1. EM Wave An electromagnetic wave is traveling at speed 1: 'm the +1: direction in a vacuum (Le. there are no charges or currents in the region). A proposed form for the electric and magnetic elds of the wave is: A E{:I:, y, z,t) =j Eacos k(:i: nt) Etc, 3;, z,t) = h Bo cos k(:c st) where E0, is, and v are positive constants and Ba is a constant that might be positive or negative. Consider a rectangular box whose sidm are aligned with the axes One corner is at the origin (0, 0,0) and the opposite corner is at (1:0, yo, :0). We will look more carefully at electromagnetic waves at the end of the course, but you have all the tools you need to start playing with them now. In particular, we will use the following four expressions: GaussI Law for electricity: e=fsii= QM 60 Gausl' Law for magnetism: is = f s 4:134 = o Faraday '5 Law: d'I'B_ _ -~ _dt _s_jscu Ampere-Maxwell Law: d'I'E fg'd]=#01end+few Note: These expressions look very similar to the ones we've used so far in the course, with a few adjustments. These four expressions are referred to as Maxwell's Equations. (a) Apply Gauss's law for the electric eld to this box and show that it is satised. (b) Apply Gauss's law for the magnetic eld to this box and show that it is satised. (c) Apply Faraday's law to a side of the box parallel to the my plane. Show that it is only satised if E0 = n80. ((1) Apply the ArnpereMaxweH law to a side of the box parallel to the 22 plane. Show that it is only satised if g; = capo. (e) Based on the conditions necessary to satisfy your results so far, nd the numerical value- of n and the ratio E0130. (f) Based on the sign of Ba, formulate a right hand rule that relates the directions of the elds to the direction of motion of the wave (here. +33)