In a single slit diffraction pattern, a light of wavelength 6000 o...

For ${n}^{th}$ minima:

$⇒ bsin\theta = n\lambda$

($\lambda$ is very small, so $sin\theta$ is very small, hence $sin\theta ≃ tan\theta$.

$⇒ btan\theta = n\lambda$

$⇒ b\dfrac{y}{D} = n\lambda$

$⇒ {y}_{n} = \dfrac{n\lambdaD}{b}$ (Position of ${n}^{th}$ minima)

$B \to {1}^{\text{st}}$ minima and $A \to {3}^{\text{rd}}$:

${y}_{3} = \dfrac{3\lambdaD}{b}, {y}_{1} = \dfrac{\lambdaD}{b}$

$⇒ \Deltay = {y}_{3} - {y}_{1} = \dfrac{2\lambdaD}{b}$

$⇒ 3 \times {10}^{- 3} = \dfrac{2 \times 6000 \times {10}^{- 10} \times 0 . 5}{b}$

$⇒ b = \dfrac{2 \times 6000 \times {10}^{- 10} \times 0 . 5}{3 \times {10}^{- 3}}$

$⇒ b = 2 \times {10}^{- 4}m$