Problem Set 1

20.309 Fall Semester 2010 Problem Set 1 Due Tuesday, September 14, 2010 before 11:59 PM

Question 1

For this question, you will need to stop by the lab and make a few measurements of lenses. The whole thing should take about twenty minutes.

Part a – measuring focal length

Measure the focal lengths of the lenses marked A, B, C and D

• What is your estimate of the focal length each lens?

Part b – image formation

• What is the relationship between d₁ (the distance from the object to the lens), d₂ (the distance from the lens to the image plane), and H (the height of the image formed)?

Measure this relationship using the 35mm bi-convex lens, illuminator, and object provided in the lab. Make a plot of your measurements that shows this relationship.

To make the measurements:

1. Make sure the power supply current is below 0.5 A (at all times), and turn on the LED illuminator. Ask a TA if you are unfamiliar with the power supply settings.
2. Log on to the computer and run IC Capture. This will bring up a window of the live image from the CCD camera.
3. Move the lens and/or imaging target to produce a focused image (the target has 50 line-space pairs per inch).
4. Measure the distance d₁ from the imaging target to the 35mm lens, the distance d₂ from the lens to the camera array, and the height H of a feature on the image.
5. Measure the height of a feature on the image in pixels. The sensor in the CCD camera has 5.6μm square pixels.
6. Repeat this measurement for several values of d₁. Make enough measurements to get a smooth plot.

• Make plots of your data showing the relationship between d1, d2, and H. Refer to your class notes or talk to a TA to find the best choices for the x- and y-axes.

Question 2

In lab, you are going to build a microscope. As discussed in lecture, a single lens can act as a magnifier. (You have certainly used a magnifying glass to look at something small.) But all modern microscopes that achieve very high magnifications use multiple lenses in series to make an image. This is clearly a much more expensive design. Why not use a single lens as a high-magnification microscope?

• List three potential problems with a single-lens, high-magnification microscope design. Explain the problems in a few sentences and include a simple sketch.

HINT: Use the information about the behavior of light that was presented in lecture. Think about the underlying assumptions of geometric optics. Under what circumstances in a real microscope design would each assumption be violated? How do these violations impact the final image? If you are stuck, go back to the lecture slides and take a close, logical look at each assumption. Where does each assumption break down? What is the effect of this breakdown in the assumption?

Extra credit

For each problem you identify with a single-lens, high-magnification microscope design, explain in a sentence or two and a brief sketch how multiple lenses might ameliorate the problem.

Question 3

Consider the two-lens system shown in the figure below, where the rectangle on the left represents an unspecified lens of focal length f₁ separated by 0.5 cm from another lens with focal length of 1 cm. Find the value of f such that all the rays incident parallel on this system will be focused at the observation plane.