Question

4. (4.5 pts) Consider a 3D electromagnetic plane wave in vacuum, described in usual complex form by, (r, t) = Ēelkr-wt) in which ło = Epein/2y. Where k = -kx is the wave vector (assume k > 0) and w > 0 is the angular frequency. As usual, the real field is Er, t) = ReLEr, t)] (a) In which direction is the wave propagating? In terms of the given values k and w, what is the speed, wavelength, and period of the wave? What does the phase of x/2 in E, affect and what does it not affect with respect to the wave? (b) Sketch the real field E(x, y = 0, z = 0,t = 0) (a 2D plot with x as the horizontal axis) and E(x = 0, y = 0, z = 0,t) (a 2D plot with t as the horizontal axis). Clearly indicate the direction of the field, the values of at least a couple of horizontal-axis intercepts, and the scale of both your axes, in terms of the given values of this problem. (C) Now sketch the real fields Ex = 0, y, z = 0,t = 0) and E(x = 32/4, y, z = 0, t = 0) in the y direction (a 2D plot with y as the horizontal axis). Why are these different than the plot vs x in part b? (d) Why is this called a plane wave? (where is (are) the plane(s)?) (e) Is this wave linearly polarized? Explain (f) Find the associated magnetic field B(r,t) for this plane electric wave. Sketch the magnetic fields, B(x, y = 0, z = 0,t = 0) and B(x = 0, y = 0, z = 0,t) indicating field direction. (g) Describe in words how B compares/contrasts with E.

Answer 1

loc E = Eo eng Ke-Kn Kode - Kn (nat yang + Zuz) Ê Cr, 4) = E ilkrowt) E to our C-KH - WAT a to el-KH - 6 + + 1 / - 2-Kn @ Wave propagating Á EX direcionsform kn value 1 k Speed vz w wavelength by K = 27 so x - 21 (urmelengths) as w - 21 ) T= 21 time period K

The phase of to affect the phase of the ware and I of to does not effect the magnitude with the respective wave Real field of Ecot) Real (Lo ellRat wt - 11) 7 e Real Chelles (Kutwx=) - sun (RM499 42). to Caus (Kutwt-I) E62,420, 220,t=0)., kocos Ku-I) = Es Sin (X n2 E(M=0,420,2-0, ) = Es los[wt-I) Eo Suislwt2 TT ST --- 1

O CIRIECH2O 4, 2:0, 620) = Es CascoR+w60) - ) - - Ed C LI)- constant = 0 - Real E(21= 345,270, =o) - Eo 63( 32 X20 +601-7) Eo cos (37-7) - E (RO)=-E

Real E ngo, y, zeo,t=0) = 0 Real E(M• 347 4, Z604:0) = - Ee -EO The real part of E is independent of y docon't vary with the variation of y gives a constant value for any value of a and t and y direction (d, This is a plane ware because it has a constant frequency and at any value of time t , H is value is constant in y z plane this yz plane in perpendicular to propogation of the wave yz is the plane in which the ware has constant Italien.and doesn't hazaif Hy2 Values changes.com

(el yes this wave is linearly polarized usare because the E only present in ŷ direction. It is has only Single direction for I vector in ŷ it is polarred in ĝ direction (& Direction of B such that Ex B= ân (ware prepagation di È in ay direction so B in az direction olso l = n Mo = pernithrity B = loh 131 = Mo Inl TEL. to n = Inpedance B = Mo to ed en twt ) n. Real Bcn, y 2o, zoo, t0) = Mo Eo Cos (Knt wco) - 11 ) - n - Moto Cos (Kn ): 1. E. Siis (kx). Real B(1207420, 72014) - Hobo cos (w(-1) LoE Sintet

utto to /29 n - Moto AL T- ittono 12 t Molo

of Bis is perpendicular direction to Ě B in & direction ein of direction B magnitude is to time ² magnitude