A ray of light suffers minimum deviation, while passing through a prism of refractive index `1.5` and refracting angle `60^@`. Calculate the angle of deviation and angle in incidence.

Explanation

To solve the problem, we will use the formula for the refractive index of a prism at minimum deviation:

\[ \mu = \frac{\sin\left(\frac{A + D_{min}}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

Where:
- \( \mu \) is the refractive index of the prism,
- \( A \) is the angle of the prism,
- \( D_{min} \) is the minimum angle of deviation.

Given:
- Refractive index \( \mu = 1.5 \)
- Angle of prism \( A = 60^\circ \)

### Step 1: Calculate \( \frac{A}{2} \)

\[ \frac{A}{2} = \frac{60^\circ}{2} = 30^\circ \]

### Step 2: Calculate \( \sin\left(\frac{A}{2}\right) \)

\[ \sin\left(30^\circ\right) = \frac{1}{2} \]

### Step 3: Substitute into the refractive index formula

Using the formula:

\[ 1.5 = \frac{\sin\left(\frac{60^\circ + D_{min}}{2}\right)}{\sin(30^\circ)} \]

Substituting \( \sin(30^\circ) = \frac{1}{2} \):

\[ 1.5 = \frac{\sin\left(\frac{60^\circ + D_{min}}{2}\right)}{\frac{1}{2}} \]

### Step 4: Rearranging the equation

Multiply both sides by \( \frac{1}{2} \):

\[ 1.5 \cdot \frac{1}{2} = \sin\left(\frac{60^\circ + D_{min}}{2}\right) \]

\[ 0.75 = \sin\left(\frac{60^\circ + D_{min}}{2}\right) \]

### Step 5: Calculate \( \frac{60^\circ + D_{min}}{2} \)

Now we need to find the angle whose sine is \( 0.75 \):

\[ \frac{60^\circ + D_{min}}{2} = \sin^{-1}(0.75) \]

Calculating \( \sin^{-1}(0.75) \):

\[ \frac{60^\circ + D_{min}}{2} \approx 48.6^\circ \]

### Step 6: Solve for \( D_{min} \)

Multiply both sides by 2:

\[ 60^\circ + D_{min} \approx 97.2^\circ \]

Now, subtract \( 60^\circ \):

\[ D_{min} \approx 97.2^\circ - 60^\circ = 37.2^\circ \]

### Step 7: Calculate the angle of incidence

Using the relation for minimum deviation in a prism:

\[ D_{min} = i + r - A \]

At minimum deviation, \( i = r \). Therefore:

\[ D_{min} = 2i - A \]

Substituting the values:

\[ 37.2^\circ = 2i - 60^\circ \]

Adding \( 60^\circ \) to both sides:

\[ 37.2^\circ + 60^\circ = 2i \]

\[ 97.2^\circ = 2i \]

Dividing by 2:

\[ i = \frac{97.2^\circ}{2} = 48.6^\circ \]

### Final Answers:
- Angle of deviation \( D_{min} \approx 37.2^\circ \)
- Angle of incidence \( i \approx 48.6^\circ \)

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