BE/APh161: Physical Biology of the CellHomework 5Due Date: Wednesday, February 8, 20121. Fleshing out the vision thing.RP to class: we assigned this last week, but I did not get to it because I decidedto change the logic of the lectures. Now this is a part of homework 5.(A) In class, I wrote an integral for the Fraunhofer diffraction from a circularaperture. Justify this integral in your own words being sure to explain why Iwas referring to stopwatches and how this captures the idea of different phasesfor different rays going out to the detection point. The key points you are tryingto get across in this part of the problem are the underlying physical basis for thediffraction from the circular aperture and making sure that the mathematicalformula is correct.(B) Go through all of the messy steps of doing the integral from part (A) toarrive at the simple formula for the angular resolution, namely,sin θ = 1.22λD. (1)What this amounts to is using the results from part (A) to carry out the relevantintegral and then interpreting the result in terms of the shape of the intensityprofile. In particular, the image plane is at a distance Z from the aperture andthe angular resolution is given bysin θ =XZ, (2)where X is the distance between the origin and the point of interest on theX-axis on the image plane.Some useful mathematical facts that will come in handy for doing this prob-lem includeZ2π0dθeixcosθ= 2πJ0(x) (3)andZxJ0(x)dx = xJ1(x). (4)What I have in mind in this part is that you carry through the entire deriva-tion that was sketched in class, explaining the approximations that you makeand the overall logic. Then, I want to make sure you explain how this tells ussomething about the resolution of an optical instrument (including the eye).(C) Plot the sum of two of the intensity curves you calculated in part (B)for several different distances between their centers and use this to explain the1argument about the diffraction limit and how objects can be resolved.(D) Do the following experiment. Draw two lines, first, 1 mm apart andsecond, 2 mm apart. Tape the paper with these lines to the wall and now walkbackwards until you reach a point where you can no longer distinguish the linesas being distinct. What is their angular separation in the two cases? How doesthis compare to the calculated limit?2. Ion channel properties.Work out problem 17.5 of PBoC2.3. Diffusion and the Fourier transform.Work out problem 13.2 of
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