Glass has refractive index `mu` with respect to air and the critical angle for a ray of light going from glass to air is `theta`. If a ray of light is incident from air on the glass with angle of incidence `theta`, the corresponding angle of refraction is

Options

A. `sin^(-1)((1)/(sqrt(mu)))`
B. `90^(@)`
C. `sin^(-1)((1)/(mu^(2)))`
D. `sin^(-1)((1)/(mu))`

Explanation

To solve the problem, we will use Snell's law, which relates the angles of incidence and refraction to the refractive indices of the two media involved.

### Step-by-Step Solution:

1. **Understanding the Critical Angle**:
The critical angle \( \theta \) is defined as the angle of incidence in the denser medium (glass) that results in the angle of refraction being \( 90^\circ \) in the less dense medium (air). According to Snell's law:
\[ \mu \sin \theta = 1 \cdot \sin 90^\circ \]
Since \( \sin 90^\circ = 1 \), we can simplify this to:
\[ \mu \sin \theta = 1 \]
This gives us:
\[ \mu = \frac{1}{\sin \theta} \]

2. **Setting Up the Problem**:
Now, we need to find the angle of refraction \( r \) when a ray of light is incident from air into glass at the angle \( \theta \). According to Snell's law for this scenario:
\[ n_1 \sin \theta = n_2 \sin r \]
Here, \( n_1 = 1 \) (refractive index of air), \( n_2 = \mu \) (refractive index of glass), and \( \theta \) is the angle of incidence.

3. **Applying Snell's Law**:
Substituting the values into Snell's law:
\[ 1 \cdot \sin \theta = \mu \sin r \]
Rearranging gives:
\[ \sin r = \frac{\sin \theta}{\mu} \]

4. **Substituting for \( \sin \theta \)**:
From the earlier step, we know that \( \mu = \frac{1}{\sin \theta} \). Therefore, we can express \( \sin \theta \) as:
\[ \sin \theta = \frac{1}{\mu} \]
Substituting this back into our equation for \( \sin r \):
\[ \sin r = \frac{\frac{1}{\mu}}{\mu} = \frac{1}{\mu^2} \]

5. **Finding the Angle of Refraction**:
To find the angle \( r \), we take the inverse sine:
\[ r = \sin^{-1} \left( \frac{1}{\mu^2} \right) \]

### Final Answer:
The corresponding angle of refraction \( r \) is:
\[ r = \sin^{-1} \left( \frac{1}{\mu^2} \right) \]