A thin prism of angle $6^{\circ}$ and refractive index for yellow light ($\mathrm{n}_{\mathrm{Y}}$) 1.5 is combined with another prism of angle $5^{\circ}$ and $\mathrm{n}_{\mathrm{Y}}=1.55$. The combination produces no dispersion. The net average deviation ($\delta$) produced by the combination is $\left(\frac{1}{\mathrm{x}}\right)^{\circ}$. The value of x is ______.

JEE Main · 2022 · 27 Jul 2022 · Shift 2 · Geometrical Optics

A thin prism of angle

6^{\circ}

and refractive index for yellow light (

n_{Y}

) 1.5 is combined with another prism of angle

5^{\circ}

and

n_{Y} = 1.55

. The combination produces no dispersion. The net average deviation (

δ

) produced by the combination is

(\frac{1}{x})^{\circ}

. The value of x is ______.

Answer: 4

Solution

For a combination of two prisms producing no dispersion (achromatic combination), the net deviation is

δ = A_{1} (μ_{1} - 1) - A_{2} (μ_{2} - 1)

, where the second prism is inverted. Substituting:

δ = 6 (1.5 - 1) - 5 (1.55 - 1) = 6 (0.5) - 5 (0.55) = 3 - 2.75 = 0.25 = \frac{1}{4}

degrees. Therefore

x = 4

JEE Main · 2022 · 27 Jul 2022 · Shift 2 · Integer answer

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