A simple periscope is built from two identical plane mirrors mounted inside a vertical tube of length 1.2m. The upper mirror is located at the very top of the tube; the lower mirror is 1.15m below it at eye level. Each mirror is oriented at an angle of 45∘ with respect to the vertical, so that their reflecting surfaces face one another.
Light from a distant streetlamp enters the top opening of the periscope, reflects from the upper mirror, then from the lower mirror, and finally travels horizontally into the observer’s eye.
(a) Draw a clearly labeled ray diagram that shows the complete path of a representative light ray from the streetlamp to the observer’s eye, including the normals at both points of reflection.
(b) Using geometrical reasoning and appropriate angle relationships, prove that the horizontal separation between the entering and exiting portions of the ray is twice the perpendicular distance between the two mirror planes.
(c) The mirrors are rectangular with width w. Derive an expression for the minimum value of w required to ensure that all rays entering the top opening with an angular spread of ±3∘ about the forward direction still emerge from the bottom mirror without being clipped.
(d) Discuss quantitatively how a 1∘ misalignment (rotation about the vertical axis) of the lower mirror would affect the direction of the emerging beam. Estimate the angular error in the beam’s final direction and comment on whether the observer would notice a displaced image.