In an intereference pattern ,the slit widths are in the ratio 1:16 .then find out the ratio of minimum and maxima intensity.

Explanation

To solve the problem of finding the ratio of minimum and maximum intensity in an interference pattern where the slit widths are in the ratio 1:16, we can follow these steps:

### Step 1: Understand the relationship between intensity and amplitude
The intensity (I) of light is proportional to the square of the amplitude (A) of the wave. Therefore, we can express the intensity in terms of the slit widths (W):
\[ I_1 \propto W_1 \quad \text{and} \quad I_2 \propto W_2 \]
Thus, we can write:
\[ \frac{I_1}{I_2} = \frac{W_1}{W_2} \]

### Step 2: Use the given ratio of slit widths
Given that the slit widths are in the ratio:
\[ \frac{W_1}{W_2} = \frac{1}{16} \]
This means:
\[ W_1 = 1 \quad \text{and} \quad W_2 = 16 \]

### Step 3: Relate amplitudes to slit widths
Since intensity is proportional to the square of the amplitude, we can relate the amplitudes as follows:
\[ \frac{A_1^2}{A_2^2} = \frac{W_1}{W_2} \]
This implies:
\[ \frac{A_1}{A_2} = \sqrt{\frac{W_1}{W_2}} = \sqrt{\frac{1}{16}} = \frac{1}{4} \]
Thus, we can say:
\[ A_1 = 1 \quad \text{and} \quad A_2 = 4 \]

### Step 4: Calculate minimum intensity
The minimum intensity (I_min) in an interference pattern is given by:
\[ I_{\text{min}} = (A_1 - A_2)^2 \]
Substituting the values:
\[ I_{\text{min}} = (1 - 4)^2 = (-3)^2 = 9 \]

### Step 5: Calculate maximum intensity
The maximum intensity (I_max) is given by:
\[ I_{\text{max}} = (A_1 + A_2)^2 \]
Substituting the values:
\[ I_{\text{max}} = (1 + 4)^2 = (5)^2 = 25 \]

### Step 6: Find the ratio of minimum intensity to maximum intensity
Now we can find the ratio of minimum intensity to maximum intensity:
\[ \frac{I_{\text{min}}}{I_{\text{max}}} = \frac{9}{25} \]

### Final Answer
Thus, the ratio of minimum intensity to maximum intensity is:
\[ \boxed{\frac{9}{25}} \]

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