Time taken by light to travel in two different materials A and B of refractive indices $\mu_{\mathrm{A}}$ and $\mu_{\mathrm{B}}$ of same thickness is $t_{1}$ and $t_{2}$ respectively. If $t_{2}-t_{1}=5 \times 10^{-10} s$ and the ratio of $\mu_{A}$ to $\mu_{B}$ is $1: 2$. Then the thickness of material, in meter is : (Given $\mathrm{v}_{\mathrm{A}}$ and $\mathrm{v}_{\mathrm{B}}$ are velocities of light in A and B materials respectively).

(A) $5 \times 10^{-10} \mathrm{~V}_{\mathrm{a}} \mathrm{m}$

(B) $5 \times 10^{-10} \mathrm{~m}$

(C) $1.5 \times 10^{-10} \mathrm{~m}$

(D) $5 \times 10^{-10} \mathrm{v}_{\mathrm{B}} \mathrm{m}$

JEE Main · 2022 · 25 Jul 2022 · Shift 1 · Geometrical Optics

Time taken by light to travel in two different materials A and B of refractive indices

μ_{A}

and

μ_{B}

of same thickness is

t_{1}

and

t_{2}

respectively. If

t_{2} - t_{1} = 5 \times 1 0^{- 10} s

and the ratio of

μ_{A}

μ_{B}

1 : 2

. Then the thickness of material, in meter is : (Given

v_{A}

and

v_{B}

are velocities of light in A and B materials respectively).

Answer: (A)

From the refractive index ratio:

\frac{μ _{A}}{μ _{B}} = \frac{v _{B}}{v _{A}} = \frac{1}{2}

, so

v_{A} = 2 v_{B}

. The time difference is

t_{2} - t_{1} = \frac{d}{v _{B}} - \frac{d}{v _{A}} = 5 \times 1 0^{- 10}

. Factoring out

d

d (\frac{1}{v _{B}} - \frac{1}{v _{A}}) = 5 \times 1 0^{- 10}

. Substituting

v_{A} = 2 v_{B}

d (\frac{1}{v _{B}} - \frac{1}{2 v _{B}}) = d (\frac{1}{2 v _{B}}) = 5 \times 1 0^{- 10}

. Therefore

d = 10 \times 1 0^{- 10} v_{B} = 5 \times 1 0^{- 10} \times 2 v_{B} = 5 \times 1 0^{- 10} v_{A}

JEE Main · 2022 · 25 Jul 2022 · Shift 1 · Single correct option