12th Class Physics Sample Paper Physics Sample Paper-15

  • question_answer What is space wave propagation? Which two communication methods make use of this mode of propagation? If the sum of the heights of transmitting and receiving antennas in line of sight of communication is fixed at h, show that the range is maximum when the two antenna have a height h /2 each.


    Space Wave Propagation (LOS) It is a mode of wave propagation in which the radio waves emitted from the transmitter antenna reach the receiving antenna through space. These radio waves are called space waves. This mode of propagation is used for television broadcast and microwave link. Maximum distance between the transmitting and receiving antenna in line of sight communication is             \[{{d}_{M}}=\sqrt{2R{{h}_{T}}}+\sqrt{2R{{h}_{R}}}\]                       ? (i) Given,   \[{{h}_{T}}+{{h}_{R}}=h\] or \[{{h}_{R}}=h-{{h}_{T}}\] Substituting \[{{h}_{R}}\] in Eq. (i), we get \[{{d}_{M}}=\sqrt{2R{{h}_{T}}}+\sqrt{2R\,(h-{{h}_{T}})}\]             Differentiating above equation w.r.t. \[{{h}_{T}},\] we get                         \[\frac{d\,({{d}_{M}})}{d{{h}_{T}}}=\sqrt{2R}\times \frac{1}{2\sqrt{{{h}_{T}}}}-\sqrt{2R}\times \frac{1}{2\sqrt{h-{{h}_{T}}}}\]             For maximum range, we have \[\frac{d\,({{d}_{M}})}{d{{h}_{T}}}=0\] or \[\frac{\sqrt{2R}}{2}\left[ \frac{1}{\sqrt{{{h}_{T}}}}-\frac{1}{\sqrt{h-{{h}_{T}}}} \right]=0\] or \[\frac{1}{\sqrt{{{h}_{T}}}}-\frac{1}{\sqrt{h-{{h}_{T}}}}=0\] or \[\frac{1}{\sqrt{{{h}_{T}}}}-\frac{1}{\sqrt{h-{{h}_{T}}}}\] or         \[h-{{h}_{T}}={{h}_{T}}\] \[\therefore \]      \[{{h}_{T}}=\frac{h}{2}={{h}_{R}}\]

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