Foca length of a concave mirror

[Steven7]

Homework Statement

A concave mirror of focal length 10cm in air is put into a beaker containing a liquid up to 7.5cm. Refractive index of the liquid is 1.2. What is the focal length of the mirror inside the beaker?

The Attempt at a Solution

I tried proportion method:
Depth Focal length
7.5cm 10cm
6.25cm(Apparent depth) 8.33cm
But I'm not convinced that this method is a legal operation. Any other way I could find the new f for the mirror?

 

[Phrak]

The angle of reflection is equal to the angle of incidence.

 

[Steven7]

Well any more hints? I can't think of any equation finding new f using angles.

 

[Phrak]

For an assigned question it's peculiar. Is it a trick question? Do you notice that the angles of reflection are independent of the index of refraction? It's a mirror not a lens.

The light rays change medium at the water surface. This is where the index of refraction counts. The distance traveled in water is variable; "up to 7.5 cm."

 

[rl.bhat]

If there were no water, the parallel beam of light would have come to focus at 10 cm from the mirror. But because of water, after 7.5 cm, the rays will refract away from the normal and come to focus nearer to the water surface. Actual distance of the focus from the water surface 2.5 cm. Apparent position of the focus is d.
So refractive index = 2.5/d.

 

[Phrak]

It can be shown, without arduous geometric or algebraic proof, that the rays passing through the water surface will not meet at a point. There is no focal point.

To show this, consider the angle of total internal reflection. For n₂/n₁=1.2, the angle of this ray within the water is 56.44 degrees from a normal line passing through the surface. In the air, the ray proceeds at 90 degrees from the normal, grazing the water surface.

Compare this to the ray reflected from the center of the mirror. It passes through the water at zero degrees from normal and is undeflected. It meets the other ray at an altitude of zero cm. above the water.

Consider any other ray between zero and 56.44 degrees. It will meet the normal ray, from the mirror's center, at some finite, non-zero distance above the water's surface. All three rays do not converge to a point.