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**ELECTROMAGNETICS AND APPLICATIONS**

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

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**Review of Fundamental Electromagnetic Laws **

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 2

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**Power Flow in Uniform Plane Waves**

z The time average is called “intensity” [W/m2] of the wave

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**Poynting Vector in Complex Notation**

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

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**Review of Fundamental Electromagnetic Laws **

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 5

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**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

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**Fields at Boundaries: Tangential Components**

Faraday’s Law: (Lim h0) L E1// Therefore: E1// = E2// h A E2// Tangential E is always continuous Ampere’s Law: Js produces a jump in tangential H: Alternatively: Note that is only possible on perfect conductors L3-7

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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). Electric Fields in perfect conductors : 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 J rs J rs J rs q J J rs J L3-8 rs

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