A point source emits sound equally in all directions in a non-absorbing medium. Two point `P` and `Q` are at distance of `2 m` and `3 m` respectively from the source. The ratio of the intensities of the wave at `P` and `Q` is :

Options

A. `9:4`
B. `2:3`
C. `3:2`
D. `4:9`

Answer

To solve the problem of finding the ratio of the intensities of sound waves at points P and Q, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Intensity**:
The intensity (I) of sound at a distance (r) from a point source is given by the formula:
\[ I = \frac{P}{A} \]
where \(P\) is the power of the source and \(A\) is the area over which the power is distributed.

2. **Area Calculation**:
For a point source emitting sound uniformly in all directions, the area \(A\) over which the sound spreads is the surface area of a sphere:
\[ A = 4\pi r^2 \]

3. **Intensity at Point P**:
For point P, which is at a distance of \(2 m\) from the source:
\[ I_P = \frac{P}{4\pi (2^2)} = \frac{P}{4\pi \cdot 4} = \frac{P}{16\pi} \]

4. **Intensity at Point Q**:
For point Q, which is at a distance of \(3 m\) from the source:
\[ I_Q = \frac{P}{4\pi (3^2)} = \frac{P}{4\pi \cdot 9} = \frac{P}{36\pi} \]

5. **Finding the Ratio of Intensities**:
Now, we need to find the ratio of the intensities at points P and Q:
\[ \frac{I_P}{I_Q} = \frac{\frac{P}{16\pi}}{\frac{P}{36\pi}} = \frac{P}{16\pi} \cdot \frac{36\pi}{P} \]
Here, \(P\) and \(4\pi\) cancel out:
\[ \frac{I_P}{I_Q} = \frac{36}{16} = \frac{9}{4} \]

6. **Final Answer**:
Therefore, the ratio of the intensities of the wave at points P and Q is:
\[ \text{Ratio} = 9 : 4 \]