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Electromagnetism INEL 4152 CH 9 Sandra Cruz-Pol, Ph. D. ECE UPRM Mayagüez, PR In summary Stationary Steady Charges currents Time-varying currents Electrostatic fields Magnetostatic fields Electromagnetic (waves!) Cruz-Pol, Electromagnetics UPRM Outline Faraday’s Law & Origin of emag Maxwell Equations explain waves Phasors and Time Harmonic fields Maxwell eqs for time-harmonic fields 9.2 Faraday’s Law Electricity => Magnetism In 1820 Oersted discovered that a steady current produces a magnetic field while teaching a physics class. This is what Oersted discovered accidentally: ò H × dl = ò J × dS L Cruz-Pol, Electromagnetics UPRM s Would magnetism would produce electricity? Eleven years later, and at the same time, (Mike) Faraday in London & (Joe) Henry in New York discovered that a time-varying magnetic field would produce an electric current! Vemf d N dt ¶ ò E × dl = -N ò ¶t B × dS L s Cruz-Pol, Electromagnetics UPRM Electromagnetics was born! This is Faraday’s Law the principle of motors, hydro-electric generators and transformers operation. *Mention some examples of em waves Cruz-Pol, Electromagnetics UPRM Faraday’s Law For N=1 and B=0 Vemf Cruz-Pol, Electromagnetics UPRM d N dt 9.3 Transformer & Motional EMF Three ways B can vary by having… 1. 2. 3. A stationary loop in a t-varying B field A t-varying loop area in a static B field A t-varying loop area in a t-varying B field Cruz-Pol, Electromagnetics UPRM 1. Stationary loop in a time-varying B field Cruz-Pol, Electromagnetics UPRM 2. Time-varying loop area in a static B field Cruz-Pol, Electromagnetics UPRM 3. A t-varying loop area in a t-varying B field Cruz-Pol, Electromagnetics UPRM Transformer Example Cruz-Pol, Electromagnetics UPRM 9.4 Displacement Current, Jd Maxwell noticed something was missing… And added Jd, the displacement current 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 9.4 Maxwell’s Equation in Final Form Summary of Terms E = electric field intensity [V/m] D = electric field density H = magnetic field intensity, [A/m] B = magnetic field density, [Teslas] J = current density [A/m2] 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 D H J t Gauss’s Law for E field. Gauss’s Law for H field. Nonexistence of monopole Faraday’s Law L E dl t s B dS D H dl J L s t dS Cruz-Pol, Electromagnetics UPRM Ampere’s Circuit Law Maxwell’s Eqs. Also the equation of continuity Maxwell v J t D t added the term 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 Relations & B.C. Cruz-Pol, Electromagnetics UPRM 9.6 Time Varying Potentials We had defined Electric Scalar & Magnetic Vector potentials: Related to B as: To find out what happens for time-varying fields Substitute into Faraday’s law: Cruz-Pol, Electromagnetics UPRM Electric & Magnetic potentials: If we take the divergence of E: We have: Taking the curl of: we get Cruz-Pol, Electromagnetics UPRM & add Ampere’s Electric & Magnetic potentials: If we apply this vector identity We end up with Cruz-Pol, Electromagnetics UPRM Electric & Magnetic potentials: We use the Lorentz condition: To get: Which are both wave equations. and: Cruz-Pol, Electromagnetics UPRM 9.7 Time Harmonic Fields Phasors Review Time Harmonic Fields Definition: is a field that varies periodically with time. Ex. Sinusoid Let’s review Phasors! Cruz-Pol, Electromagnetics UPRM 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 COMPLEX NUMBERS: Given a complex number z z x jy re j r r cos jr sin where r | z | x y is the magnitude 2 tan 1 2 y is the angle x Cruz-Pol, Electromagnetics UPRM Review: Addition, 1/ j = j= 1/ Subtraction, Multiplication, Division, Square examples : Root, Complex Conjugate e j 45o / j= e j 45o - j= 3e j 90 o 2e 2e Cruz-Pol, Electromagnetics UPRM +j= + j 45o + j 45o = +10e + j 90 o = For a Time-varying phase f = wt + q Real and imaginary parts are: Re{re } = r cos(wt + q ) jf Im{re }= rsin(wt + q ) jf Cruz-Pol, Electromagnetics UPRM PHASORS a sinusoidal current I(t) = Io cos(wt + q ) equals the real part of I o e j e jt For j I e The complex term o which results from dropping the time factor e jt is called the phasor current, denoted by I s (s comes from sinusoidal) Cruz-Pol, Electromagnetics UPRM Advantages of phasors Time derivative in time is equivalent to multiplying its phasor by j A jAs t Time integral is equivalent to dividing by the same term. As At j Cruz-Pol, Electromagnetics UPRM How to change back from Phasor to time domain The phasor is 1. multiplied by the time factor, e jt, 2. and taken the real part. A Re{ As e jt } Cruz-Pol, Electromagnetics UPRM 9.7 Time Harmonic Fields Time-Harmonic fields (sines and cosines) The wave equation can be derived from Maxwell equations, indicating that the changes in the fields behave as a wave, called an electromagnetic wave or field. Since any periodic wave can be represented as a sum of sines and cosines (using Fourier), then we can deal only with harmonic fields to simplify the equations. Cruz-Pol, Electromagnetics UPRM Maxwell Equations for Harmonic fields (phasors) 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 Ñ ´ H = s E + jwe E D H J t * (substituting Faraday’s Law Ampere’s Circuit Law D E andCruz-Pol, H Electromagnetics B) UPRM Ex. Given E, find H E = Eo cos(t-bz) ax Cruz-Pol, Electromagnetics UPRM Ex. 9.23 In free space, 50 8 E cos(10 t kz) V / m Find k, Jd and H using phasors and Maxwells eqs. Recall: é ˆ r̂ r f ê 1ê ¶ ¶ Ñ´ A = ê r ê ¶r ¶f ê Ar r Af Cruz-Pol, Electromagnetics ë UPRM ẑ ¶ ¶z Az ù ú ú ú ú ú û