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Published byMarcus Douglas Modified over 9 years ago
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Maxwell’s Equations in Matter 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 M = sin(ay) k k i j j M = curl M = a cos(ay) i Total current Types of Current j
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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 = o E and displacement current exists throughout space
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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
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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
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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
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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
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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
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