The refractive index of the material of a prism is ( A2), where A is the angle of the prism. The angle of minimum deviation m is:

Given: The refractive index is n = ( A2). Find: The angle of minimum deviation m. For a prism at minimum deviation, n = ( A + m2) ( A2) Substituting the given value of n, ( A2) = ( A + m2) ( A2) Using ( A2) = ( A2) ( A2) we get ( A2) ( A2) = ( A + m2) ( A2) Hence, ( A2) = ( A + m2) Now use ( A2) = ( 2 - A2) So, ( A + m2) = ( 2 - A2) Equating the arguments as done in the provided solution, A + m2 = 2 - A2 which gives A + m = - A and therefore m = - 2A The solution explicitly states that the correct option is B and derives m = - 2A. This value does not match any listed option exactly. Since the solution itself marks B as correct, the answer is taken as B despite the discrepancy between the derived expression and the option values.

The refractive index of the material of a prism is ( A2), where A… — Solution | JEE Physics

MCQMediumJEE 2024Prisms & Total Internal Reflection

The refractive index of the material of a prism is $\cot\left(\frac{A}{2}\right)$ , where $A$ is the angle of the prism. The angle of minimum deviation $\delta_m$ is:

A
$A$
B
$2A$
C
$\frac{A}{2}$
D
$3A$

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given: The refractive index is $n = \cot\left(\frac{A}{2}\right)$ .

Find: The angle of minimum deviation $\delta_m$ .

For a prism at minimum deviation,

n = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}

Substituting the given value of $n$ ,

\cot\left(\frac{A}{2}\right) = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}

Working and discrepancy note

Using

\cot\left(\frac{A}{2}\right) = \frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)}

we get

\frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}

Hence,

\cos\left(\frac{A}{2}\right) = \sin\left(\frac{A + \delta_m}{2}\right)

Now use

\cos\left(\frac{A}{2}\right) = \sin\left(\frac{\pi}{2} - \frac{A}{2}\right)

So,

\sin\left(\frac{A + \delta_m}{2}\right) = \sin\left(\frac{\pi}{2} - \frac{A}{2}\right)

Equating the arguments as done in the provided solution,

\frac{A + \delta_m}{2} = \frac{\pi}{2} - \frac{A}{2}

which gives

A + \delta_m = \pi - A

and therefore

\delta_m = \pi - 2A

Answer selection from given options

The solution explicitly states that the correct option is B and derives $\delta_m = \pi - 2A$ . This value does not match any listed option exactly. Since the solution itself marks B as correct, the answer is taken as B despite the discrepancy between the derived expression and the option values.

Common mistakes

Using the prism formula for a general deviation instead of the minimum deviation case is incorrect. Here the standard relation at minimum deviation must be used. Always start with the minimum deviation formula before substituting the refractive index.
Canceling $\sin\left(\frac{A}{2}\right)$ incorrectly or forgetting to rewrite $\cot\left(\frac{A}{2}\right)$ as $\frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)}$ leads to wrong simplification. Rewrite the trigonometric ratio explicitly first.
Assuming the derived expression must exactly match one option without checking the source solution can be misleading here. The solution gives $\delta_m = \pi - 2A$ but marks option B as correct. Always note such source discrepancies.

Practice more Prisms & Total Internal Reflection questions

Get unlimited AI-adaptive practice, mastery tracking, and an AI tutor that explains every step — free to start.

Start free trial More on Prisms & Total Internal Reflection

Answer

Step-by-step solution

Common mistakes

Practice more Prisms & Total Internal Reflection questions

Related questions