University of Wisconsin Green Bay

Your eyes are unable to focus on any object that is further away than 1.3 m. What shape lens do you need in order to correct your vision? What focal length lens is needed?

  • Lenses work by refracting, or bending, light. In order for an object to be in focus, the diverging rays that come off of any point on that object must be brought to a single point of focus on your retina. The amount of refraction for any shaped spherical lens is summarized as Lenz's Law. In other words, any time you are asked to understand image formation by a lens, it is likely to be a geometric optics problem.

  • In this problem, you are not told the shape of the lens and so you need to draw a picture to visualize the situation. Begin with a picture of nearby object that you are able to focus.

    If you now draw a more distant object you can see that the rays that reach your eye from that object are less divergent (they have a smaller angular spread.) Therefore, your eye has a problem of converging rays too much.

    To correct your vision, then, you need a diverging, or concave, lens to diverge light rays further before they reach your eye.

    With this picture and understanding, you have already answered the first question--you need a diverging, or concave, lens to correct your vision.

    Now that you know the shape of the lens, you can draw a ray diagram to help you further understand the relationship between the distant object and image formed by the lens.

  • There are two equations that summarize our understanding of geometric optics:

    The first equation locates the image for a given lens some distance o from the object, and the second relates the size of the object to the size of the image. In this case, you are asked for the focal length given information about image and object locations and so will use the first relationship.

  • To correct your vision, you need a lens with a focal point of -1.3 m.

  • Any time you are asked to understand what you see through a spherical lens or mirror, it is likely to be a geometric optics problem. Ray diagrams help you to visualize how the lens bends the light and how your brain understands those light rays, and also to catch mathematical mistakes. In this case, you are able to focus on close objects (highly divergent rays) but not on distant ones (less divergent rays) and so you understand that your eye converges rays too much. A lens that causes rays to diverge (a concave lens) is therefore needed to correct your vision. Both your understanding and your math agree, because concave lenses have negative focal points.

    The definition of focal point is the point at which rays coming from infinity converge and cross the optical axis. Therefore, the focal distance to correct nearsightedness should, indeed, be the far point of the eye.