Optics Homework Help: Find x for Image on Self

[kihr]

Homework Statement

Water is poured into a concave mirror of radius of curvature R up to a height h. An object is placed along the principal axis at a distance x above the level of the water. What should be the value of x so that the image of the object is formed on itself?

Homework Equations

For an image to be formed on itself the object must be placed at the centre of curvature of the concave lens.
Since water has been poured into the mirror, the optical length in the medium (water) is Mu*h (where Mu = refractive index of water).

The Attempt at a Solution

Object distance from the pole of the mirror = x + Mu*h
Or R = x + Mu*h
Or x = R - Mu*h

Since the answer given in the book is x = (R - h) / Mu, I am unable to figure out as to where I have gone wrong. Would request for help. Thanks.

 

[rl.bhat]

Focal length of the water lens is (μ-1)/R.
Light form the object once reflected from the mirror and twice refracted through water lens before returning back to the object.
Combined focal length of the combination is
1/F = 1/Fm + 1/Fw + 1/Fw

 

[kihr]

I could not quite follow the part regarding the combined focal length. Here we have only one lens (water) and one mirror. I would request for the explanation in a little more detail. Thanks.

 

[rl.bhat]

I could not quite follow the part regarding the combined focal length. Here we have only one lens (water) and one mirror. I would request for the explanation in a little more detail. Thanks.

When the rays from the object reflect back to its position they refract twice and reflect once. So the system is a combination of two lenses and one concave mirror. Its combined focal length is in my post.

 

[kihr]

By your method I don't get the answer given in the book. I guess either the answer given in the book is incorrect, or the approach to the solution may require a re-look.

 

[rl.bhat]

(x + h) can be taken as 2F of the system.
Then
2/(x + h) = 2/R + (μ-1)/R + (μ-1)/R .
Solve for x.
The answer in the book appears to be wrong.
It should be
x = (R/μ) - h