Find the angle of deviation suffered by the light ray shown in figure. The refractive index `mu= 2.0` for the prism material.

Explanation

To find the angle of deviation suffered by the light ray passing through a prism with a refractive index of \( \mu = 2.0 \) and an angle of prism \( A = 4^\circ \), we can follow these steps:

### Step 1: Understand the relationship between refractive index, angle of prism, and angle of deviation
The refractive index \( \mu \) of a prism is related to the angle of prism \( A \) and the angle of minimum deviation \( D_m \) by the formula:
\[ \mu = \frac{\sin\left(\frac{A + D_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

### Step 2: Substitute the known values into the formula
Given that \( \mu = 2.0 \) and \( A = 4^\circ \), we can substitute these values into the equation:
\[ 2 = \frac{\sin\left(\frac{4 + D_m}{2}\right)}{\sin\left(\frac{4}{2}\right)} \]
This simplifies to:
\[ 2 = \frac{\sin\left(2 + \frac{D_m}{2}\right)}{\sin(2)} \]

### Step 3: Simplify the equation
To simplify further, we can express \( \sin(2) \) as a constant value. We know that for small angles, \( \sin(x) \approx x \) in radians. So, we can approximate:
\[ \sin(2) \approx 2 \text{ (in radians)} \]
Thus, the equation becomes:
\[ 2 = \frac{\sin\left(2 + \frac{D_m}{2}\right)}{2} \]
Multiplying both sides by \( 2 \):
\[ 4 = \sin\left(2 + \frac{D_m}{2}\right) \]

### Step 4: Solve for \( D_m \)
To find \( D_m \), we need to isolate it:
\[ \sin\left(2 + \frac{D_m}{2}\right) = 4 \]
However, since the sine function cannot exceed 1, we must have made an error in our approximations or assumptions.

### Step 5: Correct the approach
Instead, we can use the formula for small angles directly:
\[ \mu = \frac{A + D_m}{A} \]
Substituting the known values:
\[ 2 = \frac{4 + D_m}{4} \]
Cross-multiplying gives:
\[ 8 = 4 + D_m \]
Thus, solving for \( D_m \):
\[ D_m = 8 - 4 = 4^\circ \]

### Conclusion
The angle of deviation \( D_m \) suffered by the light ray is \( 4^\circ \).

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