Athin slit illuminated by light of frequency f produces its first dark band at ± 38.2 ∘ in air. when the entire apparatus (slit, screen and space in between) is immersed in an unknown transparent liquid, the slit's first dark bands occur instead at ± 17.3 ∘.

find the refractive index of the liquid.

the refractive index of the liquid is 2.08

Explanation:

The diffraction formula is:

d sin(θ) =  mλ

where

Thus,

d sin(θ₁) =  m λ₁              (1)   and

d sin(θ₂) =  m λ₂              (2)

Dividing equation (1) by equation (2) you get

sin(θ₁) / sin(θ₂) = λ₁ / λ₂              (3)

λ can be expressed as

λ = v / f

where

Therefore,

λ₁ = v₁ / f

λ₂ = v₂ / f

Thus,

λ₁ / λ₂ = v₁ / v₂

Therefore, substituting the above expression for λ₁ / λ₂ into equation (3), we get

sin(θ₁) / sin(θ₂) = v₁ / v₂

where

The expression for the refractive index is

n = c / v , which can also be expressed as

n = v₁ / v₂

Therefore,

n =  sin(θ₁) / sin(θ₂)

n = sin(38.2°) / sin(17.3°)

n = 2.08

Therefore, the refractive index of the liquid is 2.08

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