Please show all work Consider a plane wave -ik.r E(r) = Ege = Eye-i(k+x+kyy+kzz) (1.1) where...
Please show all work Consider a uniform transverse electromagnetic plane wave be normally incident on an interface between region 1 and region 2, at z = 0. Let the known incident wave in region 1 be (1.20) E(1)(r) = Â Ee-ikız îkı * E")(r) у ki H*)(r) E e-ikız. (1.21) ωμι wi where kı = w M€1. Let the unknown reflected wave be (1.22) E")(r) = în Ef")eikız ku r(r) Hr)(r) = -ġ whi EX")eikiz (1.23) and the unknown transmitted...
(i) Consider the wave Ē(7,t) = Ło cos(wt – k ), where Ē, is a fixed vector. Determine the relation between w and k = \KI SO that Ē(7,t) is a solution of the wave equation -27 182 VPE = 2 ət? - What is the direction of propagation of the wave? ii) Show, by substitution of Ē(7,t) in the appropriate Maxwell's equation, that K· Ē= 0. iii) Assuming that the magnetic field B(7, t) = B, cos(wt – K:1),...
2. The electric field in a plane wave is described by the equation (k > 0): Ē(x,y,z,1)= E, sin(kz – mt)ị Answer the following questions about the wave. i. What direction is the wave traveling? Explain how you can tell from the equation for the electric field. ii. Write an expression for the magnitude of the magnetic field of the wave. iii. Calculate the average intensity of the wave if Eo = 3000 V/m. The MKS units of intensity are...
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...
Light, radiant heat (infrared radiation), X rays, and radio waves are all examples of traveling electromagnetic waves. Electromagnetic waves comprise combinations of electric and magnetic fields that are mutually compatible in the sense that the changes in one generate the other. The simplest form of a traveling electromagnetic wave is a plane wave. For a wave traveling in the x direction whose electric field is in the y direction, the electric and magnetic fields are given by Ē = E,...
b) Let a R3 be a vector of length 1. Define H={x E R3 : a·x=0). Here a x denotes the dot product of the vectors a and x. (i) Show that H is a subgroup of R (ii) For λ E R, show that : a·x= is a coset of H in R3. (ii) Is H cyclic? Prove or disprove. b) Let a R3 be a vector of length 1. Define H={x E R3 : a·x=0). Here a x...
3. Consider the electromagnetic wave k=(0.2) where .A Draw the electric field E, magnetic field B and Poynting vector S at the points = (x, y, z) that fulfill: a) th when t-0. - e condition-k = 0 Side view 3D view b) the condition F. k-2 when t-o. Side view 3D view 3.B Calculate the wavelength λ of the wave (6). Compare the magnitude of λ with the magnitude of k. 3.C Wave (6) is expressed in terms of...
Please show full workings only answer if you know how. (5) Consider the 3 x 3 matrix A - I - avv7 where a e R. I is the identity matrix and v the vector 1S 2 (a) Determine the eigenvalues and eigenvectors of A (b) Hence find a matrix which diagonalises A. (c) For which a is the matrix A singular? (d) For which α is the matrix A orthogonal ? (5) Consider the 3 x 3 matrix A...
Slove 4.3.8 please axbycz d be the equation of a plane with normal Exercise 4.3.16 a. Show that w- (u x v) = u (vxw) = v x (w x u) holds for all vectors w, u, and v. n= C w and (u x v) + (vxw) +(wxu) b. Show that v- a. Show that the point on the plane closest to Po has vector p given by are orthogonal Exercise 4.3.17 Show u x (vxw) = (u w)v-...