Question

A point source S sitting on the central axis of a positive thin lens is located (to the left) between one and two focal lengths from the lens. A concave spherical mirror is to be positioned to the right of the lens so that the final real image also lies at point S. Where should the mirror be placed? Where should a convex spherical mirror be located to accomplish the same feat?

Answer 1

Let f be the focal length of the positive thin lens (convex lens).

Since the focal length of the mirrors are not specified, I'm going to assume their focal lengths to be f as well.

Now, there are two image formations before the final image formation; first image is formed by the lens which serves as light source for the mirror. Second image is formed by the mirror which then serves as the source for the lens to form the final image.

Thin lens and concave mirror:

Now, the point source S is sitting between F and 2F point to the left of the convex lens. In this case, the image is formed beyond the 2F point on the right side of the lens. The image is magnified and inverted.

Object 2F F F 21F Image

[Image source: http://www.physicsclassroom.com/class/refrn/Lesson-5/Converging-Lenses-Object-Image-Relations]

Concave mirror produces inverted image too. We have considered the focal length of the mirror to be f as well. And, this concave mirror is placed on the right of the convex lens. We need a real image from the mirror and hence the source should be placed at a distance greater than one focal length from the mirror. If the source is placed at C (= 2f), a real, inverted image is formed at C.

ject ject Image Image Ray Diagram for Obiject Located atC Ray Diagzam for Otiject Located Betmeen CandF

If the source is place between C and F, a real and inverted image is formed beyond C.

So, if we place the mirror so that the first image formed by the lens, which acts as source to the mirror, is at C of the mirror, then, an inverted image is formed exactly at C. This image will act as source for the final image formation by the lens.

But as we have discussed earlier, this point C is beyond the point 2F of the lens on the right side and hence, the image will be formed on the left side, between F and 2F, at the same point S.

Thus, the concave mirror should be placed at a distance (4f + distance from 2F to S) from the vertical axis of the lens to get the image at point S.

Convex lens and convex mirror:

I'm not sure about this part. But as far as I know, if a source is placed close to the mirror, a virtual image is formed close to the mirror too. Thus, I guess, the convex mirror should be placed such that the first image formed by the lens is close to its surface. That makes the distance, (2f + )