The wavelength of light in two liquids ‘ x ' and ‘ y ' is 3500 Å and 7000 Å, then the critical angle of x relative to y will be

Options

A. `60^(@)`
B. `45^(@)`
C. `30^(@)`
D. `15^(@)`

Explanation

To find the critical angle of liquid 'x' relative to liquid 'y', we can follow these steps:

### Step 1: Understand the concept of critical angle
The critical angle is defined as the angle of incidence at which the angle of refraction is 90 degrees. This phenomenon occurs when light travels from a denser medium to a rarer medium.

### Step 2: Identify the refractive indices
The refractive index (n) of a medium can be related to the wavelength of light in that medium. The refractive index is given by the formula:
\[ n = \frac{\text{Wavelength in vacuum}}{\text{Wavelength in medium}} \]

### Step 3: Calculate the refractive indices for both liquids
Let:
- Wavelength in liquid 'x' = 3500 Å
- Wavelength in liquid 'y' = 7000 Å

Using the formula for refractive index:
- For liquid 'x':
\[ n_x = \frac{\text{Wavelength in vacuum}}{3500 \, \text{Å}} \]
- For liquid 'y':
\[ n_y = \frac{\text{Wavelength in vacuum}}{7000 \, \text{Å}} \]

### Step 4: Find the ratio of refractive indices
The critical angle (C) can be calculated using the formula:
\[ \sin C = \frac{n_y}{n_x} \]

### Step 5: Substitute the values
From our previous calculations, we can express the refractive indices in terms of their wavelengths:
\[ \sin C = \frac{3500 \, \text{Å}}{7000 \, \text{Å}} \]

### Step 6: Simplify the ratio
Calculating the ratio:
\[ \sin C = \frac{3500}{7000} = \frac{1}{2} \]

### Step 7: Calculate the critical angle
Now, we can find the critical angle using the inverse sine function:
\[ C = \sin^{-1}\left(\frac{1}{2}\right) \]

### Step 8: Determine the value
From trigonometric values, we know:
\[ C = 30^\circ \]

### Final Answer
The critical angle of liquid 'x' relative to liquid 'y' is \( 30^\circ \).

---