# Optics question: converging lens with virtual object

## Main Question or Discussion Point

Hi everyone,

I was thinking about an optics question in a physics textbook. I think the solutions I saw in the solution manual may be wrong.

The question is:
A lens and mirror are separated by d = 1.00 m and have focal lengths of 80.0 cm and -50.0 cm, respectively (so a converging lens, and a convex mirror). An object is placed p = 1.00 m to the left of the lens (so the object is at x = 0, the lens at x = 1.00 m, and the mirror at 2.00 m).

Considering only the light that leaves the object and travels first towards the mirror, locate the final image formed by the system.
When you do the calculations using

1/p + 1/q = 1/f

you find that the lens first creates an image 400 cm to the right of it. Using this as the object for the mirror (so, p = -300, since it is behind the mirror), you'll find q = -60 cm. This means the image is 60 cm behind the mirror, or 160 behind the lens.

So now I look at the last pass through the lens. This is where I am confused. Since the object is behind the lens, I would think that I should use p = -160 cm. With the focal length of the mirror being 80 cm, I find q = 53 cm. According to my understanding of the sign convention, this means that the object is 53 cm to the right of the lens.

However, when I look at the textbook solution, they do not get this. All of the steps are the same until the third calculation. They state that the object is 160 cm to the right of the lens (just like me), then proceed to use p = +160cm, upon which they find q = 160. They interpret this as meaning that the object is 160 cm to the left of the lens.

So, it seems that they simply interpreted the virtual object as being a real one and proceeded from there. This seems reasonable, but goes against all the info that I could find about virtual objects.

Any thoughts?

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jtbell
Mentor
So now I look at the last pass through the lens. This is where I am confused. Since the object is behind the lens, I would think that I should use p = -160 cm.

However, [they] then proceed to use p = +160cm
"In front of" and "behind" the lens are relative to the direction the light is traveling. "In front of" means the side of the lens from which the light is entering, and "behind" means the side towards which the light is emerging. The mirror reverses the direction of travel, so after the reflection, you have to switch the labels for "front" and "back" sides of the lens.

Another way to think of it: after the reflection step, mentally flip the diagram around so the light is still going from left to right. Or maybe even re-draw the diagram.

Yet another way to think of it: light rays coming from a real object diverge as they hit the lens, whereas light rays heading towards a virtual object converge as they hit the lens.

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