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. (a)`tan^(-1)(2/3)`
B. (b)`tan^(-1)(3/2)`
C. (c)`sin^(-1)(2/3)`
D. (d)`sin^(-1)(3/2)`

Answer

To find the critical angle for the medium, we can use the relationship between the wavelengths of light in air and in the medium, and the sine of the critical angle. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Wavelength in air (\( \lambda_1 \)) = 6000 Å
- Wavelength in medium (\( \lambda_2 \)) = 4000 Å

### Step 2: Use the formula for critical angle
The critical angle (\( C \)) can be calculated using the formula:
\[ \frac{1}{\sin C} = \frac{\lambda_1}{\lambda_2} \]
Where:
- \( \lambda_1 \) is the wavelength in air
- \( \lambda_2 \) is the wavelength in the medium

### Step 3: Substitute the values into the formula
Substituting the given values into the formula:
\[ \frac{1}{\sin C} = \frac{6000 \, \text{Å}}{4000 \, \text{Å}} \]

### Step 4: Simplify the equation
This simplifies to:
\[ \frac{1}{\sin C} = \frac{6000}{4000} = \frac{3}{2} \]

### Step 5: Rearranging for sine of the critical angle
Taking the reciprocal gives:
\[ \sin C = \frac{2}{3} \]

### Step 6: Calculate the critical angle
Now, to find the critical angle \( C \):
\[ C = \sin^{-1}\left(\frac{2}{3}\right) \]

### Step 7: Final answer
Using a calculator, we can find:
\[ C \approx 41.81^\circ \]

Thus, the critical angle for the medium is approximately \( 41.81^\circ \).

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