A point source is emitting sound waves of intensity \( 16 \times 10^{-8} \, Wm^{-2} \) at the origin. The difference in intensity (magnitude only) at two points located at distances of 2 m and 4 m from the origin respectively will be _____ \( \times 10^{-8} \, Wm^{-2} \).
A point source is emitting sound waves of intensity \( 16 \times 10^{-8} \, Wm^{-2} \) at the origin. The difference in intensity (magnitude only) at two points located at distances of 2 m and 4 m from the origin respectively will be _____ \( \times 10^{-8} \, Wm^{-2} \).
Correct Answer: 3
Solution and Explanation
The intensity of sound waves from a point source decreases with the square of the distance from the source. The formula for intensity \( I \) at a distance \( r \) from a point source is given by:
\[ I = \frac{P}{4\pi r^2}, \] where \( P \) is the power of the source.
Given Values: Intensity at the origin \( I_0 = 16 \times 10^{-8} \, Wm^{-2} \). Distances: \( r_1 = 2 \, m \) and \( r_2 = 4 \, m \).
Intensity at Distances \( r_1 \) and \( r_2 \): The intensity at distance \( r_1 = 2 \, m \):
\[ I_1 = I_0 \left( \frac{r_0}{r_1} \right)^2 = 16 \times 10^{-8} \times \left( \frac{1}{2} \right)^2 = 16 \times 10^{-8} \times \frac{1}{4} = 4 \times 10^{-8} \, Wm^{-2}. \]
The intensity at distance \( r_2 = 4 \, m \):
\[ I_2 = I_0 \left( \frac{r_0}{r_2} \right)^2 = 16 \times 10^{-8} \times \left( \frac{1}{4} \right)^2 = 16 \times 10^{-8} \times \frac{1}{16} = 1 \times 10^{-8} \, Wm^{-2}. \]
Calculating the Difference in Intensity: The difference in intensity \( \Delta I \) between the two points:
\[ \Delta I = I_1 - I_2 = (4 \times 10^{-8} - 1 \times 10^{-8}) \, Wm^{-2} = 3 \times 10^{-8} \, Wm^{-2}. \]
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