The maximum value of refractive index of a prism which permits the transmission of light through it when the refracting angle of the prism is `90^@,` is given by

Options

A. `1.500`
B. `1.414`
C. `2.000`
D. `1.732`

Explanation

To find the maximum value of the refractive index of a prism that allows the transmission of light when the refracting angle of the prism is \(90^\circ\), we can follow these steps:

### Step 1: Understanding the critical angle
The critical angle (\(\theta_c\)) is the angle of incidence above which total internal reflection occurs. For a prism, the maximum refractive index (\(n\)) must be such that it allows light to pass through the prism without total internal reflection.

### Step 2: Relating the refractive index to the critical angle
The relationship between the refractive index and the critical angle is given by:
\[ \sin(\theta_c) = \frac{1}{n} \]
where \(n\) is the refractive index of the material of the prism.

### Step 3: Applying the condition for the prism
For a prism with a refracting angle \(A = 90^\circ\), the condition for total internal reflection can be expressed as:
\[ A > 2 \theta_c \]
Substituting \(A = 90^\circ\):
\[ 90^\circ > 2 \theta_c \]
This implies:
\[ \theta_c < 45^\circ \]

### Step 4: Finding the sine of the critical angle
Using the relationship from Step 2, we can express the sine of the critical angle:
\[ \sin(\theta_c) < \sin(45^\circ) \]
Since \(\sin(45^\circ) = \frac{1}{\sqrt{2}}\), we have:
\[ \sin(\theta_c) < \frac{1}{\sqrt{2}} \]

### Step 5: Relating sine of the critical angle to refractive index
From the relationship \(\sin(\theta_c) = \frac{1}{n}\), we can substitute:
\[ \frac{1}{n} < \frac{1}{\sqrt{2}} \]
This implies:
\[ n > \sqrt{2} \]

### Step 6: Conclusion
The maximum value of the refractive index of the prism which permits the transmission of light through it when the refracting angle is \(90^\circ\) is:
\[ n > \sqrt{2} \approx 1.414 \]

Thus, the maximum value of the refractive index of the prism is greater than \(1.414\).

### Final Answer
The maximum value of the refractive index of the prism is greater than \(\sqrt{2}\) (approximately \(1.414\)).

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