If the amplitude of a wave at a distance `r` from a point source is `A`, the amplitude at a distance `2 r` will be

Options

A. `2 A`
B. `A`
C. `A//2`
D. `A//4`

Explanation

To solve the problem of finding the amplitude of a wave at a distance `2r` from a point source, given that the amplitude at a distance `r` is `A`, we will use the relationship between amplitude and distance from a point source.

### Step-by-step Solution:

1. **Understand the Relationship**: The amplitude of a wave from a point source decreases with distance. The intensity (I) of the wave is inversely proportional to the square of the distance (r) from the source. Mathematically, this can be expressed as:
\[ I \propto \frac{1}{r^2} \]
Since intensity is proportional to the square of the amplitude (A), we have:
\[ I \propto A^2 \]

2. **Set Up the Ratios**: Let’s denote the amplitude at distance `r` as \( A_1 = A \) and at distance `2r` as \( A_2 \). According to the relationship derived, we can write:
\[ \frac{A_1^2}{A_2^2} = \frac{r^2}{(2r)^2} \]

3. **Substitute Known Values**: Substitute \( A_1 = A \) and the distances into the equation:
\[ \frac{A^2}{A_2^2} = \frac{r^2}{4r^2} \]
Simplifying the right side gives:
\[ \frac{A^2}{A_2^2} = \frac{1}{4} \]

4. **Solve for \( A_2 \)**: Rearranging the equation to find \( A_2 \):
\[ A_2^2 = 4A^2 \]
Taking the square root of both sides:
\[ A_2 = \frac{A}{2} \]

5. **Conclusion**: Therefore, the amplitude at a distance \( 2r \) from the point source is:
\[ A_2 = \frac{A}{2} \]

### Final Answer:
The amplitude at a distance \( 2r \) will be \( \frac{A}{2} \).

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