A transparent glass plate of thickness 0.5 mm and refractive index 1.5 is placed infront of one of the slits in a double slit experiment. If the wavelength of light used is `6000 A^(@)`, the ratio of maximum to minimum intensity in the interference pattern is 25/4. Then the ratio of light intensity transmitted to incident on thin transparent glass plate is

Options

A. `9:7`
B. `9:49`
C. `3:7`
D. `7:3`

Explanation

To solve the problem step by step, we will analyze the given information and apply the principles of wave optics.

### Step 1: Understand the Problem
We have a double-slit experiment where a glass plate is placed in front of one of the slits. The thickness of the glass plate is 0.5 mm, and its refractive index is 1.5. The wavelength of the light used is 6000 Å (Angstroms). We need to find the ratio of the light intensity transmitted through the glass plate to the incident light intensity.

### Step 2: Given Data
- Thickness of the glass plate, \( t = 0.5 \, \text{mm} = 0.5 \times 10^{-3} \, \text{m} \)
- Refractive index of glass, \( n = 1.5 \)
- Wavelength of light, \( \lambda = 6000 \, \text{Å} = 6000 \times 10^{-10} \, \text{m} \)
- Ratio of maximum to minimum intensity, \( \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{25}{4} \)

### Step 3: Relationship Between Intensities
From the interference pattern, the maximum and minimum intensities can be expressed as:
\[ I_{\text{max}} = \left( \sqrt{I_1} + \sqrt{I_2} \right)^2 \]
\[ I_{\text{min}} = \left( \sqrt{I_1} - \sqrt{I_2} \right)^2 \]
Where \( I_1 \) is the intensity of light passing through the slit without the glass plate, and \( I_2 \) is the intensity of light passing through the slit with the glass plate.

### Step 4: Set Up the Intensity Ratio
Given the ratio of maximum to minimum intensity:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{25}{4} \]
Substituting the expressions for \( I_{\text{max}} \) and \( I_{\text{min}} \):
\[ \frac{\left( \sqrt{I_1} + \sqrt{I_2} \right)^2}{\left( \sqrt{I_1} - \sqrt{I_2} \right)^2} = \frac{25}{4} \]

### Step 5: Simplifying the Ratio
Taking the square root of both sides:
\[ \frac{\sqrt{I_1} + \sqrt{I_2}}{\sqrt{I_1} - \sqrt{I_2}} = \frac{5}{2} \]
Cross-multiplying gives:
\[ 2(\sqrt{I_1} + \sqrt{I_2}) = 5(\sqrt{I_1} - \sqrt{I_2}) \]
Expanding this:
\[ 2\sqrt{I_1} + 2\sqrt{I_2} = 5\sqrt{I_1} - 5\sqrt{I_2} \]
Rearranging terms:
\[ 7\sqrt{I_2} = 3\sqrt{I_1} \]

### Step 6: Squaring Both Sides
Squaring both sides gives:
\[ 49 I_2 = 9 I_1 \]
Thus, we can express the ratio of the intensities:
\[ \frac{I_2}{I_1} = \frac{9}{49} \]

### Step 7: Calculate the Ratio of Transmitted to Incident Intensity
The intensity transmitted through the glass plate \( I_2 \) is related to the incident intensity \( I_1 \) by the ratio we found:
\[ \text{Ratio of transmitted to incident intensity} = \frac{I_2}{I_1} = \frac{9}{49} \]

### Final Answer
The ratio of light intensity transmitted to the incident intensity on the thin transparent glass plate is:
\[ \frac{9}{49} \]