A prism of refractive index $\mathrm{n}_{1}$ and another prism of refractive index $\mathrm{n}_{2}$ are stuck together (as shown in the figure). $\mathrm{n}_{1}$ and $\mathrm{n}_{2}$ depend on $\lambda$, the wavelength of light, according to the relation $\mathrm{n}_{1}=1.2+\frac{10.8\times 10^{-14}}{\lambda^{2}}$ and $\mathrm{n}_{2}=1.45+\frac{1.8\times 10^{-14}}{\lambda^{2}}$ The wavelength for which rays incident at any angle on the interface BC pass through without bending at that interface will be ______ nm.

JEE Main · 2021 · 27 Jul 2021 · Shift 1 · Geometrical Optics

A prism of refractive index

n_{1}

and another prism of refractive index

n_{2}

are stuck together (as shown in the figure).

n_{1}

and

n_{2}

depend on

λ

, the wavelength of light, according to the relation

n_{1} = 1.2 + \frac{10.8 \times 1 0 ^{- 14}}{λ ^{2}}

and

n_{2} = 1.45 + \frac{1.8 \times 1 0 ^{- 14}}{λ ^{2}}

The wavelength for which rays incident at any angle on the interface BC pass through without bending at that interface will be ______ nm.

Answer: 600

A ray crosses a boundary undeviated for every angle of incidence only when the two media share the same refractive index, so the condition is

n_{1} = n_{2}

1.2 + \frac{10.8 \times 1 0 ^{- 14}}{λ ^{2}} = 1.45 + \frac{1.8 \times 1 0 ^{- 14}}{λ ^{2}}

\frac{9 \times 1 0 ^{- 14}}{λ ^{2}} = 0.25 \Rightarrow λ^{2} = 36 \times 1 0^{- 14} m^{2}

λ = 6 \times 1 0^{- 7} m

Light of wavelength

600

nm passes through the interface BC without bending.

JEE Main · 2021 · 27 Jul 2021 · Shift 1 · Integer answer