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 glass, we can follow these steps:

### Step 1: Understand the relationship between the number of waves, thickness, and wavelength.
The number of waves that pass through a medium is given by the formula:
\[ \text{Number of Waves} = \frac{\text{Thickness of Medium}}{\text{Wavelength}} \]
Let \( n_g \) be the number of waves in glass and \( n_w \) be the number of waves in water.

### Step 2: Set up the equations for the number of waves in glass and water.
Given that the thickness of the glass slab is \( 4 \, \text{cm} \) and the thickness of water is \( 5 \, \text{cm} \), we can write:
\[ n_g = \frac{4}{\lambda_g} \]
\[ n_w = \frac{5}{\lambda_w} \]
Since both media contain the same number of waves, we have:
\[ n_g = n_w \]

### Step 3: Equate the two expressions for the number of waves.
Setting the two equations equal gives us:
\[ \frac{4}{\lambda_g} = \frac{5}{\lambda_w} \]

### Step 4: Rearrange the equation to find the relationship between the wavelengths.
Cross-multiplying gives:
\[ 4 \lambda_w = 5 \lambda_g \]
Thus, we can express the wavelengths in terms of each other:
\[ \frac{\lambda_g}{\lambda_w} = \frac{4}{5} \]

### Step 5: Relate the wavelengths to the refractive indices.
The refractive index \( \mu \) is inversely proportional to the wavelength:
\[ \mu \propto \frac{1}{\lambda} \]
This means we can write:
\[ \frac{\lambda_g}{\lambda_w} = \frac{\mu_w}{\mu_g} \]

### Step 6: Substitute the known values.
We know the refractive index of water \( \mu_w = \frac{4}{3} \). Substituting this into the equation gives:
\[ \frac{4}{5} = \frac{\frac{4}{3}}{\mu_g} \]

### Step 7: Solve for the refractive index of glass.
Rearranging the equation to find \( \mu_g \):
\[ \mu_g = \frac{\frac{4}{3} \cdot 5}{4} \]
\[ \mu_g = \frac{5}{3} \]

### Final Answer:
The refractive index of glass is \( \frac{5}{3} \).

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