A thin prism $$P_{1}$$ with angle $$4^{\circ}$$ made of glass having refractive index 1.54 , is combined with another thin prism $$P_{2}$$ made of glass having refractive index 1.72 to get dispersion without deviation. The angle of the prism $$P_{2}$$ in degrees is

We need to find the angle of prism $$P_2$$ such that the combination of $$P_1$$ and $$P_2$$ produces dispersion without deviation.

- Prism $$P_1$$: angle $$A_1 = 4°$$, refractive index $$n_1 = 1.54$$

- Prism $$P_2$$: angle $$A_2 = ?$$, refractive index $$n_2 = 1.72$$

For a thin prism, the deviation is: $$\delta = (n - 1)A$$

For dispersion without deviation, the two prisms must be arranged to cancel each other's deviation. The prisms are placed with opposite orientations:

$$\delta_1 = \delta_2$$

$$(n_1 - 1)A_1 = (n_2 - 1)A_2$$

$$A_2 = \frac{(n_1 - 1)}{(n_2 - 1)} \times A_1 = \frac{(1.54 - 1)}{(1.72 - 1)} \times 4° = \frac{0.54}{0.72} \times 4°$$

$$= \frac{3}{4} \times 4° = 3°$$

The angle of prism $$P_2$$ is 3 degrees.

The correct answer is Option 1: 3.