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# Plane Waves - Plane Waves Phasor notation Euler Identity j...

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Plane Waves

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Phasor notation Euler Identity Instead of writing the sinusoidal terms of the time varying fields, we may simply express the time variation of the fields as complex exponentials To get the time domain expression of the field, we simply get the real part of the complex exponential e j cos  j sin A x , y , z ,t = A x , y , z cos  t ≡ A x , y , z e j  t  A x , y , z ,t =ℜ[ A x , y , z e j  t  ]= A x , y , z cos  t 
Maxwell's Equations Time varying form of Maxwell's Equations Valid for arbitrary time dependence To simplify discussion on plane waves, we will only consider fields with sinusoidal time variations ∇× E = −∂ B t ∇× H = D t J ∇⋅ D = ∇⋅ B = 0 A x , y , z ,t = A x , y , z cos  t 

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Maxwell's Equations Time varying form of Maxwell's Equations Maxwell's Equations using phasor notation ∇× E = −∂ B t ∇× H = D t J ∇⋅ D = ∇⋅ B = 0 ∇× E =− j B ∇× H = j D J ∇⋅ D = ∇⋅ B = 0
Helmholtz equations Maxwell's curl equations for a source free, linear, isotropic, homogenous medium 2 Equations in 2 unknowns ( E and H ) ∇× E =− j  H ∇× H = j  E

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Helmholtz Equations Taking the curl of the first equation and substituting into the second equation: Using the following vector identity, the previous equation can be further simplified ∇×∇× E =− j  ∇× H ∇×∇× E = 2  E ∇×∇× A =∇∇⋅ A −∇ 2 A
Helmholtz Equations The previous equation becomes: since for the source free case This is the wave equation, or Helmholtz equation

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Plane Waves - Plane Waves Phasor notation Euler Identity j...

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