ELECTROMAGNETICS AND APPLICATIONS Lecture 5 Normal Incidence at Media Interfaces Luca Daniel

L3-2 Review of Fundamental Electromagnetic Laws Electromagnetic Waves in Media and Interfaces oThe EM waves in homogenous Media oElectromagnetic Power and Energy oEM Fields at Interfaces between Different Media  Fields at boundaries: normal components  Fields at boundaries: tangential components  Fields inside perfect conductors  Fields at boundaries of perfect conductors oEM Waves Incident “Normally” to a Different Medium  Normal incidence to a perfect conductor  Standing waves (time domain view and energy)  Normal incidence to a dielectric  Power balance oEM Waves Incident at General Angle to a Different Medium Today’s Outline Today

Conducting Media Electric Fields in perfect conductors : Constitutive relation for conducting medium (Ohm’s Law): where σ is the conductivity [Am/V] which would instantaneously generate surface charge that immediately canceling all E. In a regular conductor charges are free to move. If E is applied, J will generate charges on the surface that start cancelling the applied E (charge relaxation). Therefore inside perfect conductors:  = 0  can only be on the surface since any charge inside would produce E and J that would instantaneously distributed it to the surface q J J J J J J ss ss ss ss ss ss L3-3

Therefore: H = 0 inside perfect conductors (if  = , and H was ever zero at any time) L3-4 Conducting Media Magnetic Fields in perfect conductors : Electric Fields in perfect conductors : Inside perfect conductors:  = 0

General Boundary Conditions:  s coulombs/m 2  J s Amperes/m  H is parallel to perfect conductors (and is terminated by surface current)  E is perpendicular to perfect conductors (and is terminated by surface charge) Inside Perfect Conductors: Summary of Boundary Conditions

Perfect conductor Surface current Amperes/m Line current over a perfect conductor Point charge over a perfect conductor Perfect conductor Surface charge Coulombs/m 2 L3-7 Two Examples: quasi-static fields from charge and current near perfect conductors

EM Waves with “Normal” Incidence to Perfect Conductors Solution Method for Boundary Value Problems: 1)Assume fields on both sides of the boundary in terms of unknown coefficients; typically a sum of terms 2)Write equations for fields that satisfy boundary conditions 3)Solve for unknowns and check answer with Maxwell Equations Example—Normal incidence on perfect conductor: 1)Incident: 2)Match B.C.: 3)Solve: c == y z 0 (given)

Standing Waves – Time Domain View and Energy Perfect Conductor W e [J/m 3 ] = (1/2)  E(t,z)| 2 = 2  E i 2 sin 2 kz sin 2  t (Where does the energy go?) t = +  /2  t =  t = -  /2  z z = 0 Standing waves oscillate without moving Never any W e here Incident: Reflected: Total: Time Domain: E = 0 every half cycle (  t = 0, , etc.) and every half wavelength for any t Electric Energy Density: y

= 0 when  t =  /2, 3 , etc. t =  t = 0 t =  Standing Waves Magnetic Field z = 0 Incident: Reflected: Total: Time Domain: Magnetic Energy Density: x z

Normal Incidence to Dielectrics Normal Incidence y z Boundary Conditions for the Electric Field Boundary Conditions for the Magnetic Field (no surface currents) x

Normal Incidence to Dielectrics – Power Balance Normal Incidence y z x Example: or