University of Wisconsin Green Bay

You can use a convex lens to magnify objects. You find a lens that focuses sunlight about 2.3 cm from the lens. How far should you hold the lens in front of a book if you want to double the size of the type?

  • Lenses work by refracting, or bending, light. For spherical lenses, the effects of refraction are summarized in how we draw and understand the principle rays. In other words, any time you are asked to understand image formation by a spherical lens, it is likely to be a geometric optics problem.

  • In this side view drawing, the arrow represents the object you are looking at--the book--and the triangle represents your eye. Note that in order for you to see the image rather than the object, light must leave the book, travel through the lens, and travel to your eye. Therefore, the lens is located between the object and your eye.

    To locate the image, we need to draw the three principle rays from the top of the object and trace them to your eye. The easiest principle ray to draw is the one that leaves the top of the object, enters the center of the lens and continues on in a straight line:

    The second principle ray leaves the top of the object traveling parallel to the optic axis. When it reaches the lens, it changes direction and goes out through the focal point:

    Finally, the third principle ray goes into the lens on a line that passes through the top of the object and the focal point. It changes direction in the lens and goes out parallel to the optic axis.

    To understand where the image is located, you need to trace the rays that reach your eye back to the point where they appear to intersect.

    You now have a picture that illustrates not only the location of the image but also shows a visual understanding of how that image is located and sized.

  • 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 to relate location to size, and so you will need both equations.

  • Because we know the magnification of the object, we are able to relate the distance from the object to the lens to the distance from the image to the lens. If you began instead with the 1/f equation, you would quickly see that you had two unknowns and would need to return to this equation before you could go further.

    In order to magnify the letters on the book to twice their size, you need to hold the magnifying lens 1.2 cm in front of the book.

  • 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. For example, note that the distance between the object and the lens is less than the focal length, as expected. If you were unsure of whether to use i = 2o or i = -2o, for example, you could catch your mistake here. In that case, make sure to understand that you need the negative (your image is upright) before you move on.