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k. M = sin(ay) k. j. i. j M = curl M = a cos(ay) i. Maxwell’s Equations in Matter. Types of Current j. Total current. Free current density from unbound conduction electrons (metals) Polarisation current density from oscillation of charges as electric dipoles
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k M= sin(ay) k j i jM= curl M = a cos(ay) i Maxwell’s Equations in Matter Types of Current j Total current Free current density from unbound conduction electrons (metals) Polarisationcurrent density from oscillation of charges as electric dipoles Magnetisation current density from space/time variation of magnetic dipoles
Maxwell’s Equations in Matter D/t is displacement current postulated by Maxwell (1862) to exist in the gap of a charging capacitor In vacuum D = eoE and displacement current exists throughout space
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
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
Maxwell’s Equations in Matter • 2E - moeoe∂2E/∂t2 = 0 E(r, t) = Eoex Re{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
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
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
