Wavelength of given light waves in air and in a medium are 6000 Å and 4000 Å respectively . The critical angle for the medium is given by

Options

A. `sin^(-1) ((3)/(4))`
B. `sin^(-1)((2)/(3))`
C. `tan^(-1) ((2)/(3))`
D. `tan^(-1) ((3)/(4))`

Explanation

To find the critical angle for the given medium based on the provided wavelengths, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- Wavelength in air, \( \lambda_{air} = 6000 \, \text{Å} \)
- Wavelength in the medium, \( \lambda_{medium} = 4000 \, \text{Å} \)

2. **Determine the Refractive Index**:
The refractive index \( \mu \) of a medium can be calculated using the formula:
\[ \mu = \frac{\lambda_{air}}{\lambda_{medium}} \]
Substituting the given values:
\[ \mu = \frac{6000 \, \text{Å}}{4000 \, \text{Å}} = \frac{6000}{4000} = \frac{3}{2} = 1.5 \]

3. **Use the Critical Angle Formula**:
The critical angle \( C \) can be found using the formula:
\[ \sin C = \frac{1}{\mu} \]
Substituting the refractive index we calculated:
\[ \sin C = \frac{1}{\frac{3}{2}} = \frac{2}{3} \]

4. **Calculate the Critical Angle**:
To find the critical angle \( C \), take the inverse sine:
\[ C = \sin^{-1}\left(\frac{2}{3}\right) \]

5. **Conclusion**:
The critical angle for the medium is \( C = \sin^{-1}\left(\frac{2}{3}\right) \).