Determination of the Focal Length of a Diverging Lens
A homework exercise due at the beginning of class on Nov. 27, 2000
Since a diverging lens (one with a negative focal length) always produces a virtual image, we can’t measure its focal length directly. However, it can be used together with a converging lens so that the combination of lenses produces a real image. The basic concept is that the image formed by the diverging lens becomes the object for the converging lens. In this exercise, you’ll analyze data collected from one of the PASCO optical benches to find the focal length of a diverging lens from the kit.
The optical bench is set up as shown in the diagram below. The object is at 0
Object fx f = 127 mm Screen
0 100 300
mm, the diverging lens (whose unknown focal length is called fx) is at 100.0 mm, and the converging lens with f = +127 mm is located at 300.0 mm. The screen is moved until a sharp image is seen, and its position is measured. Here is a summary of the results of this procedure:
Position of object 0.0 mm
Position of unknown lens 100.0 mm
Position of known lens 300.0 mm
Position of image 556.8 mm
Before beginning an analysis of these data, let’s review what’s happening on the bench. Light from the object passes through the unknown lens (fx), and an image is formed somewhere between 0 and 100 mm. The image is on the same side of the lens as the object because the unknown lens is diverging. Light from that first image then passes through the second lens, which has a known focal length of +127 mm. It forms a real image that happens to be found at 556.8 mm along the bench. From this information, we can work backwards to find where the “object” is for the 127 mm lens. That “object” is actually the image formed by the first (unknown) lens, so finding its position on the optical bench will allow us to find the image distance for the first lens. We already know that the object distance for that lens is 100 mm, so we can use the known object distance and the measured image distance to find the focal length.
(All results must have correct units.)
1. From the data given, what is the image distance for the 127 mm lens? (i.e., how far is the image from the second lens?)
Image distance for 127 mm lens
2. Use the answer to 1 to find the object distance for the 127 mm lens. This will be a position to the left of the second lens.
Object distance for 127 mm lens
3. You should have gotten an answer that corresponds to a location somewhere between 0 and 100 mm on the optical bench, which is to the left of the first (diverging) lens. This is where the image formed by the first (diverging) lens is located. Figure out how far this point is from the first (diverging) lens.
Distance from diverging lens to the “object” for the second lens
4. The object distance for the diverging lens is +100.0 mm, and the answer to 3 with a - sign is the image distance for the diverging lens. (Remember that image distances are always negative for diverging lenses!) Use the lens equation to find the focal length of the diverging lens. Give the final answer to 3 digits.
Focal length of diverging lens
5. The focal length marked on the unknown lens is -100 mm. How close is your result?