EEE340Lecture : Time-Harmonic Electromagnetics Using the rule We have the Maxwell’s equations in phasor form as The Lorentz gauge (7.98) (7.94)

EEE340Lecture 302 The nonhomogeneous wave equations where The wavenumber The phasor solutions (7.100) (7.99) (7.97) (7.96) (7.95)

EEE340Lecture 303 Note: Wavenumber (7.102)

EEE340Lecture 304 Chapter 8: Plane EM Waves 8-1: The wave equation D’Almbert equation: Where, time-domain Source free region Nonconducting, simple medium Eq. (8.1) is the d’Almbert equation Eq. (8.1) is a hyperbolic equation in mathematics The solution to (8.1) with open boundary is a uniform plane wave (8-1) (8-2)

EEE340Lecture : Plane wave in lossless media The vector Helmholtz equation Where  frequency domain Where free-space wavenumber Show Assume time convention e j  t, then It follows that Hence (8.1) becomes (8.2) (8-3) (8-4)

EEE340Lecture 306 The complex term: Phasor form which results from dropping the time factor is called the phasor. Equivalent relations:

EEE340Lecture 307 Uniform plane wave Let: Then Eq. (8.3) reduces to Assume uniform along x- and y-, i.e., (8-6) (8-5)

EEE340Lecture 308 The general solution to the 2 nd order ODE of Eq (8.6) is where E o + and E o - are arbitrary complex constants to be determined by the boundary conditions. Time-domain expression from the phasor: The solution (8.8) to (8.1) is a traveling wave. (8-7) (8-8)

EEE340Lecture 309 Phase velocity Let  t-k o z = constant Differentiating (8-9)  

EEE340Lecture 3010 t is fixed in z is fixed in Snapshot Oscilloscope

EEE340Lecture 3011 The electric field as a function of z at different times.