In vacum, to travel distance d, light takes time t and in medium to travel distance 5d, it takes time T. The critical angle of the medium is

Options

A. `sin^(-1)((5T)/(t))`
B. `sin^(-1)((5T)/(3t))`
C. `sin^(-1)((5t)/(T))`
D. `sin^(-1)((3t)/(5T))`

Explanation

To find the critical angle of the medium based on the given information, we can follow these steps:

### Step 1: Determine the speed of light in vacuum
In vacuum, the speed of light (c) can be calculated using the formula:
\[ c = \frac{d}{t} \]
where \( d \) is the distance traveled and \( t \) is the time taken.

### Step 2: Determine the speed of light in the medium
In the medium, we know that light travels a distance of \( 5d \) in time \( T \). Therefore, the speed of light in the medium (V) is given by:
\[ V = \frac{5d}{T} \]

### Step 3: Relate the refractive index to the speeds
The refractive index (μ) of the medium can be defined as the ratio of the speed of light in vacuum to the speed of light in the medium:
\[ \mu = \frac{c}{V} \]
Substituting the values we found in Steps 1 and 2:
\[ \mu = \frac{\frac{d}{t}}{\frac{5d}{T}} \]

### Step 4: Simplify the expression for refractive index
Cancelling \( d \) from the numerator and denominator:
\[ \mu = \frac{T}{5t} \]

### Step 5: Use the critical angle formula
The critical angle (\( \theta_C \)) is given by the formula:
\[ \mu \sin(\theta_C) = 1 \]
Since \( \sin(90^\circ) = 1 \), we can rearrange this to find \( \sin(\theta_C) \):
\[ \sin(\theta_C) = \frac{1}{\mu} \]

### Step 6: Substitute the value of μ
Substituting the expression for μ from Step 4 into the equation:
\[ \sin(\theta_C) = \frac{1}{\frac{T}{5t}} = \frac{5t}{T} \]

### Step 7: Find the critical angle
To find the critical angle, we take the inverse sine:
\[ \theta_C = \sin^{-1}\left(\frac{5t}{T}\right) \]

Thus, the critical angle of the medium is:
\[ \theta_C = \sin^{-1}\left(\frac{5t}{T}\right) \]