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/sqrt3)`
C. `sin^(-1)(2/sqrt3)`
D. `cos^(-1)(2/sqrt3)`

Explanation

To find the critical angle for a ray refracted from one medium to another, we can follow these steps:

### Step 1: Understand the relationship between refractive index and critical angle
The critical angle (C) is related to the refractive indices of the two media. The formula is given by:
\[ n_2 = \frac{1}{\sin C} \]
where \( n_2 \) is the refractive index of the second medium with respect to the first.

### Step 2: Use the relationship between refractive indices and wavelengths
The refractive index of a medium can also be expressed in terms of the wavelengths of light in the two media:
\[ n_1 = \frac{\lambda_1}{\lambda_2} \]
where \( \lambda_1 \) is the wavelength in the first medium and \( \lambda_2 \) is the wavelength in the second medium.

### Step 3: Plug in the given wavelengths
From the question, we know:
- \( \lambda_1 = 6000 \, \text{Å} \)
- \( \lambda_2 = 4000 \, \text{Å} \)

Now, we can calculate the refractive index:
\[ n_2 = \frac{\lambda_1}{\lambda_2} = \frac{6000 \, \text{Å}}{4000 \, \text{Å}} = \frac{3}{2} \]

### Step 4: Substitute the refractive index into the critical angle formula
Now we can substitute \( n_2 \) into the critical angle formula:
\[ \frac{1}{\sin C} = \frac{3}{2} \]

### Step 5: Solve for sin C
Taking the reciprocal gives us:
\[ \sin C = \frac{2}{3} \]

### Step 6: Calculate the critical angle C
Now we can find the critical angle by taking the inverse sine:
\[ C = \sin^{-1}\left(\frac{2}{3}\right) \]

### Final Result
Thus, the critical angle for a ray from the second medium is:
\[ C = \sin^{-1}\left(\frac{2}{3}\right) \]

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