Question

3. (30 pts) The magnetic field of a uniform plane wave propagating in a lossless medium with 8=2 is 7 (x,t)= cos(3x10't +4x)â, - sin( 3x108t + 4x)â, (Alm). Find: (1) The phasor form expression for the magnetic field (2) The direction of propagation for the wave (3) The velocity vp (4) The polarization state. If it is a circular polarization, you need to determine it is left or right handed. Provide details for your reasoning. (5) The average power density Sav.

Answer 1

کره H (xt) cos (3x10%++ 4x 1234 Sim (3x10% +4x) â [given Now, com the above equation with standered solution of Ĥ field, ie, Ħ(x)+) = cos(white) ay- Sin (ut-az معلمها So, the polarization on vibration of it (ast) in - z-y plane. By combai we pairing get, o= 34108 rad/s ke- kê =-4ê mit we can write the above equation in phasor form j (wtaron Ing where, Phason w = 3x1018 nad/s k=-4 # (x, t) = @yReed (105k-to) an e ã Form m-1 450 no Direction of vibration for x=0 (2 ) from combaining with with standered expression of oscillatory field, it is clear tat, the proparation vector k=-qê ie, The direction of propagation in along '-ve' i -axis. (3) قنا given by ze R The phase reelocity is given Now, k=151 = and = 3x108 rad/sec [By Ss, voe 3x109 m/s = 75x106m/s compairing (Ano.) 4

(4) as, Pere If, Hoy and Hoz be the con le tende at magmatice y andz gomeze fied fied, along the from the axis, we con see that, = given expression Hoy Hoze -j 1/2 ie, the equation is for Left circularly polarized Bo field. the bolarized plane is And z-y plane. (5 The mag amplidude of Electric fied and netic field is given by Eo B. 29 [ Vo: Velocity of the wave I in the Oro Eo Bov. medium Now, v u: permiability Tue vue E: permetureity Q- Tube Molo permitivity lo: Freespace permiability to: Free space .. alo, bermuterit Up: relative bermiability 20 = 29,0 = 75 x 106m/s [ from (3) € = Ep to & u= un lo [ Epi porelative Ex=2 [given So, - u E Moto ola un Ertolo lo to Super Jun Ver Ser v 13x108 75x106 on ) xlod) 62 en 75x100 up = 2 on, up=8

Now, H= B Ал where, Bis magnetic field H is field. intensity Now, from given equation of H (est) ampletude = Ho (say) of t 12+E0² 2 So, B= ura= Moupra. Moup = 852 Mo [as, p=8 Now, isa avarage power density Son zetu given by, Eo Bo Å Eo to Now, Eo = Boze Sar So, Žve Ho² M = 2 Duplo Ho² ¿ Ho B. 29 = [as, uro- et vuo 8 Ho2 on, uo 8x2 Že 11 sar My=8 [a as, Ho=22 vuo8 = 75x 106 x 471X10-7x8 X 477 x 10-7x8 [ased Mo=47X107 = 75x47x8 x 10-4 W/m2 ~ 754 W/m2 754 W/m2 (Ams.). H/m] lie, Sar