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

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

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

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

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

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

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

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

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 cc

Energy in Electromagnetic Waves