Maxwells Equations in Matter Types of Current j

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Maxwell’s Equations in Matter Types of Current j Total current k M = sin(ay)

Maxwell’s Equations in Matter Types of Current j Total current k M = sin(ay) k j i j. M = curl M = a cos(ay) i Free current density from unbound conduction electrons (metals) Polarisation current 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

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 = eo. E and displacement current exists throughout space

Maxwell’s Equations in Matter in vacuum in matter . E = r /eo .

Maxwell’s Equations in Matter in vacuum in matter . E = r /eo . D = rfree Poisson’s Equation . B = 0 No magnetic monopoles 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 - c. B)B/mo Constitutive 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,

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 / mo D = eo e E x B = moeoe ∂E/∂t x ∂B/∂t = moeoe ∂2 E/∂t 2 x (- x E) = x ∂B/∂t = moeoe ∂2 E/∂t 2 - (. E) + 2 E = moeoe ∂2 E/∂t 2 2 E - moeoe ∂2 E/∂t 2 = 0 . e E = e . E = 0 since rfree = 0

Maxwell’s Equations in Matter 2 E - moeoe ∂2 E/∂t 2 = 0 E(r,

Maxwell’s Equations in Matter 2 E - moeoe ∂2 E/∂t 2 = 0 E(r, t) = Eo ex Re{ei(k. r - wt)} 2 E = -k 2 E moeoe ∂2 E/∂t 2 = - moeoe w 2 E (-k 2 +moeoe w 2)E = 0 w 2 = k 2/(moeoe) m o e w 2 = k 2 k = ± w√(moeoe) k = ± √e w/c Let e = e 1 + ie 2 be the real and imaginary parts of e and e = (n + ik)2 We need √e = n + ik e = (n + ik)2 = n 2 - k 2 + i 2 nk e 1 = n 2 - k 2 e 2 = 2 nk E(r, t) = Eo ex Re{ ei(k. r - wt) } = Eo ex Re{ei(kz - wt)} k || ez = Eo ex Re{ei((n + ik)wz/c - wt)} = Eo ex 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,

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 / mo D = eo e E x B = mo jfree + moeoe ∂E/∂t x ∂B/∂t = mos ∂E/∂t + moeoe ∂2 E/∂t 2 x (- x E) = x ∂B/∂t = mos ∂E/∂t + moeoe ∂2 E/∂t 2 - (. E) + 2 E = mos ∂E/∂t + moeoe ∂2 E/∂t 2 2 E - mos ∂E/∂t - moeoe ∂2 E/∂t 2 = 0 . e E = e . E = 0 since rfree = 0

Maxwell’s Equations in Matter 2 E - mos ∂E/∂t - moeoe ∂2 E/∂t 2

Maxwell’s Equations in Matter 2 E - mos ∂E/∂t - moeoe ∂2 E/∂t 2 = 0 2 E = -k 2 E E(r, t) = Eo ex Re{ei(k. r - wt)} mos ∂E/∂t = mos iw E (-k 2 -mos iw +moeoe w 2 )E = 0 k || ez moeoe ∂2 E/∂t 2 = - moeoe w 2 E s >> eoe w for a good conductor E(r, t) = Eo ex 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 k. Hz d = 0. 7 mm