Question

Problem 5 (20pts). The magnetic field of an electromagnetic wave propagating in free space is given by: H(R, 0,t) = (@0.2653 +00.5305) sinėsin(wt – kR), Mo = 41 X 10-? (H/m), E. = 8.85 x 10-12 (F/m). Frequency of 150 x 106 Hz Hint: use Ě = 1 7 xÅ jwe Find: a) Ể (R, O) and E(R, 0,t), Let  component equal zero b) value of k

Answer 1

of T + to a ot Go de 1 a classical we know from the Maxwell's fourth equation electromagnetic field in free space It where it is the magnetic fiels , J is the current density and Dē is the rate of change of electric flux in free space. In this particular problem, J is zero xH JE AS H (R. O, t) is given , we will start from the Left Home San from the Curvilineas Coordinates. The Cure of physical quantity in spherical Coordinates com be written as: 5x H - Remo %0 % RHO Romio Ho where the general form is : ExĀ, 1 her h es ha es where hi ha hz ar Seale factors which takes the form e, I, har hz = rsnice in sphnical Coordinate Susten and fui, as, ug} = 42, 0, 0} there ê component is zero. Ex H = 1 emiga R² Srrio for go Rasma %R chiheen tu folus Yous h, Aus So LA %0 Rsnio to RHO where Ho= 0.2653 smo smi (wt - KR) 0.5305 smo sm (ust-KR)

Rsrio o 6 EXH %p Remo se %o Imom O Kom x 05305 K Gr 3 x H Risme 0-25 min Rx0.2653 endini (wt - KR) K R 0 40.905 m o m(wt-we) - 2060-653 6 B (wtxx)} *RÔ 40 - 10.505 masin (atka)} + {millo plane sono le § 26 Best sm (wt-ke) - 0 +R Øfo-szos tampo Gs (wt_va) x()} + Remie ô fet) 0.3653 Smio cos (wp-ke)] Gorilauttel Groxy s. Risme som en by a comp (6) 5 05305 mg cos y parte 0-2653 sm sin olos y Whire wt - KR or oxu 12 12 Gost smil ok 0.5305 sedlost- 8. old 0.2653 smidles 4 GOE st Integrating both side - G-06-08 (wt -ue)) - Ó* 05305 SMB R2 miB ( FiWvas) - 10.1653 smo sin (cut-kR) som (urt-e) to E w FR 2 Cos o sy 0.5305 smo siny - 02653 sm Omit GWR? GoWR EWR E (R.2,t) and & (2,0,0) = E(2,0) = -² cos o cos (KR) + 0 k smo sin KR 10.5305 to WR² towR dm (-6) Sino} tok 0.2653 smo si KR GOWR

(11) value of is : In free space K = Gollow to lo or k= tollo x w but know ² 2) @z toleo V. Gollo where e is the speed of light 3x108 m/s ka w > 217 f с 20* 150 x 10 3X108 >> K 211 X 15 301 3 X 10 30