The refractive index of glass plate is `3/2`. Then the correct thickness of glass plate that will permit the same number of wavelengths as that by an 18 cm long column of water is
`(mu_(w)=4/3)`

Options

A. 12cm
B. 16cm
C. 18cm
D. 24cm

Explanation

To solve the problem, we need to find the thickness of the glass plate that allows the same number of wavelengths of light to pass through it as an 18 cm long column of water. We will use the concept of refractive index and the relationship between wavelength, thickness, and refractive index.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Refractive index of glass, \( \mu_g = \frac{3}{2} \)
- Refractive index of water, \( \mu_w = \frac{4}{3} \)
- Length of the water column, \( T_w = 18 \, \text{cm} \)

2. **Understand the Relationship:**
The number of wavelengths in a medium can be expressed as:
\[ N = \frac{\mu \cdot T}{\lambda} \]
where \( N \) is the number of wavelengths, \( \mu \) is the refractive index, \( T \) is the thickness or length of the medium, and \( \lambda \) is the wavelength in vacuum.

3. **Set Up the Equation:**
Since we want the same number of wavelengths in both media (water and glass), we can set up the equation:
\[ \mu_g \cdot T_g = \mu_w \cdot T_w \]
where \( T_g \) is the thickness of the glass plate.

4. **Substitute the Known Values:**
Plugging in the known values:
\[ \frac{3}{2} \cdot T_g = \frac{4}{3} \cdot 18 \]

5. **Calculate the Right Side:**
First, calculate \( \frac{4}{3} \cdot 18 \):
\[ \frac{4 \cdot 18}{3} = \frac{72}{3} = 24 \]

6. **Solve for \( T_g \):**
Now, we have:
\[ \frac{3}{2} \cdot T_g = 24 \]
To find \( T_g \), multiply both sides by \( \frac{2}{3} \):
\[ T_g = 24 \cdot \frac{2}{3} = 16 \, \text{cm} \]

7. **Conclusion:**
The thickness of the glass plate that will permit the same number of wavelengths as that by an 18 cm long column of water is \( T_g = 16 \, \text{cm} \).

### Final Answer:
The thickness of the glass plate is **16 cm**.

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