Optics Homework Help: Find x for Image on Self

In summary, the question asks for the value of x to place an object at the center of curvature of a concave mirror filled with water, so that the image is formed on itself. The solution involves finding the combined focal length of the system, which is calculated by considering the reflection and refraction of light through the water and the mirror. The correct answer is x = (R/μ) - h, which differs from the answer given in the book.
  • #1
kihr
102
0

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.
 
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  • #2


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
 
  • #3


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.
 
  • #4


kihr said:
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.
 
  • #5


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.
 
  • #6


(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
 

1. What is the purpose of finding x for an image on self in optics?

The purpose of finding x for an image on self is to determine the location of an object's image formed by a plane mirror or a spherical mirror. This information is important in understanding the characteristics and behavior of light rays as they reflect off of different surfaces.

2. What are the steps involved in finding x for an image on self?

The steps involved in finding x for an image on self include identifying the type of mirror being used, determining the object distance from the mirror, applying the mirror equation (1/f = 1/x + 1/d), and solving for x. It is also important to remember to use the correct units and to consider the sign conventions for object and image distances.

3. Can I use the same method to find x for an image on self with any type of mirror?

No, the method for finding x for an image on self may vary depending on the type of mirror being used. For example, for a plane mirror, the image distance is equal to the object distance, while for a spherical mirror, the image distance may be positive or negative depending on the location of the object in relation to the mirror.

4. How do I know if my answer for x is correct?

To check if your answer for x is correct, you can use the magnification equation (m = -i/o) to calculate the magnification of the image. The magnification should be equal to -1 for a virtual image and +1 for a real image. Additionally, your answer should also satisfy the mirror equation (1/f = 1/x + 1/d).

5. Are there any tips for solving problems involving finding x for an image on self?

Some tips for solving problems involving finding x for an image on self include drawing accurate ray diagrams, using the correct sign conventions, and practicing with different types of mirrors. It may also be helpful to break the problem down into smaller steps and to double-check your calculations to avoid simple mistakes.

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