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Wave Optics question

2022 · 29 Jun · Shift 2 · Q69
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Wave Optics question

2022 · 29 Jun · Shift 2 · Q69

JEE MainPhysicsWave OpticsNumerical+4 / −1
In a double slit experiment with monochromatic light, fringes are obtained on a screen placed at some distance from the plane of slits. If the screen is moved by 5 ×\times× 10 −-− 2 m towards the slits, the change in fringe width is 3 ×\times× 10 −-− 3 cm. If the distance between the slits is 1 mm, then the wavelength of the light will be ‾\underline{\hspace{2cm}}​ nm.
Numerical answer
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Correct answer: 600

  1. In Young’s double slit experiment, the fringe width is β=λDd\beta = \frac{\lambda D}{d}β=dλD​ where:
  • λ\lambdaλ = wavelength of light
  • DDD = distance of screen from slits
  • ddd = slit separation

  1. If the screen is moved towards the slits by ΔD\Delta DΔD, then the change in fringe width is Δβ=λΔDd\Delta \beta = \frac{\lambda \Delta D}{d}Δβ=dλΔD​ We use magnitudes here.

  2. Given data:

  • ΔD=5×10−2 m\Delta D = 5 \times 10^{-2}\,\text{m}ΔD=5×10−2m
  • Δβ=3×10−3 cm\Delta \beta = 3 \times 10^{-3}\,\text{cm}Δβ=3×10−3cm
  • d=1 mm=10−3 md = 1\,\text{mm} = 10^{-3}\,\text{m}d=1mm=10−3m

Convert Δβ\Delta \betaΔβ into meters: 3×10−3 cm=3×10−3×10−2 m=3×10−5 m3 \times 10^{-3}\,\text{cm} = 3 \times 10^{-3} \times 10^{-2}\,\text{m} = 3 \times 10^{-5}\,\text{m}3×10−3cm=3×10−3×10−2m=3×10−5m

  1. Now use Δβ=λΔDd\Delta \beta = \frac{\lambda \Delta D}{d}Δβ=dλΔD​ So, λ=Δβ⋅dΔD\lambda = \frac{\Delta \beta \cdot d}{\Delta D}λ=ΔDΔβ⋅d​

Substitute values: λ=(3×10−5)(10−3)5×10−2\lambda = \frac{(3 \times 10^{-5})(10^{-3})}{5 \times 10^{-2}}λ=5×10−2(3×10−5)(10−3)​

λ=3×10−85×10−2=35×10−6\lambda = \frac{3 \times 10^{-8}}{5 \times 10^{-2}} = \frac{3}{5} \times 10^{-6}λ=5×10−23×10−8​=53​×10−6

λ=0.6×10−6 m=6×10−7 m\lambda = 0.6 \times 10^{-6}\,\text{m} = 6 \times 10^{-7}\,\text{m}λ=0.6×10−6m=6×10−7m

  1. Convert to nanometers: 1 nm=10−9 m1\,\text{nm} = 10^{-9}\,\text{m}1nm=10−9m So, 6×10−7 m=600×10−9 m=600 nm6 \times 10^{-7}\,\text{m} = 600 \times 10^{-9}\,\text{m} = 600\,\text{nm}6×10−7m=600×10−9m=600nm

Therefore, the wavelength of light is 600 nm\boxed{600\,\text{nm}}600nm​

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