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					                               Electromagnetic Theory

                                                     Mohammad Imran Aziz
                                                       Assistant Professor
                                                       Physics Department
                                              Shibli National Post Graduate College
                                           Azamgarh,India.(aziz_muhd33@yahoo.co.in)



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                           3 Electromagnetic Theory and Light

             Light possesses both wave-particle manifestations.
             Classical electrodynamics based on Maxwell’s electromagnetic theory unalterably leads to
             the picture of a continuous transfer of energy by way of electromagnetic waves.
             Quantum electrodynamics describes electromagnetic interaction and the transport of
             energy in terms of massless elementary “particles” known as photons, which are localized
             quanta of energy.
             One of the basic tenets of quantum mechanics is that both light and material objects each
             display both wave-particle properties.
             In physical optics light is treated as an electromagnetic wave.


             3.1 Maxwell’s equations
                 The simplest statement of Maxwell’s equations governs the behavior of the electric
             and magnetic fields in free space.
                     Maxwell’s equations are generalization of experimental results.


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                                                           B 
                                                             
                                         c E  dl   A t dS ,                                             (3.1)
                                             


                                                               E 
                                                                
                                         c B  dl  0 0 A t dS ,                                         (3.2)
                                             


                                         A E  dS  0,                                                        (3.3)
                                               


                                          B  dS  0                                                           (3.4)
                                               
                                             A



            where  0 and  0 are the permeability and the electric permittivity of free space,
            respectively.
                It should be noted that except for a multiplicative scalar, the electric and magnetic
            fields appears in above equations with a remarkable symmetry. The mathematical
            symmetry implies a good deal of physical symmetry.
                Maxwell’s equations tell us that a time-varying magnetic field generates an
            electric field and a time-varying electric field generates a magnetic field.
              Maxwell’s equations above can be written in differential form by using following
            two theorems from vector calculus.
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                              Gauss divergence theorem

                               F  dS    F dV                                  (3.5)
                                               
                                   A               V

                              and Stocke theorem

                               F  dl     F  dS                                 (3.6)
                                               
                               C               A


            By applying theorem (3.5) to Eqs. (3.1) and (3.2) and applying theorem (3.6) to Eqs
            (3.3) and (3.4), we obtain the following differential equations

                                         B
                                           
                                E                                                  ( 3 .7 )
                                    
                                         t
                                             E
                                              
                                B   0 0                                          ( 3 .8 )
                                    
                                             t
                               E  0                                              ( 3 .9 )
                                  

                               B  0                                              ( 3 . 10 )
                                  


                                     i              j     k
                                                         

                               E                                             (3.11)
                                                       
                                     x            y     z
                                     Ex            Ey     Ez
                                   E   E  E
                                E  x  y  z                                     (3.12)
                                     x   y Z


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                The consequent equations for free space are in detail as follows:

                                         E z E y     B
                                                    x,        (i )
                                          y   z        t
                                             

                                         E x E z     B
                                                    y,        (ii )                       (3.13)
                                          z   x       t
                                             

                                         E y E x     B
                                                    z,        (iii )
                                          x   y        t
                                             

                                         Bz B y           E
                                                    0 0 x ,       (i )
                                         y    z            t
                                             

                                         Bx Bz           E
                                                    0 0 y ,       (ii )                   (3.14)
                                          z   x            t
                                             

                                         B y Bx           E
                                                    0 0 z ,       (iii )
                                          x   y            t
                                             

                                         E x E y E z
                                                            0,                               (3.15)
                                          x   y     z
                                                  

                                         Bx B y Bz
                                                           0,                                   (3.16)
                                          x   y     z
                                                  




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            3.2 Electromagnetic waves
                Maxwell’s equations for free space can be manipulated into the form of two vector
            expressions:

             Taking the curl of Eq. (3.7)

                                       (  E )   (  B )                 (3.17)
                                                              
                                                            t
                                     L. H .S  (  E )   2 E     2 E
                                                               Eq.( 3.9 )      
                                                                      
                                                                2E
                                                                  
                                               Eq.( 3.8 )
                                     R.H .S    0 0 2
                                                                 t
                                                    

                                     We have
                                                      2
                                                    E
                                                          
                                     2 E  0 0 2                         (3.18)
                                         
                                                     t
                                     Similarly we have
                                                   2B
                                                          
                                     2 B  0 0 2                         (3.19)
                                         
                                                    t




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                  The Laplacian,  , operates on each component of E and B , so that the two vector
                                                                         
                                             2
                                                                                 
                  equations (3.18) and (3.19) actually represent a total of 6 scalar equations. One of
                  these expressions, in Cartesian coordinates, is

                                          2 Ex  2 Ex  2 Ex         2 Ex
                                                                0 0 2                               (3.20)
                                          x 2   y 2   z 2           t
                                                     

                Each component of the electromagnetic field ( E , E , E , B , B , B ) therefore
                                                               x   y   z   x   y   z
                obeys the scalar differential wave equation
                                                            2
                                         2  2  2   1 
                                                                ,                                      (3.21)
                                        x 2 y 2 z 2 v 2 t 2
                                                     

                  provided that
                                                     1
                                           v              .                                 (3.22)
                                                     0 0
                If we substitute the values of  0 and  0 into Eq. 3.22, the predicted speed of all
                electromagnetic waves travelling in free space would then be c= 3 x 108 m/s. This
                theoretical value was in remarkable agreement with the previously measured speed
                of light.


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                   The experimentally verified transverse character of light should be explained within the
                  context of the electromagnetic theory. To that end, consider the fairly simple case of a
                  plane wave propagating in the positive x-direction and write as E  E ( x, t ) . Eq. (3.15) is
                                                                                     

                  reduced to
                     E x
                          0                (3.23)
                      x
                  For a progressive wave, the solution of (3.23) is Ex=0. So the electric component
                  must be perpendicular to the propagation direction, x. Let’s orient the coordinate axes
                  so that the electric field is parallel to the y-axis: E  E ( x, t ) j . From Eq. (3.13
                                                                                      
                                                                             y
                  III), it follows that
                     E y          Bz
                                       .                 (3.24)
                      x            t
                             

                This implies that the time-dependent B-field
                can only have a component in the z-direction.
                Clearly then, in free space, the plane
                electromagnetic wave is indeed transverse, as
                shown in Fig. 3.1. Now let’s write

                                                                                                Fig. 3.1 Field configuration in a plane
                 E y ( x, t )  E0 y cos[ (t  x / c )   ],                   (3.25)
                                                                                                harmonic electromagnetic wave.

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                      The associated magnetic field can be
                  found by directly integrating Eq. (3.25), that
                  is,
                                     E y
                        Bz                dt
                                      x
                                    E 0 y
                             
                                       c      sin[  ( t  x      c )   ]dt

                              1
                                E 0 y cos[  ( t  x c )   ]
                              c
                             

                              E
                              y.                     (3.26)
                               c
                                                                                                      Fig. 3.2 Orthogonal harmonic E-field
                                                                                                      and B_field.
                      Clearly, E and Bz have the same time
                                  y
                   dependence, E  j ˆE y ( x, t ) and B  kBz ( x, t ) are in phase at all points inspace.
                                                        
                                                            ˆ
                   Moreover, E and B are mutually perpendicular, and their cross-product, E  B,
                                                                                                         

                   points in the propagation direction, as shown in Fig.3.2.
                      It should be noted that plane waves are not only solutions to Maxwell’s
                   equations. As we saw in the previous chapter, the differential wave equation allows
                   many solutions including spherical waves.

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          3.3 Energy

                  Energy density, u , which is the radiant energy per unit volume, is given by
                     u  0E 2                    (3.27)                                           1 2
                                                                               or           u        B .       (3.28)
                From Eqs. (3.27) and (3.28), we have
                                                                                                   0
                              1
                     u           EB                               (3.29)
                             0 c
              The amount of energy transported during a unit time and through a unit area
              perpendicular to the transport direction is (suppose the wave travels through an area
              of A and with a speed of c and with a time duration of t )
                          u ( c t A)        1
                     S                uc     EB                              (3.30)
                              t A           0
                  The corresponding vector is called Poynting vector:
                     1  
                    S    E  B.                                      (3.31)
                       0
                                                                                                                         2
                It’s along the wave propagation direction. Its SI unit is watt per square meter (W m ).




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                     E and B are so closely coupled to each other that we need to deal with only one of
                     them. Using B  E c from Eq. 3.26, we can rewrite Eq. 3.30 as
                                                                 1 2
                                                         S         E   0cE 2                                          (3.32)
                                                                c0
                       The time-averaged value of the magnitude of the Poynting vector, symbolized by
                   S , is a measure of the significant quantity known as the irradiance, I .
                                                              1     2
                                            I  S   0c E 2   0cE0                                           (3.33)
                                                              2
                  Where E0 is the peak magnitude of E. Within a linear, homogeneous, isotropic
                  medium, the expression for the irradiance becomes
                                                    1
                                    I   v E 2   vE0    2
                                                                        (3.34)
                                                    2
                                                                                 2
                  For a point light source, its irradiance is proportional to 1 r . This is well-known
                  inverse-square law. Fig. 3.3 shows that a point source emits electromagnetic
                  waves uniformly in all directions. Let us assume that the energy of the waves is

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            conserved as they spread from the source. Let
            us also center an imaginary sphere of radius r
            on the source, as shown in Fig. 3.3. If the power
            of the source is Ps , the irradiance I at the
            sphere must then be

                               Ps
                        I        2
                                    . (3.35)
                              4r
                                                                                                       Fig. 3.3 A point source emits light
                                                                                                       isotropically.



          In Quantum theory, light possesses quantum energy
                     h   hc /   12400 /                 ( 3 . 36 )
                                        is
                    is in unit of ev,  in unit of angstrom, h is Plank constant.




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            3.4 Electromagnetic-photon spectrum
               Although all forms of electromagnetic radiation propagate with the same speed in
            vacuum, they differ in frequency and wavelength. Fig. 3.4 plots the vast electromagnetic
            spectrum. The frequency range for whole electromagnetic spectrum is from a few Hz to
            1022 Hz. The corresponding wavelength range is from many kilometers to 10-14 m.
                  Radiofrequency waves: a few Hz to 109 Hz
                  Microwaves: 109 Hz to about 3 x 1011 Hz.

                  Infrared: 3 x 1011 Hz to 4 x 1014 Hz.
                         The infrared (IR) is often subdivided into four regions: the near IR (780-
                           3000 nm), the intermediate IR (3000-6000 nm), the far IR (6000-15,000 nm),
                           and the extreme IR (15,000 nm-1.0 mm).
                  Light: 3.84 x 1014 to 7.69 x 1014 Hz.
                          An narrow band of electromagnetic waves could be seen by human eye. Color is
                           not a property of light itself but a manifestation of the electrochemical sensing
                           system-eye, nerves, and brain.


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                         Fig. 3.4 Electromagnetic-photon spectrum.



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                     Ultraviolet: 8 x 1014 Hz to 3.4 x 1016 Hz.
                     X-rays: 2.4 x 1016 Hz to 5 x 1019 Hz.
                     Gamma rays: 5 x 1019 Hz to 2.5 x 1033 Hz.



                                  Table 3.1. Frequency and vacuum wavelength ranges for
                                  various colors




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                                                                                                                             120


                                                                                                                                                                  GaInP2
                                                                                                                             100




                                                                                                 Emission intensity (a.u.)
                                                                                                                             80


                                                                                                                             60


                                                                                                                             40


                                                                                                                             20


                                                                                                                              0
                                                                                                                              590   600   610   620   630   640   650   660   670

                                                                                                                                                Wavelength (nm)



                      Fig. 3.5 Spectra of sunlight and the light from                                                          Fig. 3.6 Emission spectrum of GaInP2
                      a tungsten lamp.                                                                                         semiconductor under excitation of a He-
                                                                                                                               Cd laser..




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             3.5 Light in matter
             3.5.1 Dispersion
                   In a homogeneous, isotropic dielectric, the phase velocity of light propagation becomes

                                           v 1          .                                          (3.37)

                  The ratio of the speed of electromagnetic wave in vacuum to that in matter is known as
                  the absolute index of refraction n and is given by

                                               c
                                        n                .                                                (3.38)
                                                    
                                               v
                                                 
                                                    0 0


               For most materials,              / 0    is generally equal to 1. So, the expression of n becomes


                                n   / 0 .                                                (3.39)

               Actually, n is frequency-dependent, so called dispersion. When a dielectric is
               subject to an applied electric field E, the internal charge distribution is distorted,
               which generates electric dipole moment p=Lq, with L the position vector from the
               negative charge -q to the positive charge q. The dipole moment per unit volume is
               called the electric polarization P. with
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                              P  (   0 ) E                    (3.40)
                                            


       Fig. 3.8 shows the dipole formation and a oscillator
       model for the vibration of electrons under an E-field.
       The negative electrons are fastened to a stationary
       positive nucleus. The natural frequency of the spring
       is 0  k / m with k and m being the spring constant
       and the electron mass. The force (FE) exerted on an
       electron of charge q by a harmonic wave E(t) with
       frequency  is
                 FE  qE (t )  qE0 cos(t )                        (3.41)

         Newton’s second law provides the equation of
         motion with the second term the restoring force.
                                       2d 2x
            qE0 cos(t )  m x  m 2  0                             (3.42)
                                         dt
                                                                                                            Fig. 3.8 (a) Distortion of the
            Let          x (t )  x0 cos(t )                   we have
                                                                                                            electron cloud in response to an
                        q/m                                                                                 applied E-field. (b) The
            x(t )                  E (t )                     (3.43)
                      ( 02   2 )                                                                         mechanical oscillator model for an
            For medium with electron density N                                                              isotropic medium
            the electric polarization P is
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                        q 2 N / mE ( t )
               P (t )                                       from       Eq .( 3 . 40 )
                         (  02   2 )
                                                   q2N / m
                  0  p (t ) / E (t )   0 
                                                 (  02   2 )
               From        Eq .( 3 . 39 )
                 2          Nq 2     1
               n ( )  1       ( 2       )                                  ( 3 . 44 )
                             0m  0   2


          Eq. (3.44) indicates dispersion.
          When   0 n>1; When   0
          n<1. Fig 3.9 shows dispersion
          of materials.




       Fig. 3.9 Index of refraction versus
       wavelength and frequency for several
       important optical crystals.


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Description: basic concept of electromagnetic theory