The refracting angle of a prism is A, and refractive index of the material of the prism is `cot (A)/(2)`. The angle of minimum deviation is

Options

A. `180^(@) - 3A`
B. `180^(@)` - 2A
C. `90^(@)` - A
D. `180^(@)`+2A

Answer

To find the angle of minimum deviation (D_min) for a prism with a refracting angle (A) and a refractive index (μ) given by cot(A/2), we can follow these steps:

### Step 1: Write down the formula for the refractive index of a prism
The refractive index (μ) of a prism in terms of the angle of minimum deviation (D_min) and the refracting angle (A) is given by:
\[ \mu = \frac{\sin\left(\frac{A + D_{min}}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

### Step 2: Substitute the given refractive index
According to the problem, the refractive index is given as:
\[ \mu = \cot\left(\frac{A}{2}\right) \]
We can express cotangent in terms of sine and cosine:
\[ \cot\left(\frac{A}{2}\right) = \frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

### Step 3: Set the two expressions for μ equal to each other
Now we equate the two expressions for μ:
\[ \frac{\sin\left(\frac{A + D_{min}}{2}\right)}{\sin\left(\frac{A}{2}\right)} = \frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

### Step 4: Simplify the equation
Since \(\sin\left(\frac{A}{2}\right)\) is common on both sides, we can cancel it (assuming it is not zero):
\[ \sin\left(\frac{A + D_{min}}{2}\right) = \cos\left(\frac{A}{2}\right) \]

### Step 5: Use the identity for cosine
Recall the identity:
\[ \cos\left(\frac{A}{2}\right) = \sin\left(90^\circ - \frac{A}{2}\right) \]
Thus, we can write:
\[ \sin\left(\frac{A + D_{min}}{2}\right) = \sin\left(90^\circ - \frac{A}{2}\right) \]

### Step 6: Set the arguments equal
Since the sine function is equal, we can set the arguments equal to each other:
\[ \frac{A + D_{min}}{2} = 90^\circ - \frac{A}{2} \]

### Step 7: Solve for D_min
Now, multiply both sides by 2 to eliminate the fraction:
\[ A + D_{min} = 180^\circ - A \]
Rearranging gives:
\[ D_{min} = 180^\circ - 2A \]

### Final Result
Thus, the angle of minimum deviation (D_min) is:
\[ D_{min} = 180^\circ - 2A \]

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