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