A glass slab of thickness 8 cms contains the same number of waves as 10 cms long path of water when both are traversed by the same monochromatic light. If the refractive index of water is 4/3, the refractive index of glass is

Options

A. `5/3`
B. `5/4`
C. `16/15`
D. `3/2`

Answer

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

### Step 1: Understand the relationship between thickness, wavelength, and refractive index
The number of waves traversed in a medium is related to the thickness of the medium, the wavelength of light in vacuum, and the refractive index of that medium. The relationship can be expressed as:
\[ t = n \cdot \frac{\lambda_0}{\mu} \]
where:
- \( t \) is the thickness of the medium,
- \( n \) is the number of waves,
- \( \lambda_0 \) is the wavelength of light in vacuum,
- \( \mu \) is the refractive index of the medium.

### Step 2: Set up the equations for both media
Let:
- \( t_1 = 8 \) cm (thickness of the glass slab),
- \( t_2 = 10 \) cm (thickness of the water),
- \( \mu_w = \frac{4}{3} \) (refractive index of water),
- \( \mu_g \) (refractive index of glass, which we need to find).

From the problem, we know that the number of waves in both media is the same, so we can set up the following relationship:
\[ \frac{t_1}{t_2} = \frac{\mu_w}{\mu_g} \]

### Step 3: Substitute the known values into the equation
Substituting the known values into the equation gives:
\[ \frac{8 \text{ cm}}{10 \text{ cm}} = \frac{\frac{4}{3}}{\mu_g} \]

### Step 4: Simplify the equation
This simplifies to:
\[ \frac{8}{10} = \frac{4}{3\mu_g} \]
or
\[ \frac{4}{5} = \frac{4}{3\mu_g} \]

### Step 5: Cross-multiply to solve for \( \mu_g \)
Cross-multiplying gives:
\[ 4 \cdot 3\mu_g = 4 \cdot 5 \]
which simplifies to:
\[ 12\mu_g = 20 \]

### Step 6: Solve for \( \mu_g \)
Now, divide both sides by 12:
\[ \mu_g = \frac{20}{12} = \frac{5}{3} \]

### Conclusion
The refractive index of the glass slab is:
\[ \mu_g = \frac{5}{3} \]

---