`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 arriving at a point. Let's break down the solution step by step.

### Step 1: Understand the Intensities of Coherent Waves
When `n` identical coherent waves arrive at a point with the same initial phase and identical path lengths, the resultant intensity \( I_1 \) can be expressed as:
\[ I_1 = n^2 I_0 \]
where \( I_0 \) is the intensity of one individual wave.

### Step 2: Understand the Intensities of Incoherent Waves
When the same `n` waves are incoherent, the resultant intensity \( I_2 \) is simply the sum of the individual intensities:
\[ I_2 = n I_0 \]

### Step 3: Calculate the Ratio of Intensities
Now, we can find the ratio of the intensities \( \frac{I_1}{I_2} \):
\[ \frac{I_1}{I_2} = \frac{n^2 I_0}{n I_0} \]

### Step 4: Simplify the Ratio
By simplifying the expression, we can cancel out \( I_0 \) (assuming \( I_0 \neq 0 \)) and \( n \):
\[ \frac{I_1}{I_2} = \frac{n^2}{n} = n \]

### Final Answer
Thus, the ratio of the intensities produced by coherent waves to incoherent waves is:
\[ \frac{I_1}{I_2} = n \]

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