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Electromagnetism INEL 4151 Sandra Cruz-Pol, Ph. D. ECE UPRM Mayagüez, PR Electricity => Magnetism In 1820 Oersted discovered that a steady current produces a magnetic field while teaching a physics class. Cruz-Pol, Electromagnetics UPRM Would magnetism would produce electricity? Eleven years later, and at the same time, Mike Faraday in London and Joe Henry in New York discovered that a time-varying magnetic field would produce an electric current! Vemf d N dt L E dl t s B dS Cruz-Pol, Electromagnetics UPRM Electromagnetics was born! This is the principle of motors, hydro-electric generators and transformers operation. This is what Oersted discovered accidentally: D L H dl s J t dS *Mention some examples of em waves Cruz-Pol, Electromagnetics UPRM Cruz-Pol, Electromagnetics UPRM Electromagnetic Spectrum Cruz-Pol, Electromagnetics UPRM Some terms E = electric field intensity [V/m] D = electric field density H = magnetic field intensity, [A/m] B = magnetic field density, [Teslas] Cruz-Pol, Electromagnetics UPRM Maxwell Equations in General Form Differential form Integral Form D v D dS v dv s B 0 v B dS 0 s B E t L E dl t s B dS D H J t D H dl J L s t dS Cruz-Pol, Electromagnetics UPRM Gauss’s Law for E field. Gauss’s Law for H field. Nonexistence of monopole Faraday’s Law Ampere’s Circuit Law Maxwell’s Eqs. v J t Also the equation of continuity D Maxwell added the term t to Ampere’s Law so that it not only works for static conditions but also for time-varying situations. This added term is called the displacement current density, while J is the conduction current. Cruz-Pol, Electromagnetics UPRM H J For static fields we had: But the divergence of curl of ANY vector =0 H 0 J The continuity of the current requires: v J 0 This doesn’t work for time-varying fields! t to define: H J J d We need Now when we take the divergence of curl H 0 J Jd Therefore: J J d Cruz-Pol, Electromagnetics, UPRM For dinamic fields we had: v D J d J Therefore: t t D t D Jd t Substituting in curl of H D H J t This is Maxwell’s equation for Ampere’s Law Cruz-Pol, Electromagnetics, UPRM Maxwell put them together And added Jd, the displacement current. If we didn’t have it, then: H dl J dS I L enc I S1 H dl J dS 0 L S1 I L S2 d dQ L H dl S J d dS dt S D dS dt I 2 2 S2 At low frequencies J>>Jd, but at radio frequencies both Cruz-Pol, Electromagnetics terms are comparable in magnitude. UPRM Moving loop in static field When a conducting loop is moving inside a magnet (static B field, in Teslas), the force on a charge is F Qu B F Il B Cruz-Pol, Electromagnetics UPRM Encarta® Phasors & complex #’s Working with harmonic fields is easier, but requires knowledge of phasor, let’s review complex numbers and phasors Cruz-Pol, Electromagnetics UPRM Maxwell Equations for Harmonic fields Differential form* DE v v Gauss’s Law for E field. BH 0 Gauss’s Law for H field. No monopole 0 E jH E B t H J jE D H J t * (substituting Faraday’s Law Ampere’s Circuit Law D E andCruz-Pol, H Electromagnetics B) UPRM