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JEE Main 2017 · Physics · Waves and Sound · medium

A small isotropic sound source produces an intensity level of 80 dB at a distance of 10 m in free space. Identical, incoherent sources of this kind…

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A small isotropic sound source produces an intensity level of 80 dB80\ \mathrm{dB} at a distance of 10 m10\ \mathrm{m} in free space. Identical, incoherent sources of this kind are switched on together at the same spot. The number of sources needed so that the intensity level at a distance of 40 m40\ \mathrm{m} is 80 dB80\ \mathrm{dB} is
  1. A. 1616
  2. B. 44
  3. C. 88
  4. D. 3232
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Answer: (A) 1616

For an isotropic source, I∝1/r2I \propto 1/r^2. Going from 10 m10\ \mathrm{m} to 40 m40\ \mathrm{m} reduces the intensity of one source by a factor of 1616, i.e. its level falls by 10log⁡1016=12 dB10\log_{10}16 = 12\ \mathrm{dB} (to about 68 dB68\ \mathrm{dB}). Intensities of incoherent sources add, so NN sources give NN times the intensity, raising the level by 10log⁡10N10\log_{10}N. To restore 80 dB80\ \mathrm{dB} we need N=16N = 16. The value 44 results from taking I∝1/rI\propto 1/r, and 88 from using the 12 dB12\ \mathrm{dB} loss as 10log⁡N10\log N with log⁡2\log 2 rounding.
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