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lecture11

# lecture11 - Lecture 11 Faraday's Law and Electromagnetic...

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1 ECE 303 – Fall 2007 – Farhan Rana – Cornell University Lecture 11 Faraday’s Law and Electromagnetic Induction and Electromagnetic Energy and Power Flow In this lecture you will learn: • More about Faraday’s Law and Electromagnetic Induction • The Non-uniqueness of Voltages in Magnetoquasistatics • Electromagnetic Energy and Power Flow • Electromagnetic Energy Stored in Capacitors and Inductors • Appendix (some proofs) ECE 303 – Fall 2007 – Farhan Rana – Cornell University Faraday’s Law Revisited a d H t a d B t s d E o r r r r r r . . . ∫∫ = ∫∫ = µ A closed contour B Faraday’s Law : The line integral of E- field over a closed contour is equal to –ve of the time rate of change of the magnetic flux that goes through any arbitrary surface that is bounded by the closed contour Important Note: In electroquasistatics the line integral of E-field over a closed contour was always zero In magnetoquasistatics this is NOT the case ( ) = = 0 . . s d s d E r r r φ

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2 ECE 303 – Fall 2007 – Farhan Rana – Cornell University Electromagnetic Induction and Kirchoff’s Voltage Law Now consider a circuit through which the magnetic flux is changing with time (Kirchoff’s voltage law is violated) + + - - dt d a d B t s d E λ = ∫∫ = r r r r . . dt d V R I R I s d E λ = + = 2 1 . r r 1 R 2 R I ( ) ( ) dt d R R R R V I λ 2 1 2 1 1 + + = V + - ( ) t λ + + - - 1 R 2 R I V + - Kirchoff’s voltage law comes from the electroquasistatic approximation: 0 . = s d E r r 0 . 2 1 = + = V R I R I s d E r r ( ) 2 1 R R V I + = ECE 303 – Fall 2007 – Farhan Rana – Cornell University Lenz Law + + - - Suppose an induced current I is flowing through the wire: ( ) dt d R R I λ 2 1 1 + = 1 R 2 R I The induced current in the wire produces its own magnetic field Lenz Law is just an easy way to remember in which direction the induced current flows The law states that the induced current will flow in a direction such that its own magnetic field opposes the time variation of the magnetic field that produced it Example: Suppose the magnetic flux through the wire loop shown above was increasing with time (so that d λ / dt > 0). Lenz would tell us that the induced current would flow in the clockwise direction so that its own magnetic field would oppose the increasing magnetic flux through the loop In the equation above this fact comes out from the negative sign on the right hand side ( ) t λ
3 ECE 303 – Fall 2007 – Farhan Rana – Cornell University Non-Uniqueness of Voltages in Magnetoquasistatics - I + + - - ( ) dt d R R I s d E λ = + = 2 1 . r r 1 R 2 R I 2 V 1 V Question: What is the voltages difference V 2 - V 1 ? One may be tempted to write ……. 1 1 2 2 2 1 V R I V V R I V = = 0 0 1 2 = = V V I ….. which cannot be correct since we know that: We have: What went wrong? Our usual concepts of circuit theory and potentials which are based on conservative E-fields are not valid when time varying magnetic fields are present ( ) dt d R R I λ 2 1 1 + = ( ) t λ

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