`n` identical coherent waves each with the same initial phase arrive at a point with identical path length. The intensity produced at this point is `l_(1)`. If the waves are all incoherent, the intensity produced is `l_(2)`. The ratio `(l_(1))/(l_(2))` is

Options

A. `sqrt(n)`
B. `n`
C. `n^(3)`
D. `n^(2)`

Answer

To solve the problem, we need to find the ratio of the intensities produced by coherent and incoherent waves at a point. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Understanding Coherent Waves**:
- For `n` identical coherent waves arriving at a point with the same initial phase and identical path length, the intensity produced at this point is denoted as \( l_1 \).
- The intensity for coherent waves can be expressed in terms of amplitude. If the intensity of each wave is \( I \), then the resultant intensity \( l_1 \) can be calculated as:
\[ l_1 = (n \cdot A)^2 \]
where \( A \) is the amplitude of each wave.

2. **Intensity Relation**:
- Since intensity \( I \) is proportional to the square of the amplitude, we can express \( l_1 \) as:
\[ l_1 = n^2 I \]
- Here, \( I \) is the intensity of one wave.

3. **Understanding Incoherent Waves**:
- For incoherent waves, the intensity produced is simply the sum of the individual intensities. Thus, if each wave has the same intensity \( I \), the total intensity \( l_2 \) for `n` incoherent waves is:
\[ l_2 = I + I + ... + I \quad (n \text{ times}) = nI \]

4. **Finding the Ratio**:
- Now, we need to find the ratio \( \frac{l_1}{l_2} \):
\[ \frac{l_1}{l_2} = \frac{n^2 I}{n I} \]
- Simplifying this gives:
\[ \frac{l_1}{l_2} = \frac{n^2}{n} = n \]

5. **Final Answer**:
- Therefore, the ratio \( \frac{l_1}{l_2} \) is:
\[ \frac{l_1}{l_2} = n \]