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Wave Optics question

2018 · Shift 0 · Q46
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Wave Optics question

2018 · Shift 0 · Q46

JEE MainPhysicsWave OpticsMCQ+4 / −1
The angular width of the central maximum in a single slit diffraction pattern is 60°. The width of the slit is 1 μ\muμ m. The slit is illuminated by monochromatic plane waves. If another slit of same width is made near it, Young’s fringes can be observed on a screen placed at a distance 50 cm from the slits. If the observed fringe width is 1 cm, what is slit separation distance? (i.e. distance between the centres of each slit.)
  1. A
    100 μ\muμ m
  2. B
    25 μ\muμ m
  3. C
    50 μ\muμ m
  4. D
    75 μ\muμ m
View written solutionFree

Correct answer: B

  1. Use single-slit diffraction to find the wavelength

For a single slit of width aaa, the angular width of the central maximum is

2θ=angular width2\theta = \text{angular width}2θ=angular width

Given angular width of central maximum =60∘= 60^\circ=60∘, so

2θ=60∘⇒θ=30∘2\theta = 60^\circ \Rightarrow \theta = 30^\circ2θ=60∘⇒θ=30∘

For first minima in single-slit diffraction,

asin⁡θ=λa\sin\theta = \lambdaasinθ=λ

Given:

  • a=1 μm=10−6 ma = 1\,\mu m = 10^{-6}\,ma=1μm=10−6m
  • θ=30∘\theta = 30^\circθ=30∘

So,

λ=asin⁡30∘=10−6×12=5×10−7 m\lambda = a\sin 30^\circ = 10^{-6}\times \frac{1}{2} = 5\times 10^{-7}\,mλ=asin30∘=10−6×21​=5×10−7m

Thus,

λ=500 nm\lambda = 500\,nmλ=500nm

  1. Use Young's double slit formula

Fringe width is

β=λDd\beta = \frac{\lambda D}{d}β=dλD​

where:

  • β=1 cm=10−2 m\beta = 1\,cm = 10^{-2}\,mβ=1cm=10−2m
  • D=50 cm=0.5 mD = 50\,cm = 0.5\,mD=50cm=0.5m
  • λ=5×10−7 m\lambda = 5\times 10^{-7}\,mλ=5×10−7m
  • d=d =d= slit separation

So,

d=λDβd = \frac{\lambda D}{\beta}d=βλD​

Substitute values:

d=(5×10−7)(0.5)10−2d = \frac{(5\times 10^{-7})(0.5)}{10^{-2}}d=10−2(5×10−7)(0.5)​

d=2.5×10−710−2=2.5×10−5 md = \frac{2.5\times 10^{-7}}{10^{-2}} = 2.5\times 10^{-5}\,md=10−22.5×10−7​=2.5×10−5m

d=25×10−6 m=25 μmd = 25\times 10^{-6}\,m = 25\,\mu md=25×10−6m=25μm

  1. Match with options

d=25 μmd = 25\,\mu md=25μm

So the correct option is:

B: 25 μm25\,\mu m25μm

  1. Comparison with stored answer

Stored correct answer: B

Our derived answer: B

So they agree.

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