(b)The electric field intensityisE(=) = e,20e-j2元 V/mThemagnetic fieldintensityis-j2元H() =A/me.xE=Z.6元(c)The energy flowdensityvectoris10W/m?S.-ExH'=e.3元(d) The phase and energy velocities are0=3×10° m/s=Ve2kU√
( ) 20e V/m j2πz x z − E = e (b) The electric field intensity is e A/m 6π 1 1 ( ) j2π 0 z z y Z z − H = e E = e The magnetic field intensity is * 2 c W/m 3π 10 z S = E H = e (c) The energy flow density vector is 3 10 m/s 8 p = e = = k v v (d) The phase and energy velocities are
3.PlaneWavesinConductingMediaIf o +o,thefirstMaxwell'seuation becomesV×H=oE + jOE = jo(c-j-)Es.=6-jαIf let0Then theabove equation can be rewritten asV×H= j0s.Ewhere is calledthe equivalent permittivityIn this way, a sinusoidal electromagnetic field then satisfies thefollowinghomogeneousvectorHelmholtzequation:VE+0"us.E=0V?H+°ue,H = 0U√
3. Plane Waves in Conducting Media If 0 , the first Maxwell’s equation becomes H =E + jE If let j e = − H = j e E Then the above equation can be rewritten as where e is called the equivalent permittivity. In this way, a sinusoidal electromagnetic field then satisfies the followinghomogeneous vectorHelmholtz equation: + = + = 0 0 e 2 2 e 2 2 H H E E j ( j )E = −
k,=oVue.=o(s-j-)LetCWe obtain?E+k?E=0?H +kH = 0EEIf we let E = E,eas before, andQthen the solution oftheaxequationis the same as that in the lossless case as long as k is replacedby ke, so thatE, =Exoe-ik.:Because k is a complex number, we definek。=k'- jk"We find1ek'= k"=008U
( j ) c e Let k = = − We obtain + = + = 0 0 2 c 2 2 c 2 H H E E k k If we let as before, and , then the solution of the equation is the same as that in the lossless case as long as k is replaced by kc , so that Ex x E = e = 0 = y E x Ex x k z x x c E E j 0 e − = Because kc is a complex number, we define k = k− jk c We find 1 1 2 2 + = + k 1 1 2 2 − = + k
In this way,the electricfield intensity can be expressed asE, =Eroe-k*e-ikswherethefirstexponent leadsto an exponentialdecayofthe amplitudeof the electric field intensityin the z-direction,and the second exponentgivesriseto aphasedelayThe real part k'is called the phase constant, with the unit of rad/mwhile the imaginary part k" is called the attenuation constant and has aunit of Np/m.0Thephasevelocityis08It depends not only on the parameters of the medium but also on thefrequencyA conductingmediumis a dispersivemediumUV
k z k z Ex Ex − − = j 0 e e In this way, the electric field intensity can be expressed as where the first exponent leads to an exponential decay of the amplitude of the electric field intensity in the z-direction, and the second exponent gives rise to a phase delay. The phase velocity is + + = = 1 1 2 1 2 p k v It depends not only on the parameters of the medium but also on the frequency. A conductingmedium is a dispersive medium. The real part k is called the phase constant, with the unit of rad/m, while the imaginary part k is called the attenuation constant and has a unit of Np/m
2元2元The wavelength is2k'Le一008The wavelengthis related to the propertiesofthe medium, andit hasa nonlineardependence onthefrequencyThe intrinsic impedanceisuuZ.808which is a complex number.Since theintrinsicimpedanceis a complex number, andit leadsto a phase shiftbetween electric field and the magnetic field.U
The wavelength is + + = = 1 1 2 2π 2π 2 k The wavelength is related to the properties of the medium, and it has a nonlineardependence on the frequency. The intrinsic impedance is e c 1 j = − Z = which is a complex number. Since the intrinsic impedance is a complex number, and it leads to a phase shift between electric field and the magnetic field