## Lens Report

Tuesday, August 10th, 2004

Today was quite a revolutionary day. I came into today with what I now look back upon as a weak understanding of lenses and the thin lens formula. I've learned many things now that they've finally been beaten into my head. I also began to gain a new perspective on high school education; we learn formulas and concepts for tests but we never put them into practice. So, these concepts are inevitably molded and changed until one is left with a weak understanding at best. In any case, I have now learned many concepts that I will never forget.

I'll try to explain the entire discussion chronologically to the best of my ability. Dr. Noé started off by asking me to calculate the diameter of the sun's image when using the large magnifying glass (f = 20 cm). I made this calculation with the following formula:

But it turned out that I was off by a factor of ten because of an error in defining my variables. Then Dr. Noé showed me another way of solving the problem using the angular diameter of the sun (.009 radians). As a part of this same discussion, I made the grave error of writing the equation s = r(theta) incorrectly, which caused me to make the equally grave error of defining the units of radians as square meters. I soon discovered how wrong I was. Radians are a dimension-less unit that describe the relationship between two things. It was a valuable digression I must say. So, the angular diameter of the sun (9 mrad) times the focal length of the lens (200 mm) gave a value of the image size of the sun. Then, Dr. Noé explained that our two methods of solving the problem were just different ways of expressing the same thing.

After this, we talked about the distance a quarter should be held away from one's eye to block out the moon (angular diameter = .01 rad). We found the distance away from the eye to be 250 cm.

Then we began to discuss the concepts that I had learned in terms of the microscope and the CCD camera. We assumed that a particle was a micron in diameter (d = 1u). So the angular diameter of the particle was calculated to be 6.67 X 10^-4 rad (based on the 16 mm focus of the lens). And based on the Magnification of the microscope, the particle is magnified to 100um. However, it turned out that the camera lens was actually minifying the particle, so the 100um particle would have a size of maybe 10um on the CCD element.

Then we decided to remove the lens and look at the rainbow glasses with just the CCD camera. This was a very productive exercise because I was able to figure out the magnification of the particle all the way from the slide to the TV screen. I took measurements of the number of dots on the television (going based on the valid assumption that the TV screen "saw" everything that the CCD element "saw"). So, starting from a distance (for the CCD element) of 120 mm above the slide, I counted the number of dots. For the measurement of 160 mm (the focal length of the 100X objective), I counted 9 dots (horizontally), which equated to 58um of the actual rainbow glass. This worked out very well because it turned out that the CCD element had a horizontal length of 6mm. So, with these two pieces of data, I was able to discern that there was a magnification of about 103.4X (3.4% error based on the accepted value of 100X magnification at 160mm from the object). I took values for the number of dots at 10mm intervals. I began collecting data at 120mm and went all the way up to 270mm above the slide. As I went beyond 160mm, there was a steady increase in magnification (I was able to see fewer and fewer dots as the camera went higher up). So, at the final data point (270mm), I was able to count 3 dots, which equates to 18um and a 333X magnification. As I moved the camera higher and higher, I of course had to refocus the objective using the fine adjustment knob. So, the focal length of the objective was not held constant. But, I can calculate the values of the focal length of the microscope that I needed using the magnification and the image distance. First, I would need to find the object distance (M = i/o). Using the thin lens formula, this distance would virtually be equal to the focal length of the objective (because 1 divided by large numbers like from 120-270 (image distances) is so close to zero that it is negligible). So using this logic, the focal length of the microscope objective at 270mm is .8mm.

Data

 Object Size (microns) Image Size (mm) Magnification Image Distance (mm) Objective Focal Length/Object Distance 58 6.0 103.4 160 1.55 42 8874 42.5 4512 45 9 (off-axis) 22.86 14.12 0.717 25 63.5 20.04 1.02 93.5 237.5 60.84 3.09 157.25 399.4 102.3 5.20

 Anirudh Ramesh