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5. Plane Electromagnetic Waves Dr. Rakhesh Singh Kshetrimayum 1 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.1 Introduction Electromagnetic Waves Poynting vector Plane waves Polarization Lossless medium Lossy conducting medium Plane waves in various media Good conductor Good dielectric Fig. 5.1 Plane Waves 2 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves 5.2.1 What are plane waves? What are waves? Waves are a means for transferring energy or information from one place to another What are EM waves? Electromagnetic waves as the name suggests, are a means for transferring electromagnetic energy Why it is named as plane waves? Mathematically assumes the following form r r r j ( kr •rr −ωt ) F ( r , t ) = F0 e 3 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves r where k is the wave vector and it points in the direction of 4 wave propagation, r r is the general position vector, ω is the angular frequency, and r F 0 is a constant vector r F 0 denotes either an electric or magnetic field ( F is a notation for field not for the force) r For example, in electromagnetic waves, F 0 is either vector r r electric ( E 0 ) or magnetic field ( H 0 ) Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves In rectangular or Cartesian coordinate system r ) ) ) k = kxx + ky y + kzz r ) ) ) r = xx + yy + zz r r 2 2 2 ⇒ k = k • k = ( k x ) + ( k y ) + ( k z ) = ω 2 µε 2 Note that the constant phase surface for such waves r r ) ) ) ) ) ) k • r = ( k x x + k y y + k z z ) • ( xx + yy + zz ) = k x x + k y y + k z z = con tan t 5 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves defines a plane surface and hence the name plane waves Since the field strength is uniform everywhere it is also known as uniform plane waves A plane wave is a constant-frequency wave whose wavefronts (surfaces of constant phase) are infinite parallel planes of constant amplitude normal to the phase velocity vector For plane waves from the Maxwell’s equations, the following relations could be derived (see Example 4.3) r r r r r r r r r r k × E = ωµ H ; k × H = −ωε E ; k • E = 0; k • H = 0 6 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves Properties of a uniform plane wave: Electric and magnetic field are perpendicular to each other No electric or magnetic field in the direction of propagation (Transverse electromagnetic wave: TEM wave) The value of the magnetic field is equal to the magnitude of the electric field divided by η0 (~377 Ohm) at every instant (magnetic field amplitude is much smaller than the electric field amplitude) 7 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves The direction of propagation is in the same direction as Poynting vector The instantaneous value of the Poynting vector is given by E2/η0, or H2η0 The average value of the Poynting vector is given by E2/2η0, or H2η0/2 The stored electric energy is equal to the stored magnetic energy at any instant 8 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves 5.2.2 Wave polarization Polarization of plane wave refers to the orientation of electric field vector, which may be in fixed direction or may change with time Polarization is the curve traced out by the tip of the arrow representing the instantaneous electric field The electric field must be observed along the direction of propagation 9 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves Types of polarization Linear polarized (LP) LHCP 10 Circularly polarized (CP) RHCP Electromagnetic Field Theory by R. S. Kshetrimayum Elliptically polarized (EP) RHEP LHEP 3/19/2014 5.2 Plane waves If the vector that describes the electric field at a point in space varies as function of time and is always directed along a line which is normal to the direction of propagation the field is said to be linearly polarized If the figure that electric field trace is a circle (or ellipse), then, the field is said to be circularly (or elliptically) polarized 11 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves Besides, the figure that electric field traces is circle and anticlockwise (or clockwise) direction, then, electric field is also said to be right-hand (or left-hand) circularly polarized wave (RHCP/LHCP) Besides, the figure that electric field traces is ellipse and anticlockwise (or clockwise) direction, then, electric field is also said to be right-hand (or left-hand) elliptically polarized (RHEP/LHEP) 12 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves Let us consider the superposition of a x- linearly polarized wave with complex amplitude Ex and a y- linearly polarized wave with complex amplitude Ey, both travelling in the positive z-direction Note that Ex and Ey may be varying with time for general case so we may choose it for a particular instant of time Note that since the electric field is varying with both space and time 13 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves Easier to analyze at a particular instant of time first And add the time dependence later The total electric field can be written as r jφ E (z ) = (E x xˆ + E y yˆ )e − jβz = E x 0 e jφ x xˆ + E y 0 e y yˆ e − jβz ( ) Note Ex and Ey may be complex numbers and Ex0 and Ey0 are the amplitudes of Ex and Ey 14 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves φx and φy are the phases of Ex and Ey Putting in the time dependence and taking the real part, we have, r E (z , t ) = E x 0 cos(ωt − βz + φ x )xˆ + E y 0 cos(ωt − β z + φ y )yˆ A number of possibilities arises: Linearly polarized (LP) wave: If both Ex and Ey are real (say Ex = Eox and Ey = Eoy), then, r ELP ( z ) = (E x xˆ + E y yˆ )e − jβz = (E0 x xˆ + E0 y yˆ )e − jβz 15 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves Putting in the time dependence and taking the real part, we r have, ELP ( z , t ) = E0 x cos(ωt − β z )xˆ + E0 y cos(ωt − βz ) yˆ The amplitude of the electric field vector is given by r ELP (z , t ) = (E0 x )2 + (E0 y )2 cos(ωt − βz ) which is a straight line directed at all times along a line that makes an angle θ with the x-axis given by the following relation Ey Ex θ LP = tan −1 16 Electromagnetic Field Theory by R. S. Kshetrimayum E0 y −1 = tan E 0x 3/19/2014 5.2 Plane waves If Ex ≠ 0 and Ey = 0, we have a linearly polarized plane wave in x- direction r ELP ( z , t ) = Eox cos(ωt − βz )xˆ 17 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves Easier to fix space to see the polarization For a fixed point in space (say z=0), r ELP ( z , t ) z =0 = Eox cos(ωt )xˆ For all times, electric field will be directed along x-axis hence, the field is said to be linearly polarized along the x- direction 18 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves Fig. 5.2 (a) LP wave 19 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves Circularly polarized (CP) wave: Now consider the case Ex = j Ey = Eo, where Eo is real so that Ex = E0 e j 0 ; E y = E0 e −j π 2 ; r ERHCP = Eo ( xˆ − jyˆ )e− jβ z The time domain form of this field is (putting in the time dependence and taking the real part) r π ERHCP ( z , t ) = Eo [ xˆ cos(ωt − β z ) + yˆ cos(ωt − β z − )] 2 20 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves Note that x- and y-components of the electric field have the same amplitude but are 900 out of phase Let us choose a fixed position (say z=0), then, sin ωt −1 = tan tan (ωt ) = ωt cos ωt θ RHCP = tan −1 which shows that the polarization rotates with uniform angular velocity ω in anticlockwise direction for propagation along positive z-axis 21 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves An observer sitting at z=0 will see the electric field rotating in a circle and the field never goes to zero Since the fingers of right hand point in the direction of rotation of the tip of the electric field vector when the thumb points in the direction of propagation, this type of wave is referred to as right hand circularly polarized wave (RHCP wave) 22 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves Fig. 5.2 (b) RHCP wave x y 23 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves Elliptically polarized (EP) wave: Now, consider a more general case of EP wave, when the amplitude of the electric field in the x- and ydirections are not equal in amplitude and phase unlike CP wave, so that, r EEP (z ) = xˆ + Ae jφ yˆ e − jβz ( ) Putting in the time dependence and taking the real part, we have, 24 r E (z , t )EP = cos(ωt − β z )xˆ + A cos(ωt − β z + φ ) yˆ Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves If φ is in the upper half of the complex plane then the wave is LHEP whereas φ is in the lower half of the complex plane, then the wave is RHEP Let us choose a fixed position (say z=0) like in the CP case, then, r EEP z =0 = cos (ωt ) xˆ + A cos (ωt + φ ) yˆ Some particular cases: 25 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves (a) (b) (c ) (d ) 26 r A = 1, φ = 0; E = E0 cos (ωt )( xˆ + yˆ ) z =0 r A = 1, φ = π ; E = E0 cos (ωt )( xˆ − yˆ ) z =0 A = 1, φ = π 2 A = 1, φ = − r ; E π 2 z =0 r ; E ( LP ) ( LP ) = E0 {cos (ωt ) xˆ − yˆ sin (ωt )} z =0 = E0 {cos (ωt ) xˆ + yˆ sin (ωt )} Electromagnetic Field Theory by R. S. Kshetrimayum ( LHCP ) ( RHCP ) 3/19/2014 5.2 Plane waves (e ) (f) (g) ( h) 27 A = 3, φ = π 2 r ; E A = 0.5, φ = − π z =0 = E0 {cos (ωt ) xˆ − yˆ 3sin (ωt )} r ; E ( LHEP ) = E0 {cos (ωt ) xˆ + yˆ 0.5sin (ωt )} ( RHEP ) 2 r π π A = 1, φ = ; E = E0 cos (ωt ) xˆ + yˆ cos ωt + ( LHEP ) z =0 4 4 r π π A = 1, φ = −3 ; E = E0 cos (ωt ) xˆ + yˆ cos ωt − 3 ( RHEP ) z =0 4 4 z =0 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.2 Plane waves Fig. 5.2 (c) LHEP wave Direction of propagation Electric field x Magnetic field at each point is orthogonal to the electric field y 28 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.3 Poynting vector & power flow in EM fields The rate of energy flow per unit area in a plane wave is described by a vector termed as Poynting vector which is basically curl of electric field intensity vector and magnetic field intensity vector r r r* S = E×H The magnitude of Poynting vector is the power flow per unit area and it points along the direction of wave propagation vector 29 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.3 Poynting vector & power flow in EM fields The average power per unit area is often called the intensity of EM waves and it is given by r r r* 1 S avg = Re E × H 2 ( ) Let us try to derive the point form of Poynting theorem from two Maxwell’s curl equations r r ∂H ∇ × E = −µ ∂t 30 Electromagnetic Field Theory by R. S. Kshetrimayum r r ∂E r ∇× H =ε +J ∂t 3/19/2014 5.3 Poynting vector & power flow in EM fields From vector analysis, r r r r r r r r r r ∂H ∂E r ∇ • ( E × H ) = H • (∇ × E ) − E • (∇ × H ) = H • (− µ ) − E • (ε + J) ∂t ∂t We can further simplify r r ∂A 1 ∂ r r Q A• = A• A ∂t 2 ∂t ( ) r r µ ∂ r r ε ∂ r r r r ∴∇ • ( E × H ) = − (H • H ) − (E • E) − E • J 2 ∂t 2 ∂t Basically a point relation It should be valid at every point in space at every instant of time 31 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.3 Poynting vector & power flow in EM fields The power is given by the integral of this relation of Poynting vector over a volume as follows r r r r r r r µ ∂ r r ε = − ( H • H ) dv − ∇ • E × H dv = E × H • d s = S • d s ∫ ∫ ∫ ∫ ( V ) ( S ) S 2 ∂t V r r ∂ r r ( E • E )dv − ∫ E • Jdv 2 ∫ ∂t V V We can interchange the volume integral and partial derivative w.r.t. time r r ∂ 1 ∂ 1 2 2 2 S • d s = − µ H dv − ε E dv − σ E dv ∫S ∫ ∫ ∫ ∂t V 2 ∂t V 2 V 32 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.3 Poynting vector & power flow in EM fields This is the integral form of Poynting vector and power flow in EM fields Poynting theorem states that the power coming out of the closed volume is equal to the total decrease in EM energy per unit time i.e. power loss from the volume which constitutes of rate of decrease in magnetic energy stored in the volume rate of decrease in electric energy stored in the volume Ohmic power loss (energy converted into heat energy per unit time) in the volume 33 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.3 Poynting vector & power flow in EM fields Now going back to the last four points of plane waves: The direction of propagation is in the same direction as of Poynting vector The instantaneous value of the Poynting vector is given by E2//η0, or H2η0 The average value of the Poynting vector is given by E2/2η0, or H2η0/2 The stored electric energy is equal to the stored magnetic energy at any instant 34 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.3 Poynting vector & power flow in EM fields Let us assume a plane wave traveling in the +z direction in free space, then r r − jβ z r − jk z r E = E0 e = E0e 0 ; H = ) r z × E0 η0 e− j β z The instantaneous value of the Poynting vector: r r r r∗ r r − jβz zˆ × E0 jβz 1 r S = E × H = E0 e × e = E0 × zˆ × E0 η0 η0 r 2 r r r r r r E0 zˆ zˆ E0 • E0 − E0 E0 • zˆ zˆ E0 • E0 = = = ( ( ) ) ( ( )( ) ( η0 35 Electromagnetic Field Theory by R. S. Kshetrimayum ) ) η0 η0 3/19/2014 5.3 Poynting vector & power flow in EM fields o Note that the direction of Poynting vector is also in the z- direction same as that of the wave vector o The average value of the Poynting vector: r 2 r 2 r r r ∗ 1 E0 zˆ E0 zˆ 1 S avg = Re E × H = Re = 2 2 η0 2η 0 1 2 w = ε E o Stored Electric Energy: e 0 2 o Stored Magnetic Energy: ( ) 1 1 E2 1 ε0 2 1 2 wm = µ0 H = µ0 2 = µ0 E = ε 0 E 2 = we 2 2 η0 2 µ0 2 36 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.4 Plane waves in various media A media in electromagnetics is characterized by three parameters: ε, µ and σ 5.4.1 Lossless medium In a lossless medium, 2 ε and µ are real, σ=0, so β is real Q γ = jωµ (σ + jωε ) γ 2 = j 2ω 2 µε = ( jβ )2 ⇒ β = ω µε Assume the electric field with only x- component, no variation along x- and y-axis and propagation along z-axis, i.e., 37 Electromagnetic Field Theory by R. S. Kshetrimayum r r ∂E ∂E = =0 ∂x ∂y 3/19/2014 5.4 Plane waves in various media Helmholtz wave equation reduces to ∂2 ∂z 2 Ex + β 2 Ex = 0 whose solution gives waves in one dimension as follows Ex = E + e − j β z + E − e + j β z where E+ and E- are arbitrary constants 38 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.4 Plane waves in various media Putting in the time dependence and taking real part, we get, E x ( z , t ) = E + cos(ωt − βz ) + E − cos(ωt + βz ) For constant phase, ωt-βz=constant=b(say) Since phase velocity, dz d ωt − b) ω vp = = ( )= = dt dt β β 1 µε = 1 µ r µ 0ε r ε 0 Q β = ω µε 39 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.4 Plane waves in various media For free space, vp = 1 µ 0ε 0 = c = 3 × 10 8 m / s which is the speed of light in free space This emergence of speed of light from electromagnetic considerations is one of the main contributions from Maxwell’s theory The magnetic field can be obtained from the source free Maxwell’s curl equation 40 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.4 Plane waves in various media r r ∇ × E = − jωµH r r r ∇ × E j∇ × E j H =− = = jωµ ωµ ωµ xˆ ∂ ∂x E + e− jβ z + E − e+ jβ z yˆ ∂ ∂y 0 zˆ ∂ j ∂ + − jβ z − + jβ z = yˆ E e + E e ) ( ∂z ωµ ∂z 0 r − j β ( E + e− jβ z ) + ( E − e+ jβ z ) j β − j β {( E + e − j β z ) − ( E − e + j β z )} H= ( j ) yˆ = ( j ) yˆ ωµ = 41 β {( E + e− jβ z ) − ( E − e + jβ z )} ωµ ωµ yˆ = 1 η Electromagnetic Field Theory by R. S. Kshetrimayum [ E + e − j β z − E − e + j β z ] yˆ 3/19/2014 5.4 Plane waves in various media η is the wave impedance of the plane wave Ex ωµ µ = = η= β ε Hy For free space, ηo = µo = 120π = 377Ω εo 5.4.2 Lossy conducting medium If the medium is conductive with a conductivity σ, then the Maxwell’s curl equations can be written as 42 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.4 Plane waves in various media r r ∇× E = − jωµ H ; r r r r r ∇ × H = jω ε E + σ E = ( jω ε + σ ) E = jω ε eff E ; ε eff (ω ) = ε + σ jσ jσ =ε − = ε 1 − jω ω ωε The effect of the conductivity has been absorbed in the complex frequency dependent effective permittivity 43 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.4 Plane waves in various media r r r 2 r 2 2 ⇒ ∇ E + ω µε eff (ω ) E = ∇ E + ( jγ ) E = 0 2 We can define a complex propagation constant γ = jω µε eff (ω ) = α + j β where α is the attenuation constant and β is the phase constant 44 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.4 Plane waves in various media What is implication of complex wave vector? The wave is exponentially decaying (see example 4.4). The dispersion relation for a conductor (usually nonmagnetic) is γ = jω µε eff (ω ) = jω µ0ε 0 ε eff (ω ) ω = jω µ0ε 0 neff (ω ) = j neff (ω ) ε0 c where neff is the complex refractive index 45 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.4 Plane waves in various media 1-D wave equation for general lossy medium becomes ∂ 2 Ex ∂z 2 − γ 2 Ex = 0 whose solution is 1-D plane waves as follows E x ( z ) = E + e −γz + E − e +γz = E + e −αz e − jβz + E − eαz e jβz 46 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.4 Plane waves in various media Putting the time dependence and taking real part, we get, E x ( z, t ) = E + e −αz cos(ωt − βz ) + E − eαz cos(ωt + βz ) The magnetic field can be found out from Maxwell’s equations as in the previous section H y (z) = 47 1 ηeff [ E + e − γ z − E − eγ z ] Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.4 Plane waves in various media where useful expression for intrinsic impedance is ηeff = jωµ0 γ = jωµ0 jω µ0ε eff (ω ) = µ0 ε eff (ω ) The electric field and magnetic field are no longer in phase as εeff is complex Poynting vector or power flow for this wave inside the lossy conducting medium is 48 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.4 Plane waves in various media r r r* E e e + −α z − j β z ˆ S = E × H = E e e x× ηeff + −α z − j β z * e e + 2 −α z − j β z ˆ × y = E e e * η eff −α z + j β z zˆ = E η + 2 * e−2α z zˆ eff it is decaying in terms of square of an exponential function 5.4.3 Good dielectric/conductor Note that σ/ωε is defined as loss tangent of a medium A medium with σ/ωε <0.01 is said to be a good insulator whereas a medium with σ/ωε >100 is said to be a good conductor 49 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.4 Plane waves in various media For good dielectric, Q σ << wε ∴ γ = jω µε ( 1 − jσ ωε ) can be approximated using Taylor’s series expansion obtain α and β as follows: α= σ 2 µ ε β = ω µε For a good conductor, σ >> ωε Therefore, γ ≅ (1 + j ) 50 ωµσ 2 Electromagnetic Field Theory by R. S. Kshetrimayum ⇒α = β = wµσ 2 3/19/2014 5.4 Plane waves in various media Skin effect The fields do attenuate as they travel in a good dielectric medium α in a good dielectric is very small in comparison to that of a good conductor As the amplitude of the wave varies with e-αz, the wave amplitude reduces its value by 1/e or 37% times over a distance of δ= 51 1 α = 1 β = 2 ωµσ Electromagnetic Field Theory by R. S. Kshetrimayum = 2 1 = 2π f µσ π f µσ 3/19/2014 5.4 Plane waves in various media which is also known as skin depth This means that in a good conductor (a) higher the frequency, lower is the skin depth (b) higher is the conductivity, lower is the skin depth and (c) higher is the permeability, lower is the skin depth Let us assume an EM wave which has x-component and traveling along the z-axis Then, it can be expressed as E x ( z , t ) = E0 e −αz e − j ( βz −ωt ) 52 Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 5.4 Plane waves in various media Taking the real part, we have, E x ( z , t ) = E0 e −αz cos(ωt − βz ) Substituting the values of α and β for good conductors, we have, E x ( z , t ) = E0 e − πfµσ z ( cos ωt − πfµσ z ) Now using the point form of Ohm’s law for conductors, we can write J x = σE x (z , t ) = σE0 e − 53 Electromagnetic Field Theory by R. S. Kshetrimayum πfµσ z ( cos ωt − πfµσ z ) 3/19/2014 5.4 Plane waves in various media What is the phase velocity and wavelength inside a good conductor? vp = 54 ω 2π = ωδ ; λ = = 2πδ β β Electromagnetic Field Theory by R. S. Kshetrimayum 3/19/2014 Electromagnetic Waves 5.5 Summary Plane waves Plane waves in various media r r ) ) ) ) ) ) k • r = ( k x x + k y y + k z z ) • ( xx + yy + zz ) = k x x + k y y + k z z = con tan t Polarization Lossless medium r ELP (z ) = (E0 x xˆ + E0 y yˆ )e − jβz r ERHCP = Eo ( xˆ − jyˆ )e− j β z β = ω µε r E EP ( z ) = xˆ + Ae jφ yˆ e − jβz ( Lossy conducting medium ) vp = Poynting vector η= ω 1 = β µε ωµ µ = β ε ε eff (ω ) = ε 1 − 55 2 ∂t (H • H ) − 2 ∂t α =β = γ = jω µε eff (ω ) = α + j β δ = ηeff = r r ∂ 1 ∂ 1 2 2 S • d s = − µ H dv − εE dv − ∫ σE 2 dv ∫S ∫ ∫ ∂t V 2 ∂t V 2 V r r r r ε ∂ r r r r µ ∂ ∴∇ • ( E × H ) = − jσ ωε Good conductor (E • E) − E • J Electromagnetic Field Theory by R. S. Kshetrimayum jωµ 0 γ = µ0 ε eff (ω ) vp = 1 α = 1 β Good dielectric ωµσ α= 2 = 1 πfµσ σ 2 µ ε β = ω µε ω 2π = ωδ ; λ = = 2πδ β β J x = σE x ( z , t ) = σE0 e − πfµσ z ( cos ωt − πfµσ z Fig. 5.3 Plane waves in a nutshell 3/19/2014 )