Electromagnetic Fields ELC205B-Spring 2012 Department of Electronics and Electrical Communications Engineering Faculty of Engineering – Cairo University

Outline  Mathematical Operators  Fundamental Laws of Electro and Magneto Statics  Faraday’s Experiment  Lenz’s Law  Faraday’s Law  Induced Electric Field  Electric Field in terms of Potential Functions  Ampere’s Law and the Continuity Equation  Retarded Potentials © Mamdouh Hassan Abbas,2012

Outline  Mathematical Operators  Fundamental Laws of Electro and Magneto Statics  Faraday’s Experiment  Lenz’s Law  Faraday’s Law  Induced Electric Field  Electric Field in terms of Potential Functions  Ampere’s Law and the Continuity Equation  Retarded Potentials © Mamdouh Hassan Abbas,2012

Mathematical Operators  © Mamdouh Hassan Abbas,2012 Positive Source Negative Sink Zero No charge S S S n n n

Mathematical Operators  n Always Zero Circulating quantity S © Mamdouh Hassan Abbas,2012

Mathematical Operators  Always Zero Flowing quantity Non-zero Circulating quantity E B © Mamdouh Hassan Abbas,2012

Outline  Mathematical Operators  Fundamental Laws of Electro and Magneto Statics  Faraday’s Experiment  Lenz’s Law  Faraday’s Law  Induced Electric Field  Electric Field in terms of Potential Functions  Ampere’s Law and the Continuity Equation  Retarded Potentials © Mamdouh Hassan Abbas,2012

Fundamental Laws of Electro and Magneto Statics  © Mamdouh Hassan Abbas,2012

Fundamental Laws of Electro and Magneto Statics  © Mamdouh Hassan Abbas,2012

Fundamental Laws of Electro and Magneto Statics  © Mamdouh Hassan Abbas,2012

Outline  Mathematical Operators  Fundamental Laws of Electro and Magneto Statics  Faraday’s Experiment  Lenz’s Law  Faraday’s Law  Induced Electric Field  Electric Field in terms of Potential Functions  Ampere’s Law and the Continuity Equation  Retarded Potentials © Mamdouh Hassan Abbas,2012

Faraday’s Experiment battery switch toroidal iron core compass primary coil secondary coil © Mamdouh Hassan Abbas,2012

Faraday’s Experiment battery switch toroidal iron core compass primary coil secondary coil © Mamdouh Hassan Abbas,2012

Faraday’s Experiment battery switch toroidal iron core compass primary coil secondary coil © Mamdouh Hassan Abbas,2012

Outline  Mathematical Operators  Fundamental Laws of Electro and Magneto Statics  Faraday’s Experiment  Lenz’s Law  Faraday’s Law  Induced Electric Field  Electric Field in terms of Potential Functions  Ampere’s Law and the Continuity Equation  Retarded Potentials © Mamdouh Hassan Abbas,2012

Lenz’s Law  © Mamdouh Hassan Abbas,2012

Outline  Mathematical Operators  Fundamental Laws of Electro and Magneto Statics  Faraday’s Experiment  Lenz’s Law  Faraday’s Law  Induced Electric Field  Electric Field in terms of Potential Functions  Ampere’s Law and the Continuity Equation  Retarded Potentials © Mamdouh Hassan Abbas,2012

Faraday’s Law  © Mamdouh Hassan Abbas,2012

Faraday’s Law  Illustrative examples © Mamdouh Hassan Abbas,2012

Faraday’s Law  © Mamdouh Hassan Abbas,2012

Faraday’s Law  © Mamdouh Hassan Abbas,2012

Faraday’s Law  B(Outward flux) Conducting path Conducting rod © Mamdouh Hassan Abbas,2012

Faraday’s Law  - - B(Outward flux) U F Induced (inward flux) © Mamdouh Hassan Abbas,2012

Faraday’s Law  © Mamdouh Hassan Abbas,2012

Faraday’s Law  © Mamdouh Hassan Abbas,2012

Outline  Mathematical Operators  Fundamental Laws of Electro and Magneto Statics  Faraday’s Experiment  Lenz’s Law  Faraday’s Law  Induced Electric Field  Electric Field in terms of Potential Functions  Ampere’s Law and the Continuity Equation  Retarded Potentials © Mamdouh Hassan Abbas,2012

Induced Electric Field  © Mamdouh Hassan Abbas,2012

Induced Electric Field  Faraday’s law states that a changing magnetic field induces an electric field.  The induced electric field is non- conservative. © Mamdouh Hassan Abbas,2012

Outline  Mathematical Operators  Fundamental Laws of Electro and Magneto Statics  Faraday’s Experiment  Lenz’s Law  Faraday’s Law  Induced Electric Field  Electric Field in terms of Potential Functions  Ampere’s Law and the Continuity Equation  Retarded Potentials © Mamdouh Hassan Abbas,2012

Electric Field in terms of Potential Functions  Electric scalar potential © Mamdouh Hassan Abbas,2012

Electric Field in terms of Potential Functions  © Mamdouh Hassan Abbas,2012

Outline  Mathematical Operators  Fundamental Laws of Electro and Magneto Statics  Faraday’s Experiment  Lenz’s Law  Faraday’s Law  Induced Electric Field  Electric Field in terms of Potential Functions  Ampere’s Law and the Continuity Equation  Retarded Potentials © Mamdouh Hassan Abbas,2012

Ampere’s Law and the Continuity Equation  Q S n J © Mamdouh Hassan Abbas,2012

Ampere’s Law and the Continuity Equation  © Mamdouh Hassan Abbas,2012

Ampere’s Law and the Continuity Equation  Conduction current Displacement current © Mamdouh Hassan Abbas,2012

Ampere’s Law and the Continuity Equation  Continuity equation is satisfied © Mamdouh Hassan Abbas,2012

Ampere’s Law and the Continuity Equation  © Mamdouh Hassan Abbas,2012

Ampere’s Law and the Continuity Equation  Displacement current is the type of current that flows between the plates of a capacitor.  Displacement current is the mechanism which allows electromagnetic waves to propagate in a non-conducting medium. © Mamdouh Hassan Abbas,2012

Ampere’s Law and the Continuity Equation  Current in a capacitor © Mamdouh Hassan Abbas,2012

Outline  Mathematical Operators  Fundamental Laws of Electro and Magneto Statics  Faraday’s Experiment  Lenz’s Law  Faraday’s Law  Induced Electric Field  Electric Field in terms of Potential Functions  Ampere’s Law and the Continuity Equation  Retarded Potentials © Mamdouh Hassan Abbas,2012

Retarded Potentials  The retarded potential formula describes the potentials due to time varying currents or charges distributions. © Mamdouh Hassan Abbas,2012

Retarded Potentials

© Mamdouh Hassan Abbas,2012 Retarded Potentials

© Mamdouh Hassan Abbas,2012 Retarded Potentials

© Mamdouh Hassan Abbas,2012 Retarded Potentials

© Mamdouh Hassan Abbas,2012 Retarded Potentials

© Mamdouh Hassan Abbas,2012 Retarded Potentials

© Mamdouh Hassan Abbas,2012 Retarded Potentials

© Mamdouh Hassan Abbas,2012 Retarded Potentials

© Mamdouh Hassan Abbas,2012 Retarded Potentials

© Mamdouh Hassan Abbas,2012 Retarded Potentials

 Modeling Source Observer (Origin) © Mamdouh Hassan Abbas,2012

Retarded Potentials  Retarded potentials Variable Constant © Mamdouh Hassan Abbas,2012