FWLec17 - A. F. Peterson: Notes on Electromagnetic Fields &...

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 11/04 Fields & Waves Note #17 Maxwell’s Equations Objectives: Present Ampere’s Law as it was extended by Maxwell to incorporate a displacement current term. Summarize the entire set of Maxwell’s equations and related equations. Show that Maxwell’s equations admit waves as solutions and discuss the characteristics of electromagnetic waves. James Clerk Maxwell After Michael Faraday discovered in 1831 that a change in magnetic flux caused an electrical effect, the equations describing electricity and magnetism remained basically the same for about 30 years. Then, in the 1860s, James Clerk Maxwell extended Ampere’s Circuital Law to include a new term that took into account a time-varying electric flux and the associated magnetic effect. Maxwell organized the existing knowledge base into a collection of about 20 different equations describing electricity and magnetism. Using these equations, he predicted the existence of electromagnetic waves, and also postulated that light was of an electromagnetic nature. Because of Maxwell’s discoveries, today the primary equations of electromagnetic fields are named after him. In 1888, after Maxwell’s death, Hertz demonstrated that electromagnetic waves could be launched and received using simple antennas. Around 1900, Oliver Heaviside simplified Maxwell’s equations into the 4 primary equations we still work with today. Heaviside also developed the vector notation for divergence and curl that we use today, more than 100 years later! In this Note, we review these developments and walk through a mathematical exercise that may be similar to the path James Clerk Maxwell followed to obtain electromagnetic waves. Ampere’s Law as modified by Maxwell The integral form of Ampere’s Circuital Law, as modified by James Clerk Maxwell in the 1860s, is given by mmf = = + ÚÚ ÚÚ Ú Hd Jd S d dt Dd S SS C l (17.1) where S is an open surface that terminates on the closed contour C , and the right-hand screw convention (Note #13) is used to link the direction of dS with the orientation of C . The abbreviation “mmf” was originally used to denote “magneto-motive force,” which actually isn’t a force at all. In fact, the units of (17.1) are Amperes. We observe that if the fields are static, the new term on the right side drops out, leaving Ampere’s Law as it was presented in Note #13.
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A. F. Peterson: Notes on Electromagnetic Fields & Waves 11/04 The term on the right side of (17.1), I d dt Dd S S displacement = ÚÚ (17.2) was introduced by Maxwell and given the name “displacement” current. Observe that the units of electric flux are Coulombs, and the time derivative converts the units to Amperes, so the term is equivalent to a current. Maxwell reasoned that in certain parts of an electrical system, a time varying electric field played the role of a current.
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This note was uploaded on 01/27/2011 for the course ECE 3025 taught by Professor Citrin during the Spring '08 term at Georgia Institute of Technology.

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FWLec17 - A. F. Peterson: Notes on Electromagnetic Fields &...

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