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# Maxwell’s Equations in Vacuum

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1. Maxwell’s Equations in Vacuum (1) .E = r /eo Poisson’s Equation (2) .B = 0 No magnetic monopoles (3)  x E = -∂B/∂t Faraday’s Law (4)  x B = moj + moeo∂E/∂t Maxwell’s Displacement In vacuum with r = 0andj = 0 (1’) .E = 0 (4’)  x B = moeo∂E/∂t

2. Maxwell’s Equations in Vacuum Take curl of both sides of 3’ (3)  x ( x E) = -∂ ( x B)/∂t = -∂ (moeo∂E/∂t)/∂t = -moeo∂2E/∂t2  x ( x E) = (.E) - 2E - 2E = -moeo∂2E/∂t2 (.E = 0) 2E -moeo∂2E/∂t2 = 0 Vector wave equation

3. Maxwell’s Equations in Vacuum Plane wave solution to wave equation E(r, t) = Re {Eo ei(k.r-wt)}Eoconstant vector 2E =(∂2/∂x2 + ∂2/∂y2 +∂2/∂z2)E = -k2E .E = ∂Ex/∂x + ∂Ey/∂y +∂Ez/∂z = ik.E= ik.Eo ei(k.r-wt) If Eo || k then .E≠ 0 and  x E = 0 If Eo┴k then .E= 0 and  x E≠0 For light Eo┴k and E(r, t) is a transverse wave

4. l r|| r k r Consecutive wave fronts Maxwell’s Equations in Vacuum Eo Plane waves travel parallel to wave vector k Plane waves have wavelength 2p /k

5. Maxwell’s Equations in Vacuum • Plane wave solution to wave equation • E(r, t) = Eo ei(k.r-wt) Eoconstant vector • moeo∂2E/∂t2 = - moeow2Emoeow2 =k2 w =±k/(moeo)1/2 = ±ck w/k = c = (moeo)-1/2 phase velocity w(k) w= ±ck Linear dispersion relationship k

6. Maxwell’s Equations in Vacuum Magnetic component of the electromagnetic wave in vacuum From Faraday’s law  x ( x B) = moeo∂( x E)/∂t = moeo∂(-∂B/∂t)/∂t = -moeo∂2B/∂t2  x ( x B) = (.B) - 2B - 2B = -moeo∂2B/∂t2 (.B = 0) 2B -moeo∂2B/∂t2 = 0 Same vector wave equation as for E

7. Maxwell’s Equations in Vacuum If E(r, t) = Eo ei(k.r-wt) and k || z and Eo || x (x,y,z unit vectors)  x E = ik Eox ei(k.r-wt) y = -∂B/∂t From Faraday’s Law ∂B/∂t = -ik Eox ei(k.r-wt) y B = (k/w) Eo ei(k.r-wt) y = (1/c) Eo ei(k.r-wt) y For this wave Eo || x, Bo || y, k || z, cBo = Eo

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

9. ct b a Energy in Electromagnetic Waves Average energy over one cycle of light wave Distance travelled by light over one cycle ct = 2pc/w Average energy in volume ab ct

10. Energy in Electromagnetic Waves