ps2soln - ELEC 262 2009 Problem set 2 solutions 1(a The...

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ELEC 262 2009 Problem set 2 solutions 1(a) The complex amplitude is everything other than exp{j(kz- ω t)}. It is given by ( )( ) 0.4 0 ˆ ˆ 200 / = + j E x y V m e . (b) The B field is the same as the E field except that the direction is perpendicular, and the magnitude is smaller by a factor of the speed. So: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 7 -1 15 8 7 7 -1 15 200 / ˆ ˆ , , , cos 1.307 10 m 2.772 10 sec 0.4 2.121 10 / ˆ ˆ 9.43 10 cos 1.307 10 m 2.772 10 sec 0.4 = − + × × + × = − + × × × + / V m B x y z t x y z t m s x y Tesla z t This field is less than 1 microTesla, so it is more than 50 times smaller than the earth’s static field. (c) The divergence is given by: + + y x z E E E x y z . This field has E z = 0, and furthermore E x and E y depend only on z, not on x or y. Thus all three derivatives in this sum vanish. Obviously, the field has to satisfy 0 ∇⋅ = E because that’s one of Maxwell’s equations. 2(a) To find the peak wavelength, we start with the blackbody formula (lecture 5, slide 14), differentiate, and set this equal to zero: ( ) 3 3 2 3 2 3 8 8 3 0 1 1 1 ν ν ν ν π ν π ν ν = = = ν ν h kT h kT h kT h kT dI d h c h e h kT d d e c e e which looks much simpler if we define = ν x h kT , and cancel out the constants in front: ( ) 2 3 2 3 0 1 1
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