Two transparent media A and B are separated by a plane boundary. The speed of light in medium A is `2.0xx10^(8)ms^(-1)` and in medium B is `2.5xx10^(8)ms^(-1)`. The critical angle for which a ray of light going from A to B suffers total internal reflection is

Options

A. `sin^(-1)1//2`
B. `sin^(-1)2//5`
C. `sin^(-1)4//5`
D. `sin^(-1)3//4`

Answer

To find the critical angle for total internal reflection when light travels from medium A to medium B, we can use Snell's law. Here’s a step-by-step solution:

### Step 1: Understand the concept of critical angle
The critical angle (C) is defined as the angle of incidence beyond which light cannot pass through the boundary between two media and instead is totally internally reflected. For total internal reflection to occur, light must travel from a medium with a higher refractive index to a medium with a lower refractive index.

### Step 2: Identify the refractive indices
The refractive index (n) of a medium can be calculated using the formula:
\[ n = \frac{c}{v} \]
where:
- \( c \) is the speed of light in a vacuum (approximately \( 3.0 \times 10^8 \) m/s),
- \( v \) is the speed of light in the medium.

For medium A:
\[ n_A = \frac{c}{v_A} = \frac{3.0 \times 10^8 \, \text{m/s}}{2.0 \times 10^8 \, \text{m/s}} = 1.5 \]

For medium B:
\[ n_B = \frac{c}{v_B} = \frac{3.0 \times 10^8 \, \text{m/s}}{2.5 \times 10^8 \, \text{m/s}} = 1.2 \]

### Step 3: Apply Snell's Law
According to Snell's law:
\[ n_A \sin(C) = n_B \sin(90^\circ) \]
Since \( \sin(90^\circ) = 1 \), we can simplify this to:
\[ n_A \sin(C) = n_B \]

### Step 4: Substitute the values of refractive indices
Substituting the values we calculated:
\[ 1.5 \sin(C) = 1.2 \]

### Step 5: Solve for sin(C)
Rearranging the equation gives:
\[ \sin(C) = \frac{1.2}{1.5} = \frac{4}{5} \]

### Step 6: Find the critical angle
To find the critical angle \( C \), we take the inverse sine:
\[ C = \sin^{-1}\left(\frac{4}{5}\right) \]

### Final Answer
Thus, the critical angle for which a ray of light going from A to B suffers total internal reflection is:
\[ C = \sin^{-1}\left(\frac{4}{5}\right) \]