When a ray is refracted from one medium into another, the wavelegths changes from `6000Å` to `4000Å`. The critical angle for a ray from the second medium will be

Options

A. `cos^(-1)(2/3)`
B. `sin^(-1)(2/sqrt(3))`
C. `sin^(-1)(2/3)`
D. `cos^(-1)(2/sqrt(3))`

Answer

To solve the problem of finding the critical angle for a ray transitioning from one medium to another, given the change in wavelength from \(6000 \, \text{Å}\) to \(4000 \, \text{Å}\), we can follow these steps:

### Step 1: Understand the relationship between wavelength and refractive index
The refractive index (\(\mu\)) of a medium can be related to the wavelengths in two different media using the formula:

\[ \mu_1 = \frac{\lambda_1}{\lambda_2} \]

where \(\lambda_1\) is the wavelength in the first medium and \(\lambda_2\) is the wavelength in the second medium.

### Step 2: Substitute the given wavelengths
From the problem, we have:

- \(\lambda_1 = 6000 \, \text{Å}\)
- \(\lambda_2 = 4000 \, \text{Å}\)

Substituting these values into the formula gives:

\[ \mu_1 = \frac{6000 \, \text{Å}}{4000 \, \text{Å}} = \frac{3}{2} \]

### Step 3: Determine the refractive indices
Since \(\mu_1\) is the refractive index of the first medium and \(\mu_2\) is the refractive index of the second medium, we can express \(\mu_2\) in terms of \(\mu_1\):

\[ \mu_2 = \frac{3}{2} \]

### Step 4: Use Snell's Law to find the critical angle
The critical angle (\(\theta_c\)) can be found using Snell's Law, which states:

\[ \mu_2 \sin(\theta_c) = \mu_1 \sin(90^\circ) \]

Since \(\sin(90^\circ) = 1\), we can simplify this to:

\[ \mu_2 \sin(\theta_c) = \mu_1 \]

Rearranging gives us:

\[ \sin(\theta_c) = \frac{\mu_1}{\mu_2} \]

### Step 5: Substitute the values of refractive indices
Substituting the values we found:

\[ \sin(\theta_c) = \frac{\frac{3}{2}}{\mu_2} \]

However, we need to recognize that \(\mu_2\) is greater than \(\mu_1\) since the ray is moving from a denser medium to a rarer medium. Thus, we can express \(\mu_2\) in terms of \(\mu_1\):

\[ \sin(\theta_c) = \frac{2}{3} \]

### Step 6: Calculate the critical angle
Now, we can find \(\theta_c\):

\[ \theta_c = \sin^{-1}\left(\frac{2}{3}\right) \]

### Final Answer
Thus, the critical angle for a ray from the second medium is:

\[ \theta_c = \sin^{-1}\left(\frac{2}{3}\right) \]