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Maxwell’s Equations in Vacuum (1) .E = / o Poisson’s Equation (2) .B = 0No magnetic monopoles (3) x E = -∂B/∂t Faraday’s Law (4) x B = o j + o o ∂E/∂t Maxwell’s Displacement In vacuum with = 0 and j = 0 (1’) .E = (4’) x B = o o ∂E/∂t

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Maxwell’s Equations in Vacuum Take curl of both sides of 3’ (3) x ( x E) = -∂ ( x B)/∂t = -∂ ( o o ∂E/∂t)/∂t = - o o ∂ 2 E/∂t 2 x ( x E) = ( .E) - 2 E - 2 E = - o o ∂ 2 E/∂t 2 ( .E = 0) 2 E - o o ∂ 2 E/∂t 2 = 0 Vector wave equation

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Maxwell’s Equations in Vacuum Plane wave solution to wave equation E(r, t) = Re {E o e i (k.r- t) }E o constant vector 2 E =(∂ 2 /∂x 2 + ∂ 2 /∂y 2 + ∂ 2 /∂z 2 )E = -k 2 E .E = ∂E x /∂x + ∂E y /∂y + ∂E z /∂z = i k.E = ik.E o e i (k.r- t) If E o || k then .E ≠ 0 and x E = 0 If E o ┴ k then .E = 0 and x E ≠ 0 For light E o ┴ k and E(r, t) is a transverse wave

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r r || rr k Consecutive wave fronts Plane waves travel parallel to wave vector k Plane waves have wavelength 2 /k Maxwell’s Equations in Vacuum EoEo

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Plane wave solution to wave equation E(r, t) = E o e i(k.r- t) E o constant vector o o ∂ 2 E/∂t 2 = - o o 2 E o o 2 =k 2 =±k/( o o ) 1/2 = ±ck /k = c = ( o o ) -1/2 phase velocity = ±ck Linear dispersion relationship (k) k

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Maxwell’s Equations in Vacuum Magnetic component of the electromagnetic wave in vacuum From Faraday’s law x ( x B) = o o ∂( x E)/∂t = o o ∂(-∂B/∂t)/∂t = - o o ∂ 2 B/∂t 2 x ( x B) = ( .B) - 2 B - 2 B = - o o ∂ 2 B/∂t 2 ( .B = 0) 2 B - o o ∂ 2 B/∂t 2 = 0 Same vector wave equation as for E

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Maxwell’s Equations in Vacuum If E(r, t) = E o e i(k.r- t) and k || z and E o || x (x,y,z unit vectors) x E = ik E ox e i(k.r- t) y = -∂B/∂t From Faraday’s Law ∂B/∂t = -ik E ox e i(k.r- t) y B = (k/ ) E o e i(k.r- t) y = (1/c) E o e i(k.r- t) y For this wave E o || x, B o || y, k || z, cB o = E o

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Energy in Electromagnetic Waves Energy density Average obtained over one cycle of light wave

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Energy in Electromagnetic Waves Average energy over one cycle of light wave Distance travelled by light over one cycle c = 2 c/ Average energy in volume ab c a b cc

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Energy in Electromagnetic Waves

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