c)} Attenuated wave with phase velocity vp = c/n

10. Maxwell’s Equations in Matter Solution of Maxwell’s equations in matter for m= 1, rfree= 0,jfree= s(w)E Maxwell’s equations become  x E = -∂B/∂t  x H = jfree + ∂D/∂t H = B /moD = eoeE  x B = mojfree+ moeoe∂E/∂t  x ∂B/∂t = mos∂E/∂t + moeoe∂2E/∂t2  x (- x E) =  x ∂B/∂t = mos∂E/∂t+moeoe∂2E/∂t2 -(.E) + 2E = mos∂E/∂t+moeoe∂2E/∂t2 . e E= e . E = 0 since rfree= 0 2E - mos∂E/∂t - moeoe∂2E/∂t2 = 0

11. Maxwell’s Equations in Matter 2E - mos∂E/∂t - moeoe∂2E/∂t2 = 0 E(r, t) = EoexRe{ei(k.r- wt)}k|| ez 2E = -k2E mos∂E/∂t= mosiwEmoeoe∂2E/∂t2 = - moeoe w2E (-k2-mosiw+moeoew2 )E = 0 s >> eoe w for a good conductor E(r, t) = Eoex Re{ ei(√(wsmo/2)z - wt)e-√(wsmo/2)z} NB wave travels in +z direction and is attenuated The skin depth d = √(2/wsmo) is the thickness over which incident radiation is attenuated. For example, Cu metal DC conductivity is 5.7 x 107 (Wm)-1 At 50 Hz d = 9 mm and at 10 kHz d = 0.7 mm