Question

A thin prism \( P_1 \), with angle \( 4^\circ \) and made from glass of refractive index 1.54, is combined with another thin prism \( P_2 \), made from glass of refractive index 1.72 to produce dispersion without deviation. The angle of prism for \( P_2 \) is

• A. \( 4^\circ \)
• B. \( 5.33^\circ \)
• C. \( 2.6^\circ \)
• D. \( 3^\circ \)

Hint:

For dispersion without deviation, the angles of the two prisms must produce equal and opposite deviations. Use the refractive indices and angles to set up an equation for the unknown angle.

Answer 1

The Correct Option is D

Step 1: Condition for dispersion without deviation.
For dispersion without deviation, the total deviation produced by the two prisms must be zero. The deviation produced by a prism is given by: \[ \delta = (\mu - 1) A \] where \( \mu \) is the refractive index and \( A \) is the angle of the prism.
Step 2: Setting up the equation.
For dispersion without deviation, the deviation produced by \( P_1 \) should be equal and opposite to that produced by \( P_2 \). Hence, we have: \[ (\mu_1 - 1) A_1 = (\mu_2 - 1) A_2 \] Substituting the known values \( \mu_1 = 1.54 \), \( A_1 = 4^\circ \), and \( \mu_2 = 1.72 \), we can solve for \( A_2 \).
Step 3: Solving for \( A_2 \).
\[ (1.54 - 1) \times 4 = (1.72 - 1) \times A_2 \] \[ 0.54 \times 4 = 0.72 \times A_2 \] \[ A_2 = \frac{0.54 \times 4}{0.72} = 3^\circ \]
Step 4: Conclusion.
Thus, the angle of prism \( P_2 \) is \( 3^\circ \), which corresponds to option (D).