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 (\delta) produced by the combination is (\frac1x)^\circ . The value of x is \underline\hspace2cm .

Geometrical Optics PYQ — JEE Main 2022 · 27 Jul · Shift 2 · Q58 with Solution

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Correct answer: 4

\delta=(n-1)A

where $n$ is refractive index and $A$ is prism angle.

For no dispersion, the angular dispersions produced by the two prisms must cancel each other. That means the prisms are placed oppositely, and

(\omega_1 A_1)=(\omega_2 A_2)

where $\omega$ is dispersive power.

Using the standard prism combination result, when there is no dispersion, the net deviation is

\delta=(n_1-1)A_1-(n_2-1)A_2

with opposite signs because the prisms are oppositely oriented.

A_1=6^\circ,\quad n_1=1.5

A_2=5^\circ,\quad n_2=1.55

So,

\delta=(1.5-1)6^\circ-(1.55-1)5^\circ

(1.5-1)6^\circ=0.5\times 6^\circ=3^\circ

(1.55-1)5^\circ=0.55\times 5^\circ=2.75^\circ

Hence,

\delta=3^\circ-2.75^\circ=0.25^\circ

\delta=\left(\frac{1}{x}\right)^\circ

\frac{1}{x}=0.25=\frac{1}{4}

Therefore,

x=4