A point source is emitting sound waves of intensity 16 × 10⁻⁸ Wm⁻² at the origin. The difference in intensity (magnitude only) at two points located at a distances of 2 m and 4 m from the origin respectively will be _________ × 10⁻⁸ Wm⁻².

Solution

Physics sir · 4 steps

Intensity falls as 1/r², so take the given value at 1 m and scale it out to 2 m and 4 m.

1. Why inverse square. A point source sends its power P out equally in all directions. At distance r that power is spread over a sphere of area 4π r², so I = P/(4π r²), which means I ∝ 1/r².
2. Read the given value. Intensity exactly at a point source would be infinite. So the given 16 × 10⁻⁸ W m⁻² is taken as the intensity at the reference distance r = 1 m, which is how this question is meant to be read.

   I(r) = I₁/r², I₁ = 16 × 10⁻⁸ W m⁻²

3. Put in numbers. At r = 2 m: I = (16 × 10⁻⁸)/4 = 4 × 10⁻⁸ W m⁻². At r = 4 m: I = (16 × 10⁻⁸)/16 = 1 × 10⁻⁸ W m⁻².
4. So. The difference is (4 - 1) × 10⁻⁸ = 3 × 10⁻⁸ W m⁻², so the answer is 3.

The common slip: 4. Using I ∝ 1/r instead of 1/r² gives intensities of 8 and 4, so the difference comes out as 4, not 3.

The idea to keep. For a point source, intensity goes as 1/r² because the same power covers a sphere of area 4π r². Amplitude goes as 1/r. Do not mix the two up.

Answer from the NTA final answer key. Solution by PrepWiser.