Chapter8 Plane Electromagnetic WavesPlane waves in perfect dielectricPlane waves in conducting mediaPolarizations of plane wavesNormal incidence on a planar surfacePlane waves in arbitrary directionsOblique incidence at boundaryPlane waves in anisotropic mediaV
Chapter 8 Plane Electromagnetic Waves Plane waves in perfect dielectric Plane waves in conducting media Polarizations of plane waves Normal incidence on a planar surface Plane waves in arbitrary directions Oblique incidence at boundary Plane waves in anisotropic media
1.WaveEguations2.PlaneWavesinPerfectDielectric3.Plane Waves in Conducting Media4.Polarizationsof PlaneWaves5.Normal Incidence on A Planar Surface6.Normal Incidence at Multiple Boundaries7. Plane Waves in Arbitrary Directions8.ObligueIncidenceatBoundarybetweenPerfectDielectrics9.Null and Total Reflections1o.Obliguelncidenceat ConductingBoundary11.ObliqueIncidenceat Perfect ConductingBoundary12. Plane Waves in Plasma13. Plane Waves in FerriteV
1. Wave Equations 2. Plane Waves in Perfect Dielectric 3. Plane Waves in Conducting Media 4. Polarizations of Plane Waves 5. Normal Incidence on A Planar Surface 6. Normal Incidence at Multiple Boundaries 7. Plane Waves in Arbitrary Directions 8. Oblique Incidence at Boundary between Perfect Dielectrics 9. Null and Total Reflections 10. Oblique Incidence at Conducting Boundary 11. Oblique Incidence at Perfect Conducting Boundary 12. Plane Waves in Plasma 13. Plane Waves in Ferrite
1.Wave EquationsIn infinite, linear homogeneous,isotropic media, a time-varyingelectromagneticfield satisfiesthefollowing equations:o"E(r,t)aJ(r,t)?E(r,t)-μeVp(r,t)uat?ato"H(r,t)V?H(r,t)-μs-VxJ(r,t)at?which are called inhomogeneous wave equations,andJ(r,t)= J'(r,t)+oE(r,t)whereJ'(ristheimpressedsource.u√
1. Wave Equations In infinite, linear, homogeneous, isotropic media, a time-varying electromagnetic field satisfies the following equations: = − − + = − ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) 1 ( , ) 2 2 2 2 2 2 t t t t t t t t t t J r H r H r r E r J r E r which are called inhomogeneous wave equations,and J(r,t) = J(r,t) +E(r,t) where J (r, is the impressed source. t)
The relationship between the charge density p(r, t) and the conductioncurrentoE(r,t)isapV- (cE) =atIn a region without impressed source, J = O. If the medium is aperfect dielectric, then, = O . In this case, the conduction current iszero, and p= O. The aboveeguation becomeso’E(r,t)VE(r,t)-u8=0at?o"H(r,t)v?H(r,t)- μe=0at?Whichare called homogeneous wave equations.To investigate the propagation of plane waves, we first solve thehomogeneouswave equations.u7
t = − (E) In a region without impressed source, J ' = 0. If the medium is a perfect dielectric, then, = 0 . In this case, the conduction current is zero, and = 0. The above equation becomes = − = − 0 ( , ) ( , ) 0 ( , ) ( , ) 2 2 2 2 2 2 t t t t t t H r H r E r E r Which are called homogeneous wave equations. To investigate the propagation of plane waves, we first solve the homogeneous wave equations. The relationship between the charge density (r, t) and the conduction current is E(r,t)
For a sinusoidal electromagnetic field, the above equation becomes?E(r)+k?E(r) = 0V?H(r)+k’H(r) = 0which are called homogeneous vectorHelmholtzeguations,and herek=ovucInrectangularcoordinatesystem,wehave?H.(r)+k?H.(r) = 0V?E,(r)+k?E(r)=0V?E,(r)+k'E,(r) =0?H,(r)+k"H,(r)= 0?E.(r)+k?E.()= 0V?H.(r)+k"H.(r)= 0whicharecalledhomogeneous scalarHelmholtzeguationsAll ofthese equations have the same form, and the solutions aresimilar.u
For a sinusoidal electromagnetic field,the above equation becomes + = + = ( ) ( ) 0 ( ) ( ) 0 2 2 2 2 H r H r E r E r k k which are called homogeneous vectorHelmholtz equations, and here k = In rectangular coordinate system, we have + = + = + = ( ) ( ) 0 ( ) ( ) 0 ( ) ( ) 0 2 2 2 2 2 2 r r r r r r z z y y x x E k E E k E E k E + = + = + = ( ) ( ) 0 ( ) ( ) 0 ( ) ( ) 0 2 2 2 2 2 2 r r r r r r z z y y x x H k H H k H H k H which are called homogeneous scalarHelmholtz equations. All of these equations have the same form, and the solutions are similar
In a rectangular coordinate system,if the field depends on onevariable only, the field cannot have a component along the axis of thisvariable.If the fieldis related to the variablez only,we can showE, =H.=0Since the field is independent of the variablesx and y, we haveOE,OEXOE.OE.PV.EOzaxOzayaHaHah.aHV.HOzazayaxu7
In a rectangular coordinate system, if the field depends on one variable only, the field cannot have a component along the axis of this variable. If the field is related to the variable z only, we can show E z = H z = 0 = + + = = + + = z H z H y H x H z E z E y E x E x y z z x y z z H E Since the field is independent of the variables x and y, we have
Due to V. E - o, V. H =O, from the above equations we obtainOE..oH.= 0azOz福Consideringa'E.a?Ea"EOE.V?E.=0Ozax?ay?az2a?Hα?HH.a"HV?H.=00z?O2ax?ay?Substituting that into Helmholtz equations:V?E.(r)+k E.(r) = 0v?H.(r)+k"H.(r) = 0E. =H. =0WefindU
Due to , from the above equations we obtain E = 0, H = 0 = 0 = z H z Ez z Considering 0 2 2 2 2 2 2 2 2 2 = = + + = z H z H y H x H H z z z z z 0 2 2 2 2 2 2 2 2 2 = = + + = z E z E y E x E E z z z z z Substituting that into Helmholtz equations: ( ) ( ) 0 2 2 E z r + k E z r = ( ) ( ) 0 2 2 H z r + k H z r = We find E z = H z = 0
2.PlaneWaves in PerfectDielectricIn a region withoutimpressedsourcein a perfect dielectric,asinusoidal electromagneticfield satisfiesthe following homogeneousvectorHelmholtzequation[V?E(r)+k’E(r) = 0VH(r)+k’H(r) = 0Wherek-o usIf the electric fieldintensityE is related to the variablez only,and independentof the variablesx and y, then the electric field hasno z-component.Let E = e, E, , then the magnetic fieldintensity HisIVxE-IVx(e,E,)H=ououI-[(VE)xe, +E,Vxe,]=(VE,)xe,ououu7
2. Plane Waves in Perfect Dielectric In a region without impressed source in a perfect dielectric, a sinusoidal electromagnetic field satisfies the following homogeneous vector Helmholtz equation + = + = ( ) ( ) 0 ( ) ( ) 0 2 2 2 2 H r H r E r E r k k If the electric field intensity E is related to the variable z only, and independent of the variables x and y, then the electric field has no z-component. Let , then the magnetic field intensity H is x Ex E = e ( ) j j x Ex H = E = e x x x ( ) j [( ) ] j = e + e = e Ex Ex Ex Where . k =
aE.aEaE.OE.Due toVE, =exteeyaxayOzOzj EH-j EWehave=e,H,H = eoμ azouozFromlastsection, we know that each componentof the electricfieldintensitysatisfiesthe homogeneous scalarHelmholtzequationaEaEConsideringo,wehaveaxayd’E.1+k’E,=0dz?which is an ordinary differential eguation of second order, and thegeneralsolutionisE,= Exoe-ie+ EloeleThe first term stands for a wave traveling along the positivedirection of the z-axis, while the second term leads to the opposite巴
z E z E y E x E E x z x z x y x x x = + + Due to = e e e e z E H x y = j y y x y H z E H e = e = We have j From last section, we know that each component of the electric field intensity satisfies the homogeneous scalar Helmholtz equation. Considering , we have = 0 = y E x Ex x 0 d d 2 2 2 + x = x k E z E which is an ordinary differential equation of second order, and the general solution is kz x kz Ex Ex E j 0 j 0 = e + e − The first term stands for a wave traveling along the positive direction of the z-axis, while the second term leads to the opposite
Here only the wave traveling along with the positive direction ofz-axis is consideredE.(2) = Exoe-ikewhere Ero is the effective value of the electric field intensityat z - OThe instantaneousvalue E,(z,t) isE,(z,t) = ~2Ero sin(0 t - kz)E(z, t)Anillustrationof theelectric field intensity varyingover space at different timesis shown in the left figure.The wave is traveling alongTthe positivez-directionti=0t242u
Here only the wave traveling along with the positive direction of z-axisis considered kz x Ex E z j 0 ( ) e − = where Ex0 is the effective value of the electric field intensity at z = 0 . ( , ) 2 sin( ) 0 E z t E t kz x = x − The instantaneous value E (z,t) is x An illustration of the electric field intensity varying over space at different times is shown in the left figure. Ez (z, t) z O 2 2 3 t1 = 0 4 2 T t = 2 3 T t = The wave is traveling along the positive z-direction