A source S and a detector D are separated by a distance d . As shown in the figure, a large cardboard is positioned at a distance 2 d from the source and the detector. The source emits a wave with a wavelength of d/2 , which reaches the detector after reflecting off the cardboard. This reflected wave is observed to be in phase with the direct wave arriving from the source. Determine the minimum distance the cardboard

Step-by-step solution

Path length of the direct wave from S to D is x₁ = d . Path length of the reflected wave is x₂ = 2 ( 2 d)^2 + ( d 2 )^2 = 2 2d^2 + d^2 4 = 3d . Initial path difference is x = x₂ - x₁ = 3d - d = 2d . Given wavelength = d 2 , the path difference is x = 4 . This confirms the waves are initially in phase. To make the waves out of phase by shifting the cardboard away by a minimum distance, the new path difference must be x' = 4.5 = 4.5 ( d 2 ) = 2.25d . Let the new distance of the cardboard be y . The new path length of