Using the expression 2 d sin θ = λ , one calculates the values of d by measuring the corresponding angle θ in the range 0 to 90 ° . The wavelength λ is exactly known and the error in θ is constant for

Question

Using the expression 2 d sin θ = λ , one calculates the values of d by measuring the corresponding angle θ in the range 0 to 90 ° . The wavelength λ is exactly known and the error in θ is constant for all the values of θ . As θ increases from 0 ° ,

TestHub · Accepted Answer

2 d sin θ = λ d = λ 2 sin θ Differentiate on both the sides, ∂ d = λ 2 ∂ cosec θ ∂ d = λ 2 - cosec θ cot θ ∂ θ ∂ d = - λ cos θ 2 sin 2 θ ∂ θ θ = increases and λ cos θ 2 sin 2 θ decreases. Alternate solution d = λ 2 sin θ ℓ n d = ℓ n λ - ℓ n 2 - ℓ n sin θ Δ d d = 0 - 0 - 1 sin θ × cos θ Δ θ Fractional error, + d = cot θ Δ θ Absolute error, Δ d = d cot θ Δ θ d 2 sin θ × cos θ sin θ Δ d = cos θ sin 2 θ

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