Two identical concave mirrors each of focal length f are facing each other as shown in the schematic diagram. The focal length f is much larger than the size of the mirrors. A glass slab of thickness t and refractive index n0 is kept equidistant from the mirrors and perpendicular to their common principal axis. A monochromatic point light source S is embedded at the center of the slab on the principal axis, as shown in the schematic diagram. For the image to be formed on S itself, which of the following distances between the two mirrors is/are correct: [figure]
- 4f+ (1- 1n0 )t
- 2f+ (1- 1n0 )t
- 4f+(n0-1)t
- 2f+(n0-1)t
Answer
(1) 4f+ (1- 1n0 )t, (2) 2f+ (1- 1n0 )t
Solution
S is at depth t2 in the slab, so seen from outside it appears shifted towards each mirror by t2 (1- 1n0 ).
Case 1: apparent source at the centre of curvature of each mirror: rays retrace back to S.
Distance of each mirror from S =2f+ t2 (1- 1n0 ), separation =4f+ (1- 1n0 )t. (A)
Case 2: apparent source at the focus of each mirror: one mirror sends a parallel beam (unshifted by the slab) to the other, which focuses it at its focus, i.e. back at S.
Each distance =f+ t2 (1- 1n0 ), separation =2f+ (1- 1n0 )t. (B)
Answer: (A), (B)