Figure 13.4.7 Period
13-17 At time t = 0 , E y (0,0) = 0 . Let T be the time that it takes the electric field to complete one oscillation on the plane x = 0 . Then E y (0, T ) = ! E 0 sin(2 " cT / # ) = 0 . This condition is satisfied when T = ! c , (13.4.43) then E y (0, T ) = ! E 0 sin(2 " ) = 0 . The time T is called the period. The frequency f is defined as the inverse of the period, f ! 1 T . (13.4.44) The angular frequency ! is defined by ! " 2 # f = 2 # T = 2 # c $ . (13.4.45) The SI units of angular frequency are [rad ! s -1 ] . With these definitions we can rewrite our electric and magnetic fields as ! E = E y ( x , t ) ˆ j = E 0 sin( kx ! " t ) ˆ j ! B = B z ( x , t ) ˆ k = B 0 sin( kx ! " t ) ˆ k . (13.4.46) In empty space, the wave propagates at the speed of light c . The characteristic behavior of the sinusoidal electromagnetic wave is illustrated in Figure 13.4.8. Figure 13.4.8 Plane electromagnetic wave propagating in the + x - direction. We see that the E ! and B ! fields are always in phase (attaining maxima and minima at the same time.) To obtain the relationship between the field amplitudes 0 E and 0 B , we make use of Eqs. (13.4.6) and (13.4.12). We first calculate the following the partial derivatives
13-18 ! E y ! x = kE 0 cos( kx " # t ) , (13.4.47) ! B z ! t = " # B 0 cos( kx " # t ) , (13.4.48) Eq. (13.4.6) then implies that 0 0 E k B ! = , which we rewrite as 0 0 E c B k ! = = . (13.4.49) Because the sinusoidal behavior of the fields are the same, the magnitudes of the fields at any instant are related by E c B = . (13.4.50) Let us summarize the important features of electromagnetic waves described in Eq. (13.4.38): 1. The wave is transverse since both E ! and B ! fields are perpendicular to the direction of propagation, which points in the direction of the cross product ! E B ! ! . 2. The E ! and B ! fields are perpendicular to each other. Therefore, their dot product vanishes, 0 ! = E B ! ! . 3. The speed of propagation in vacuum is equal to the speed of light, and 0 0 1/ c μ ! = . 4. The ratio of the magnitudes and the amplitudes of the fields is c = E B = E 0 B 0 = ! k = " T . 13.5 Standing Electromagnetic Waves Let us examine the situation where there are two sinusoidal plane electromagnetic waves, one traveling in the positive x -direction, with E 1 y ( x , t ) = E 10 sin( k 1 x ! " 1 t ), B 1 z ( x , t ) = B 10 sin( k 1 x ! " 1 t ) (13.5.1)
13-19 and the other traveling in the negative x -direction, with E 2 y ( x , t ) = + E 20 sin( k 2 x + ! 2 t ), B 2 z ( x , t ) = " B 20 sin( k 2 x + ! 2 t ) . (13.5.2) For simplicity, we assume that these electromagnetic waves have the same amplitudes, E 0 ! E 10 = E 20 and B 0 ! B 10 = B 20 , and wavelengths, ! " ! 1 = ! 2 . Using the superposition principle, the electric field and the magnetic fields can be written as E y ( x , t ) = E 1 y ( x , t ) + E 2 y ( x , t ) = E 0 [sin( kx ! " t ) + sin( kx + " t )] , (13.5.3) B z ( x , t ) = B 1 z ( x , t ) + B 2 z ( x , t ) = B 0 [sin( kx ! " t ) ! sin( kx + " t )] . (13.5.4) Using the identities sin( ! ± " ) = sin( ! )cos( " ) ± cos( ! )sin( " ) (13.5.5) The above expressions may be rewritten as E y ( x , t ) = E 0 [(sin( kx )cos( ! t ) " cos( kx )sin( ! t )) + (sin( kx )cos( ! t ) + cos( kx )sin( ! t ))] = 2 E 0 sin( kx )cos( ! t ), (13.5.6) and B z ( x , t ) = B 0 [(sin( kx )cos( ! t ) + cos( kx )sin( ! t )) " (sin( kx )cos( ! t ) " cos( kx )sin( ! t ))] = 2 B 0 cos( kx )sin( ! t ), (13.5.7) One may verify that the total fields E y ( x , t ) and B z ( x , t ) still satisfy the wave equation stated in Eqs. (13.4.15) and (13.4.17) even though they no longer have the form of functions of k ( x ± ct ) = ( kx ± ! t ) . The waves described by Eqs. (13.5.6) and (13.5.7) are standing waves , which do not propagate but simply oscillate independently in space and time.
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- Fall '14
- Magnetic Field, ........., electromagnetic waves