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Discussion Question 13B P212, Week 13 Electromagnetic Waves A laser beam travels through vacuum. The electric field of the plane electromagnetic wave produced by the laser has the form given below. The wavelength of the beam is λ = 514 nm, and the amplitude of the electric field is E0 = 2.5 x 104 N/C. E (x, y,z,t) = yˆ E 0 cos(kz + ωt − 45 ) (a) In what direction is this wave propagating? What does “propagating wave” mean anyway? It means that the shape of the wave remains the same as a function of time, it just moves. The shape of the wave is determined by that cosine. As time rolls forward, how do you have to change position so that the shape looks the same? G ˆ 0 cos(−450 ). This spot obeys Lets follow a spot on the wave corresponding to E = yE kz + ωt = 0 or z = - ω ω t. Hence this spot moves along the -zˆ axis and evidently = c k k (b) What are the magnitudes of the wave number k and angular frequency ω? 8 2π 2π 2πc 2π ( 3 ×10 ) 7 -1 k= = = 1.22 × 10 m ; ω = ck = = = 3.66 × 1015 Hz 514 × 10−9 λ 514 × 10−9 λ (c) Make a sketch of what the E field looks like at some moment in time. π⎞ ⎛ At t = 0, we have E y = E0 cos ⎜ kz − ⎟ 4⎠ ⎝ which is a cosine curve that maximizes π at z = + 4k Ey t =0 z (d) Write down an expression for the magnetic field B(x,y,z,t). Express your answer algebraically (i.e. no numbers) in terms of the symbols k, ω, and B0 (the latter being the magnetic field amplitude). Be sure to indicate the direction of the magnetic field. The E and B fields of an electromagnetic wave have a fixed relationship to the direction of propagation of the wave. You know the direction of the wave, and the direction of E, so you can figure out the direction of B with the help of your sketch. Add the B-field to your drawing in part (c). The B field has the same position and time dependence as the E field. It is directed G G such that E × B is along the direction of propagation (marked S in the figure) G B ( x, y, z, t ) = xˆ B0 cos(kz + ωt − 45D ) E z y x B S Page 1 of 2 E (x, y,z,t) = xˆ E 0 cos(kz + ωt − 45 ) where E0 = 2.5 x 104 N/C and λ = 514 nm (e) What is the amplitude B0 of the magnetic field? E0 2.5 × 104 B0 = = = 8.33 × 10−5 T 8 c 3 ×10 (f) What is the intensity I of the wave? Be sure to indicate the units of your answer. 2 2.5 × 104 ) ( Erms I = = = 8.29 × 105 W/m 2 or alternatively Z0 2( 377 Ω) 2 −5 4 Erms Brms Emax Bmax ( 2.5 × 10 )( 8.33 × 10 ) I = = = = 8.29 × 105 W/m 2 −7 µ0 2µ 0 2 ( 4π×10 ) (g) A solar cell is an example of a photo-sensitive detector that absorbs the energy of incident electromagnetic radiation and converts it into useful power (with which to operate your solar-cell calculator, for example). Suppose our electromagnetic wave is incident on a small, square photo-sensitive detector of side 2 mm. You can assume that this detector is smaller than the wave itself, and is oriented so that its surface is perpendicular to the incoming beam. You can also assume that the detector is 100% efficient at converting all the radiative energy incident on it into useful power. What, then, is the average power <P> that our detector would generate? As always, units are an enormous help. The units of your answer to part (e) tell you how to determine the answer to this question! detector The intensity is the power per unit area. P = I A = ( 8.29 × 105 W/m 2 )( 0.0022 m 2 ) = 3.32 W Page 2 of 2