Two optical media of refractive indices `n_1` and `n_2` contain x and y waves of the same colour in the same thickness. Then their relative refractive index `n_1//n_2`

Options

A. xy
B. `y/x`
C. `x/y`
D. `(y-x)/x`

Answer

To find the relative refractive index \( \frac{n_1}{n_2} \) of two optical media containing waves of the same color, we can follow these steps:

### Step 1: Understand the relationship between refractive index and wave properties
The refractive index \( n \) of a medium is defined as:
\[ n = \frac{c}{v} \]
where \( c \) is the speed of light in vacuum and \( v \) is the speed of light in the medium.

### Step 2: Relate speed of light to wavelength and frequency
The speed of light in a medium can also be expressed in terms of wavelength \( \lambda \) and frequency \( f \):
\[ v = f \cdot \lambda \]
Thus, we can rewrite the refractive index in terms of wavelength:
\[ n = \frac{c}{f \cdot \lambda} \]

### Step 3: Write the relationship for both media
For the first medium with refractive index \( n_1 \) and thickness \( x \):
\[ x = \frac{T}{\frac{\lambda}{n_1}} \implies x = \frac{T \cdot n_1}{\lambda} \]
For the second medium with refractive index \( n_2 \) and thickness \( y \):
\[ y = \frac{T}{\frac{\lambda}{n_2}} \implies y = \frac{T \cdot n_2}{\lambda} \]

### Step 4: Set the thicknesses equal
Since the thicknesses \( x \) and \( y \) are the same, we can equate the two expressions:
\[ \frac{T \cdot n_1}{\lambda} = \frac{T \cdot n_2}{\lambda} \]
Here, \( T \) and \( \lambda \) cancel out, leading to:
\[ n_1 = n_2 \cdot \frac{x}{y} \]

### Step 5: Rearrange to find the relative refractive index
Rearranging the above equation gives us:
\[ \frac{n_1}{n_2} = \frac{x}{y} \]

### Conclusion
Thus, the relative refractive index \( \frac{n_1}{n_2} \) is equal to the ratio of the thicknesses of the two media:
\[ \frac{n_1}{n_2} = \frac{x}{y} \]

### Final Answer
The relative refractive index \( \frac{n_1}{n_2} \) is \( \frac{x}{y} \).

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