ELECTROMAGNETICS AND APPLICATIONS Lecture 3 Waves in Conducting / Lossy Medium. Electromagnetic Power & Energy. Luca Daniel

L3-2 Review of Fundamental Electromagnetic Laws Electromagnetic Waves in Media and Interfaces oWaves in homogeneous lossless and lossy media oPower flow and energy balance (Poynting Theorem) oWaves at interfaces Digital & Analog Communications Microwave Communications Optical Communications Wireless Communications Acoustics Course Outline

L3-3 Course Overview and Motivations Maxwell Equations (review from 8.02) EM waves in homogenous media –EM Wave Equation –Solution of the EM Wave equation  Uniform Plane Waves (UPW)  Complex Notation (phasors) –EM Waves in homogeneous conducting/lossy media Electromagnetic Power and Energy –The Poynting Theorem –Wave Intensity –Poynting Theorem in Complex Notation EM Fields at Interfaces between Different Media Today’s Outline Today

L3-4 Waves in Conducting/Lossy Medium the imaginary part is the “lossy” party For example wave in good conductor z x

L3-5 Course Overview and Motivations Maxwell Equations (review from 8.02) EM waves in homogenous media –EM Wave Equation –Solution of the EM Wave equation  Uniform Plane Waves (UPW)  Complex Notation (phasors) –EM Waves in homogeneous conducting/lossy media Electromagnetic Power and Energy –The Poynting Theorem –Wave Intensity –Poynting Theorem in Complex Notation EM Fields at Interfaces between Different Media Today’s Outline Today

L3-6 Power and Energy Units of Power: [Joule]=[W s]=[V A s] [Watts]=[V A] Units of Energy: at steady state: non-steady state: Net power flow INTO the surface Power dissipated inside volume Non-zero power balance generates an increase of stored energy What is the relation between Power p(t) and Energy w(t)?

L3-7 Electromagnetic Power Flow Non-zero INCOMING power balance generates an increase of stored energy

L3-8 Electromagnetic Power and Energy Vector Identity using Faraday and Ampere’s Laws using Gauss Divergence Theorem

L3-9 The Poynting Theorem Energy Stored in Magnetic Field w m Energy Stored in Electric Field w e Power dissipated w d Net power flow INTO the surface The Poynting vector: gives both the magnitude of the power density and the direction of its flow.

L3-10 Uniform Plane Wave: EM fields EM Wave in z direction: Linearity implies superposition many wave solutions for different ,k,  Magnetic energy density Electric energy density y z z x Wavelength

L3-11 Power Flow in Uniform Plane Waves Note: is typically called “intensity” [W/m 2 ] of the wave 0 z

L3-12 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): Thus, we can define and Note: (by definition)

L3-13 Course Overview and Motivations Maxwell Equations (review from 8.02) EM waves in homogenous media –EM Wave Equation –Solution of the EM Wave equation  Uniform Plane Waves (UPW)  Complex Notation (phasors) –EM Waves in homogeneous conducting/lossy media Electromagnetic Power and Energy –The Poynting Theorem –Wave Intensity –Poynting Theorem in Complex Notation EM Fields at Interfaces between Different Media Today’s Outline Next Time