Chapter 24, Problem 57 In a traveling electromagnetic wave, the electric field i

Question

Chapter 24, Problem 57

In a traveling electromagnetic wave, the electric field is represented mathematically as E = E0 sin[(1.8 x 1010 s-1)t - (7.5 x 101 m-1)x] where E0 is the maximum field strength. (a) What is the frequency of the wave? (b) This wave and the wave that results from its reflection can form a standing wave, in a way similar to that in which standing waves can arise on a string (see Section 17.5). What is the separation between adjacent nodes in the standing wave?

Explanation / Answer

E = EoSin(wt-kx) .this mean the wave is moving in the positive x direction. ? in the above wave, w = 2.1 X 1010 s-1 . So, 2 pi f = 2.1 X 1010 s-1 ?f = 337.56 s-1 the reflected wave is moving in the opposite direction so the wave will be expressed as E=EoSin(wt+kx), this means that the wave is moving in the negative x direction now . so the resulting wave is the superposition => E= Eo (Sin(wt-kx) +Sin(wt+kx)) => 2*Eo Sin(kx) Cos(wt). hence nodes are at positions where Sin(kx)=0. => kx=n pi. hence distance between two consecutive nodes = pi/k = pi/(6.8 x 101) = 0.457 cm or 4.57 x 10^-3 m .

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