An optical arrangement is depicted in the figure. The distance D is significantly greater than the separation d between the slits. Determine the minimum value of d such that a dark fringe appears at point O .

Step-by-step solution

Let the source slit be S , the slits in the second plane be S₁ and S₂ , and the point on the screen be O . From the given figure, the optical axis passes through the source slit S , the lower slit S₁ , and the point O on the screen. The upper slit S₂ is at a distance d from the optical axis. Light reaches point O via two paths: Path 1: S S₁ O Path 2: S S₂ O The length of Path 1 is: L₁ = SS₁ + S₁O = D + D = 2D The length of Path 2 is: L₂ = SS₂ + S₂O = D^2 + d^2 + D^2 + d^2 = 2 D^2 + d^2 The path difference x at poin