A concave mirror of radius R is kept on a horizontal table. Water (refractive index =mu.) is poured into it up to a height h. Where should an object be placed so that its image is formed on itself ?

Explanation

Light ray from object is first refracted into water and then incident on mirror. After refraction from water the ray should fall normal on the surface of mirror of that light ray may retrace its path to form image on the location of object. See the following figure for the ray diagram. Let the object be placed at a is distance x from the water surface.

To fall normally on the surface of mirror light ray should appear as if coming from the centre of curvature of mirror. In other words we can say that image formed due to refraction from water surface should be at a distance `R-h` from the water surface. We can use refraction formula for plane surfaces also by substituting R as infinity.
`(mu_(2))/(v)-(mu_(1))/(u=(mu_(2)-mu_(1))/(R )`
`(mu)/(-(R-h))-(1)/(-x)=(mu_(2)-mu_(1))/(oo)`
`rArr" "x=(R-h)/(mu)`
Hence object should be placed at a distance `(R-h)/(mu)`
above the surface of water.