Elements of electromagnetic field theory and guided waves Joule’s law in differential and integral form Time-varying current: Inductance Time-varying currents: capacitance Method of phasors (complex amplitudes) Complex impedances Maxwell’s equations for field phasors Time-dependent wave equation Helmholtz’ wave equation Plane wave

Joule’s law Elemental volume dV=Sdl J 2 1 l Differential form Area S Elemental volume dV=Sdl J Differential form 2 1 l Integral form

Time-varying current. Capacitance EMFext R 1 2 Process of recharging:

Time-varying current. Inductance BI eext Definition of L Process of self-inductance I(t)

Method of phasors Series LRC-circuit with external EMF e: Let it be cos wt Finally, one can find both amplitude and phase of the real current i(t) So simple!

Complex impedances, Ohm’s and Kirchhoff laws for current and voltage phasors ZL=jwL ZC=1/jwC ZR=R or General notation Impedance Notation r V=IZ Es Z

Maxwell equations for field phasors

Wave equation for source-free regions (r=J=0) Apply axbxc=b(ac)-c(ab) (Recall 3d Maxwell’s equation) Time-harmonic case: k=w/u - wave number Wave equation For phasors it is the so-called Helmholtz’ equation phase velocity

Plane wave in free space Wave number in free space Forward wave Backward wave y E=Ex z=ct - wave front z z=ray t=0 t=p/2w