A thin prism with angle 5° of refractive index 1.72 is combined with another prism of refractive index 1.9 to produce dispersion w ithout deviation. The angle of second prism is ____ .

We need to find the angle of the second prism when two thin prisms are combined to produce dispersion without deviation.

We first state the deviation formula for a thin prism.

For a thin prism with small angle $$A$$ and refractive index $$n$$, the deviation is given by:

$$ \delta = (n - 1)A $$

This formula is derived from Snell's law under the small-angle approximation ($$\sin\theta \approx \theta$$).

Next, we set up the condition for zero net deviation.

When two prisms are combined in opposite orientations, their deviations act in opposite directions. For the net deviation to be zero:

$$ \delta_1 + \delta_2 = 0 $$

This implies:

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

(The deviations must be equal in magnitude but opposite in direction.)

Substituting the given values $$n_1 = 1.72$$, $$A_1 = 5°$$, and $$n_2 = 1.9$$ into this equation gives:

$$ (1.72 - 1) \times 5 = (1.9 - 1) \times A_2 $$

$$ 0.72 \times 5 = 0.9 \times A_2 $$

$$ 3.6 = 0.9 \times A_2 $$

Solving for $$A_2$$ yields:

$$ A_2 = \frac{3.6}{0.9} = 4° $$

The correct answer is Option B: $$4°$$.