Question

The angle of minimum deviation for a prism of apex angle 60° and refractive index of \(\sqrt{2}\) is:

• A. 45°
• B. 90°
• C. 30°
• D. 60°
• E. 15°

Answer 1

The Correct Option is C

Given:

• Apex angle of prism, \( A = 60^\circ \)
• Refractive index, \( \mu = \sqrt{2} \)

Step 1: Formula for Minimum Deviation (\( \delta_{\text{min}} \))

The angle of minimum deviation is given by:

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

Step 2: Substitute Known Values

\[ \sqrt{2} = \frac{\sin\left(\frac{60^\circ + \delta_{\text{min}}}{2}\right)}{\sin\left(30^\circ\right)} \]

Since \( \sin 30^\circ = 0.5 \):

\[ \sqrt{2} = \frac{\sin\left(30^\circ + \frac{\delta_{\text{min}}}{2}\right)}{0.5} \]

\[ \sin\left(30^\circ + \frac{\delta_{\text{min}}}{2}\right) = \frac{\sqrt{2}}{2} = \sin 45^\circ \]

Step 3: Solve for \( \delta_{\text{min}} \)

\[ 30^\circ + \frac{\delta_{\text{min}}}{2} = 45^\circ \]

\[ \frac{\delta_{\text{min}}}{2} = 15^\circ \]

\[ \delta_{\text{min}} = 30^\circ \]

Conclusion:

The angle of minimum deviation is \( 30^\circ \).

Answer: \(\boxed{C}\)

Answer 2

1. Define variables and given information:

• A = 60° (apex angle of the prism)
• n = √2 (refractive index)
• δ_m = ? (angle of minimum deviation)

2. Recall the formula for the angle of minimum deviation:

The formula relating the angle of minimum deviation (δ_m), apex angle (A), and refractive index (n) is:

\[n = \frac{\sin(\frac{A + \delta_m}{2})}{\sin(\frac{A}{2})}\]

3. Substitute the given values and solve for δ_m:

\[\sqrt{2} = \frac{\sin(\frac{60^\circ + \delta_m}{2})}{\sin(\frac{60^\circ}{2})}\]

\[\sqrt{2} = \frac{\sin(\frac{60^\circ + \delta_m}{2})}{\sin(30^\circ)}\]

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

\[\sqrt{2} = 2\sin(\frac{60^\circ + \delta_m}{2})\]

\[\frac{\sqrt{2}}{2} = \sin(\frac{60^\circ + \delta_m}{2})\]

\[\frac{1}{\sqrt{2}} = \sin(\frac{60^\circ + \delta_m}{2})\]

Since \(\sin(45^\circ) = \frac{1}{\sqrt{2}}\):

\[45^\circ = \frac{60^\circ + \delta_m}{2}\]

\[90^\circ = 60^\circ + \delta_m\]

\[\delta_m = 90^\circ - 60^\circ = 30^\circ\]