As educators, David H. Staelin, Ann W. Morgenthaler, and Jin Au Kong saw a need for a book presenting electromagnetic theory and applications in a clear, compact, and. Electromagnetic Waves: Solutions Manual to Accompany K6636, 2000, Umran Inan, Aziz S. Inan,, 617, Prentice Hall PTR, 2000.

Electromagnetic Waves! M A N M O H A N D A S H, P H Y S I C I S T, T E A C H E R Physics for ‘Engineers and Physicists’ Lecture - 3 Electromagnetic Waves • I remember its close to 2 decades ago, when I saw the Maxwell's Equations and see those charge and currents and think to myself; if these things vanish why still we have the fields, I mean we read in the text books the sources of charges is what produces the electric fields and the currents are what produces magnetic fields, why the fields survive when the charges vanish. I was still thinking like the +2 (PU/American Highschool Senior) student that I was, because that's when I studied the fields being produced by the charges and currents.

Somewhere down the line, when I became well versed with what these equations are actually doing, I understood; the fields are produced by charges and currents, that are not necessarily explicitly seen in these equations, but the charges and currents that interact with these fields are explicitly placed in the equations. Its like we are produced by our parents who are not visible much in our lives although they impact us, but its our girlfriends and boyfriends who impact us explicitly, trying to pull the strings here and there! It was just a passe analogy. But Physics is often fun if you imagine how they correspond to our day to day moorings.

Lets discover what electromagnetic phenomena are entailed by the Maxwell’s equations. • “A concise course of important results” Lectures delivered around 12.Nov.2009 + further content developments this week; 28 Aug - 02 Sept 2015! • Electromagnetic Waves Electromagnetic Waves are 3 dimensional propagation of vibrations of electric and magnetic fields. In the last two lectures we discussed what are vector fields. Electric fields and magnetic fields are vector fields.

>>Lecture-1 >Lecture-2 >00 1  emv0 1 x )1( 2 2 22 2       tv  The wave equations we obtained are vector equations which we can cast into individual scalar equations for 3 scalar components of the fields; x, y, z, for both fields; E and B. 0 0 0 2 z 2 00z 2 2 y 2 00y 2 2 x 2 00x 2             t E E t E E t E E    0,0,0 2 z 2 00z 2 2 y 2 00y 2 2 x 2 00x 2           t B B t B B t B B  SISI 104,1085.8 7 0 12 0    • Wave equation in charge-free, non-conducting media Its easy to see the forms in which they pertain to charge-free but non-conducting media, eg in (1), (2)! Eps, mu; electro- magnetic properties of the media. In last slide we saw the wave equations for the freespace, by freespace we meant charge-free and current-free space which are essential conditions of vacuum. 0)2( 0)1( 2 2 2 2 2 2         t B B t E E       1 ) 1 (, 00 00        v c vcv There is a drop in speed of light when it enters any media from vacuum, speed of light in vacuua is maximum and drops by a factor eta known as refractive index of the media and the refractive index is related to eps and mu in freespace and in media.

• Wave eqn, charge-free, conducting media; Telegraph Eqn Sigma is called the conductivity; of the given medium. ‘Charge-free and Conducting’ media is a special case of the Maxwell’s Equations where we represent this condition by the equations on right! Ej     ,0 Lets take curl of the 3rd equation and interchange curl and time differentiation!

From the steady-state and time-varying Maxwell’s Equations given in slide 9, for any medium given by eps, mu and the conditions above, we have E t EB t B EBE                )4(,0)3(,0(2),0)1( • Wave equation; The Telegraph Equations! This leads to the telegraph equation in terms of E field, (5). We apply the same treatment on equation (4) and obtain the telegraph equation for the B field, (6).

With the step mentioned in last slide we obtain (3’); E t E B t B EE              )'4(,0 )( -)()'3( 2 The terms on the right hand side of telegraph equations are dissipative terms. T B t B B t E t E E                  2 2 2 2 2 2 )6(,)5( Telegraph Equations • Vector Potential and Scalar Potential In Lecture-1 we saw every vector field is associated with a scalar field, the gradient of the scalar is the vector and we called the scalar field as the potential.

We saw that for many vectors, there are potentials associated with each of them. Torrent Fahrenheit 451 Pdf. These vectors were always a space gradient (or any space derivative) of the scalar potentials. Hence the name vector field and (scalar) potential. When it’s a scalar whose gradient creates a vector field it’s a scalar potential and when it’s a vector whose curl (another way to create space derivative of a vector) creates a vector field the former vector is called a vector potential. • Magnetic Vector Potential In Lecture-1 we saw for every vector whose divergence vanishes the vector is necessarily a curl of another vector and for every vector that is a curl of another vector the divergence of the former vector vanishes. Since the divergence of the magnetic field is zero it implies from the above, magnetic field vector B is curl of another vector A, the magnetic vector potential.

FGthen0G 0GthenFGif 0;)(       if FSince potentialvectormagneticA 0)Cf(asCfAA AB0;B       Since • Electro-Magnetic Scalar Potential We take the 3rd of Maxwell equations (check slide 15), eqn (1) below, use the fact that the magnetic field is the curl of the vector potential A we just defined, in last slide. Interchange the order of the curl and the time derivative;!, t - - t 0;)(,0) t (,ABwith0, t E(1) potentialscalar A E A E fSince A E B                             We already saw in Lecture-1 that curl of any gradient is always 0. Phi, the scalar potential, is partly electric and partly magnetic potential.

• Gauge Transformations, Lorentz and Coulomb’s Gauge! We saw that vector potentials and scalar potentials that we just discussed are arbitrary, that is, there is not a single one of them. Raspberry Pi Mpeg 2 Cracked.

0;, 1..,; ),(,,; 2         AGaugeCoulomb tc AGaugeLorentzge ConditionGaugecalleddconstraineAandGauged trff t f fAAtionsTransformaGauge       Gauge Conditions are therefore a particularly chosen definitions of vector or scalar potentials. The Gauge Transformations are a change of the definition of these potentials, so that the E and B field stand unchanged, thus Physics stays unique and not arbitrary; • Wave Equation in terms of Scalar Potential We saw that vector potential and scalar potential, that we just discussed are now unique, given to a particular Gauge Condition. 0 11; 0 )(;0)(.0;,'x 2 2 2 2                tctc AGaugeLorentzApply t A or t A E ELawGaussEqnswellMaFreespace         Using this, we can write the wave equation in terms of the scalar potential, instead of just in E and B fields. 0 1 2 2     tc   wave equation in terms of the scalar potential • Wave Equation in terms of Vector Potential We saw that scalar potential has its own wave equation just as E and B fields did. Lets find the wave equation for the vector potential. 0][][ 0 )( )( 0)()(,0(1) 2 2 00 2 00 2 2 0000 2 0000                         t A A t A t A t AA t A t A t E B                Using Freespace Ampere- Maxwell Law (1), we can write the wave equation in terms of the vector potential. 2 2 2 2 1 t A c A      wave equation in terms of the vector potential 1st [term] is 0; Lorentz Gauge • Transverse nature of electromagnetic waves In the beginning we claimed that electromagnetic waves are transverse in nature, that is the propagation of the energy and wave as such, occurs in a direction perpendicular to the oscillations of the vector fields of E and B.

Lets see this. We see that the wave equations (1) are differential equations, 2nd order in space and 2nd order in time. Plane waves (2) and (3) given above are their solutions. Lets prove that E and B fields are perpendicular to each other as well to the wave propagation direction; k vector. Note that E is along e and B along b, the unit vectors.

)( 0 )( 02 2 00 2 ˆ),()3(,ˆ),()2(,0 ),( ),()1( trkitrki eBbtrBeEetrE t BE BE            • Transverse nature of electromagnetic waves e and b are thus unit vectors, constant in space and time. K is vector along which wave propagates, called wave vector and its magnitude is called as wave number and gives wavelength as well as momentum of the plane wave. Omega is the angular frequency and gives the time period and frequency of the wave, as well as the energy.

E, b, k form a right hand trio. E_zero and B_zero are the amplitudes or maximum value of the fields. Below we see e and k are perpendicular. 0ˆ0))(iEˆ(or0)(ˆ.ˆ;0ˆ),(ˆ)(ˆ0 )()(Use0;)ˆ(,0 )( 0 )( 0 )( 0 )( 0 )( 0       keekeeEe constiseeeEeeEe VAVAVAeEeE trkitrki trkitrki trki          0ˆ ke  e and k are perpendicular • Homework Homework; (1) prove the result we used in last slide; )( 0 )( 0 )(iE)( trkitrki ekeE      Homework; (2) prove that b and k are also perpendicualar to each other just like e and k. 0ˆ0  kbB  In the following slide we will prove from Maxwell’s relations, e, b, k are all mutually perpendicular in a right handed way.