A glass slab of thickness `4 cm` contains the same number of waves as `5 cm` of water, when both are traversed by the same monochromatic light. If the refractive index of water is `4//3,` then refractive index of glass is

Options

A. `5//3`
B. `5//4`
C. `16//15`
D. `1.5`

Explanation

To find the refractive index of the glass slab, we can follow these steps:

### Step 1: Understanding the Problem
We know that a glass slab of thickness \( T_g = 4 \, \text{cm} \) contains the same number of waves as \( T_w = 5 \, \text{cm} \) of water. The refractive index of water \( n_w = \frac{4}{3} \). We need to find the refractive index of the glass slab \( n_g \).

### Step 2: Relating Thickness and Wavelength
The number of waves \( n \) that pass through a medium can be expressed as:
\[ n = \frac{T}{\lambda} \]
where \( T \) is the thickness of the medium and \( \lambda \) is the wavelength of light in that medium.

### Step 3: Wavelength in Different Media
The wavelength in a medium is related to the wavelength in vacuum \( \lambda_0 \) by:
\[ \lambda = \frac{\lambda_0}{n} \]
Thus, for the glass slab and water, we have:
- For glass: \( \lambda_g = \frac{\lambda_0}{n_g} \)
- For water: \( \lambda_w = \frac{\lambda_0}{n_w} \)

### Step 4: Setting Up the Equation
Since the number of waves is the same in both media, we can set up the equation:
\[ \frac{T_g}{\lambda_g} = \frac{T_w}{\lambda_w} \]
Substituting the expressions for \( \lambda_g \) and \( \lambda_w \):
\[ \frac{T_g}{\frac{\lambda_0}{n_g}} = \frac{T_w}{\frac{\lambda_0}{n_w}} \]
This simplifies to:
\[ \frac{T_g \cdot n_w}{T_w} = n_g \]

### Step 5: Plugging in the Values
Now, substituting the known values:
- \( T_g = 4 \, \text{cm} \)
- \( T_w = 5 \, \text{cm} \)
- \( n_w = \frac{4}{3} \)

We get:
\[ n_g = \frac{4 \cdot \frac{4}{3}}{5} \]

### Step 6: Calculating \( n_g \)
Calculating the right-hand side:
\[ n_g = \frac{16/3}{5} = \frac{16}{15} \]

### Final Answer
Thus, the refractive index of the glass slab is:
\[ n_g = \frac{16}{15} \]

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