A glass beaker has a solid, plano-convex base of refractive index 1.60, as shown in the figure. The radius of curvature of the convex surface (SPU) is 9 cm, while the planar surface (STU) acts as a mirror. This beaker is filled with a liquid of refractive index n up to the level QPR. If the image of a point object O at a height of h (OT in the figure) is formed onto itself, then, which of the following option(s) is(are) correct? [figure]
- For n=1.42, h=50 cm.
- For n=1.35, h=36 cm.
- For n=1.45, h=65 cm.
- For n=1.48, h=85 cm.
Answer
(1) For n=1.42, h=50 cm., (2) For n=1.35, h=36 cm.
Solution
Key idea: the image coincides with the object if the rays, after refraction into the glass, fall normally on the plane mirror STU, i.e. travel parallel to the axis (image at infinity). (Paraxial: base thickness negligible.)
Refraction at the convex surface SPU from liquid (n) into glass (1.60), R=+9 cm. Object in air at height h above P; the plane liquid surface at P does not change the reduced distance un1, so effectively nu=- 1h.
1.60 + 1h= 1.60-n9 ⇒ h= 91.60-n cm.
n=1.42: h= 90.18=50 cm ✓; n=1.35: h= 90.25=36 cm ✓.
n=1.45: h=60 cm (not 65); n=1.48: h=75 cm (not 85).
Answer: (A), (B)