A monochromatic beam of light is travelling from medium A of refractive index `n_1` to a medium B of refractive index . `n_2` In the medium A, there are x number of waves in certain distance. In the medium B, there are y numbers of waves in the same distance. Then, refractive index of medium A with respect to medium B is …

Options

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

Explanation

To find the refractive index of medium A with respect to medium B, we can follow these steps:

### Step 1: Understand the relationship between wavelength and refractive index
The refractive index (n) of a medium is inversely proportional to the wavelength (λ) of light in that medium. This can be expressed as:
\[ n \propto \frac{1}{\lambda} \]
Thus, if we denote the wavelengths in medium A and medium B as \( \lambda_1 \) and \( \lambda_2 \) respectively, we can write:
\[ n_1 \propto \frac{1}{\lambda_1} \]
\[ n_2 \propto \frac{1}{\lambda_2} \]

### Step 2: Relate the number of waves to wavelength
In medium A, there are \( x \) waves in a distance \( D \), which gives us the wavelength in medium A:
\[ \lambda_1 = \frac{D}{x} \]
In medium B, there are \( y \) waves in the same distance \( D \), which gives us the wavelength in medium B:
\[ \lambda_2 = \frac{D}{y} \]

### Step 3: Find the ratio of wavelengths
Now, we can find the ratio of the wavelengths:
\[ \frac{\lambda_1}{\lambda_2} = \frac{D/x}{D/y} = \frac{y}{x} \]
This means:
\[ \frac{\lambda_1}{\lambda_2} = \frac{y}{x} \]

### Step 4: Relate the refractive indices using the wavelengths
Since the refractive indices are inversely proportional to the wavelengths, we can write:
\[ \frac{n_1}{n_2} = \frac{\lambda_2}{\lambda_1} \]
Substituting the ratio of wavelengths we found:
\[ \frac{n_1}{n_2} = \frac{x}{y} \]

### Step 5: Find the refractive index of medium A with respect to medium B
To express the refractive index of medium A with respect to medium B, we need to find \( \frac{n_1}{n_2} \):
\[ n_1 = n_2 \cdot \frac{x}{y} \]
Thus, the refractive index of medium A with respect to medium B is:
\[ \frac{n_1}{n_2} = \frac{x}{y} \]

### Final Answer
The refractive index of medium A with respect to medium B is:
\[ n_1 : n_2 = \frac{x}{y} \]

---