Question

Calculate the angle of minimum deviation of an equilateral prism. The refractive index of the prism is \(\sqrt{3}\). Calculate the angle of incidence for this case of minimum deviation also.

Answer 1

Prism: Minimum Deviation and Angle of Incidence Calculation

Given:

• Prism refractive index, \( \mu = \sqrt{3} \)
• Prism angle, \( A = 60^\circ \) (equilateral prism)

Solution:

The formula for refractive index at minimum deviation is:

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

Substitute \( A = 60^\circ \) and \( \mu = \sqrt{3} \):

\[ \sqrt{3} = \frac{\sin \left( \frac{60^\circ + D_m}{2} \right)}{\sin 30^\circ} \]

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

\[ \sqrt{3} = \frac{\sin \left( \frac{60^\circ + D_m}{2} \right)}{0.5} \]

Multiply by \( 0.5 \):

\[ \sin \left( \frac{60^\circ + D_m}{2} \right) = \frac{\sqrt{3}}{2} \]

Since \( \sin 60^\circ = \frac{\sqrt{3}}{2} \):

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

Solve for \( D_m \):

\[ 60^\circ + D_m = 120^\circ \quad \Rightarrow \quad D_m = 60^\circ \]

Step 2: Calculate Angle of Incidence for Minimum Deviation

The angle of incidence at minimum deviation \( i \) is calculated as:

\[ i = \frac{A + D_m}{2} \]

Substitute \( A = 60^\circ \) and \( D_m = 60^\circ \):

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

Final Answer:

• Angle of minimum deviation, \( D_m = 60^\circ \)
• Angle of incidence for minimum deviation, \( i = 60^\circ \)