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Wave Optics question

2020 · 3 Sep · Shift 2 · Q47
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Wave Optics question

2020 · 3 Sep · Shift 2 · Q47

JEE MainPhysicsWave OpticsMCQ+4 / −1
Two light waves having the same wavelength λ\lambdaλ in vacuum are in phase initially. Then the first wave travels a path L1 through a medium of refractive index n1 while the second wave travels a path of length L2 through a medium of refractive index n2 . After this the phase difference between the two waves is :
  1. A
    2πλ(n1L1−n2L2){{2\pi } \over \lambda }\left( {{n_1}{L_1} - {n_2}{L_2}} \right)λ2π​(n1​L1​−n2​L2​)
  2. B
    2πλ(n2L1−n1L2){{2\pi } \over \lambda }\left( {{n_2}{L_1} - {n_1}{L_2}} \right)λ2π​(n2​L1​−n1​L2​)
  3. C
    2πλ(L1n1−L2n2){{2\pi } \over \lambda }\left( {{{{L_1}} \over {{n_1}}} - {{{L_2}} \over {{n_2}}}} \right)λ2π​(n1​L1​​−n2​L2​​)
  4. D
    2πλ(L2n1−L1n2){{2\pi } \over \lambda }\left( {{{{L_2}} \over {{n_1}}} - {{{L_1}} \over {{n_2}}}} \right)λ2π​(n1​L2​​−n2​L1​​)
View written solutionFree

Correct answer: A

  1. Phase gained by a wave in a medium

For a wave of wavelength λ\lambdaλ in vacuum, the vacuum wave number is

k0=2πλ.k_0 = \frac{2\pi}{\lambda}.k0​=λ2π​.

If the wave travels a distance LLL in a medium of refractive index nnn, its optical path length is

OPL=nL.\text{OPL} = nL.OPL=nL.

Hence the phase gained is

ϕ=k0(nL)=2πλ nL.\phi = k_0 (nL) = \frac{2\pi}{\lambda} \, nL.ϕ=k0​(nL)=λ2π​nL.
  1. Phase gained by each wave
  • For the first wave:
ϕ1=2πλn1L1\phi_1 = \frac{2\pi}{\lambda} n_1 L_1ϕ1​=λ2π​n1​L1​
  • For the second wave:
ϕ2=2πλn2L2\phi_2 = \frac{2\pi}{\lambda} n_2 L_2ϕ2​=λ2π​n2​L2​
  1. Initial condition

The waves are initially in phase, so initial phase difference is zero.

Therefore, after traveling through the media, the phase difference is just the difference in phases accumulated:

Δϕ=ϕ1−ϕ2\Delta \phi = \phi_1 - \phi_2Δϕ=ϕ1​−ϕ2​

so

Δϕ=2πλ(n1L1−n2L2).\Delta \phi = \frac{2\pi}{\lambda}(n_1L_1 - n_2L_2).Δϕ=λ2π​(n1​L1​−n2​L2​).
  1. Match with options

This matches:

2πλ(n1L1−n2L2)\boxed{\frac{2\pi}{\lambda}(n_1L_1 - n_2L_2)}λ2π​(n1​L1​−n2​L2​)​

which is Option A.

  1. Check other options briefly
  • B incorrectly mixes indices with the wrong path lengths.
  • C and D use L/nL/nL/n, but phase depends on optical path length nLnLnL, not L/nL/nL/n.

Therefore, the correct answer is A.

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