<p>Consider two antennas radiating 7.7 MHz radio waves in phase with each other.

Question

<p>Consider two antennas radiating 7.7 MHz radio waves in phase with each other. They are located at points S1 and S2, separated by a distance d = 175 m, Fig. 24-61. What are the farthest two points on the positive y axis where there will be destructive interference? (Destructive interference in this case means that a radio receiver placed at that point would pick up no signal).<br />farthest y ?? m</p>
<p>Second farthest y ?? m</p>
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<p>S1 at origin and s2 at some point to the right of S1 on the x-axis</p>

Explanation / Answer

   distance d = 175m ------------------------------------------------------------    wavelength = c / f                      = (3*108 m/s)/( 6.0*106Hz) = 38.96 m    Let us take P be the point on the y-axis. At this point destructive interference occures.     the path difference is            = S2 P - S1 P             = ( y2+ d2) - y condition for disturctive inerfernce is           = ( m + 1/2)    ( m + 1/2) =  ( y2+ d2) - y     ( m + 1/2)    + y=  ( y2+ d2)      squaring on both sides      [( m + 1/2) ]2 + y2 + 2y( m + 1/2) = (y2+ d2)                  [( m + 1/2) ]2 + 2y ( m + 1/2)=   d2                     y = [d2-[( m + 1/2) ]2]/  [2( m + 1/2) ]   For m=0               y = [d2-[( 1/2) ]2]/  [2( 1/2) ]                     = [(175 m)2-[( 0+ 1/2)(38.96 m) ]2]/  [2( 0+ 1/2) (38.96 m)]                   =  776.32 m For m=1                       y = [d2-[( 1+ 1/2) ]2]/  [2(1+ 1/2) ]                   =  [(175 m)2-[( 1+ 1/2)(38.96 m) ]2]/  [2( 1+ 1/2) (38.96 m)]                    = [(175 m)2-[( 0+ 1/2)(38.96 m) ]2]/  [2( 1+ 1/2) (38.96 m)]                   =  232.800m For m =2         y = [d2-[( 2+ 1/2) ]2]/  [2(2+ 1/2) ]                   =  [(175 m)2-[( 2+ 1/2)(38.96 m) ]2]/  [2( 2+ 1/2) (38.96 m)]                                =  108.5 m   For m =2         y = [d2-[( 2+ 1/2) ]2]/  [2(2+ 1/2) ]                   =  [(175 m)2-[( 2+ 1/2)(38.96 m) ]2]/  [2( 2+ 1/2) (38.96 m)]                                =  108.5 m                  =  [(175 m)2-[( 2+ 1/2)(38.96 m) ]2]/  [2( 2+ 1/2) (38.96 m)]                                =  108.5 m  

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