Optics - Images with two convave mirrors

[kihr]

Homework Statement

Two concave mirrors of equal focal length f are placed a distance d apart in air. A point source is placed midway between the two mirrors. What should d be so as to get only a single image?

Homework Equations

Mirror formula (1/f = 1/u + 1/v)

The Attempt at a Solution

The image distance v1 of the first mirror is calculated using the mirror formula.
1/-f = -2/d + 1/v1
1/v1 = 2/d - 1/f
v1 = df/2f - d
Similarly the image distance v2 of the same object from the second mirror is
v2 = df/2f -d (distances measured from the pole of the second mirror)
The distance of the image formed by the second mirror from the pole of the first mirror is

v3 = d - v2
= d - df/2f -d
= df - d*2 / 2f - d

If the images formed by the two mirrors are to coincide

v1 = v3
df / 2f - d = df - d*2 / 2f - d
This leads to d =0 which is not possible. I would like to know where I have gone wrong. The answer given in the book is 2f and 4f.

 

[rl.bhat]

In a concave mirror, if you keep an object on the focus the image will be at infinity, and if you keep it at 2f, the image will be on the object i.e. at 2f. So d should be ...?

 

[kihr]

Yes in this case the answer should be 4f (i.e. 2f + 2f), and 2f should not be one of the solutions as given in the book. Would you also let me know in what way I have erred in applying the mirror formula to solve the problem. Thanks.

 

[rl.bhat]

Yes in this case the answer should be 4f (i.e. 2f + 2f), and 2f should not be one of the solutions as given in the book. Would you also let me know in what way I have erred in applying the mirror formula to solve the problem. Thanks.

2f is also the correct answer. Because if you keep the object on the focus of the first mirror, the reflected rays from that mirror will be parallel. These parallel rays fall on the second mirror which reflect them back to the focus.

 

[kihr]

Thanks a lot.