Lecture 4 Poynting Vector in Complex Notation. EM Fields and Interfaces.

1. Lecture 4Poynting Vector in Complex Notation.EM Fields and Interfaces. 6.013 ELECTROMAGNETICS AND APPLICATIONS Luca Daniel

2. Today’s Outline • Review of Fundamental Electromagnetic Laws • Electromagnetic Waves in Media and Interfaces • The EM waves in homogenous Media • Electromagnetic Power and Energy • The Poynting Theorem • Wave Intensity • Poynting Theorem in Complex Notation • EM 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 • EM Waves Incident “Normally” to a Different Medium • EM Waves Incident at General Angle to a Different Medium Today

3. Power Flow in Uniform Plane Waves z 0 The time average is called “intensity” [W/m2] of the wave

4. Poynting Vector in Complex Notation Note: Defining a meaningful and relating it to is not obvious. It is easier to relate it to the intensity (time average): (by definition) Thus the Intensity can put computed directly from the field phasors and

5. Today’s Outline • Review of Fundamental Electromagnetic Laws • Electromagnetic Waves in Media and Interfaces • The EM waves in homogenous Media • Electromagnetic Power and Energy • The Poynting Theorem • Wave Intensity • Poynting Theorem in Complex Notation • EM 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 • EM Waves Incident “Normally” to a Different Medium • EM Waves Incident at General Angle to a Different Medium

6. Fields at Boundaries: Normal Components Gauss’s Law: A h (assuming Lim h 0) surface charge density rs surface S Therefore: Similarly i.e. normal B is always continuous! L3-4

7. Note that is only possible on perfect conductors Fields at Boundaries: Tangential Components Faraday’s Law: (Lim h0) L E1// E1// = E2// Therefore: h A E2// Tangential E is always continuous Ampere’s Law: Js produces a jump in tangential H: Alternatively: L3-7

8. Electric Fields in perfect conductors : Conducting Media Constitutive relation for conducting medium (Ohm’s Law): where σ is the conductivity [Am/V] 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). which would instantaneously generate surface charge that immediately canceling all E. Therefore inside perfect conductors: Eout// = Ein// = 0 => E fields can only be  to a perfect metal surface r = 0 rs r can only be on the surface since any charge inside would produce E and J that would instantaneously distributed it to the surface rs J J rs rs J q J J J rs rs L3-8