Physics 234 1. Spring 2015 Homework #6 Where's the image? In the figure at the right is shown a thin lens
(indicated by a gray rectangle) and a coordinate
system. The x axis passes through the center of the
lens and runs along its axis of symmetry with the
positive x direction indicated by the arrowhead. The
lens may be treated as of negligible thickness and has
a focal length f.
The points {x,y} = {f,0} and {-f,0} are marked by
black dots.
A small object is placed at the position {x,0}. For
each of the four cases (i)-(iv) below, indicate whether
the location of the image formed (= x') is: on the positive or negative side of the axis closer to the lens than the focal point or
farther away. Thus your answers should take a form such as (x' > f > 0) to indicate (the image is on the positive side of
the axis and further away than the focal point) or (0> x >f) to indicate the image is on the negative side of
the axis between the lens and the focal point) respectively. Note that in some cases the focal length
specified is negative and in some cases it is positive.
i. -f < x < 0 ii. f<0<x iii. x<f<0 iv. 0<f<x 2. On the mirror
While doing a tutorial lesson on "How to find out where something is by looking," two students, Ethelred
and Guinivere, answered the question, "Where does the image [in the mirror] appear to be located?" by
saying "On the mirror." Do you agree with them or not? If you disagree, where would you say the image
is and how would you justify your answer to them? If you agree, propose different plausible position for
the image in the mirror and explain why you prefer Ethelred and Guinivere's answer. 3. Modeling color – chromophores
Many biological molecules give color to objects by
absorbing light of particular colors and reflecting
other light. Some notable examples are betacarotene (orange -- found in carrots and egg
yolks), lycopene (red -- found in tomatoes), and
chlorophyll (green -- found in many plants). The
absorption spectra of beta-carotene is shown in the
figure at the right. It absorbs strongly in the blue -at about 450 nm -- so it leaves the red, yellow, and
orange light that hits it to reflect back.*
* The figure is from a very old paper -- 1942 -- so
the units given are Ã…ngstroms (Ã…), not the SI units
of nanometers (nm). 10 Ã… = 1 nm.)
The part of the molecule that is primarily
responsible for the absorption of visible light
hitting the molecule is called a chromophore. For
beta-carotene, the molecular structure is shown
below, and the chromophore is indicated in red on
the chemical formula shown below that. Source: F. Zscheile et al., Plant Physiology, 17(3),
331-346 (1942) The chromophore consists of a long string of double bonds. A reasonable starting point might be to model
the energy levels as if the electrons just could run back and forth freely along the molecule. The quantum
physics tells us (see the reading Quantum string for a full derivation) if that's the case, the energy levels
can be gotten assuming the electron waves are like the normal modes of an oscillating string. This model
of the electron states gives the energy of the n-th level of a string of length L as To model the excited states of the system we have to know a bit of chemistry. The electrons in the
chromophore fill up the energy levels. You can only put two electrons in each state. There are 22 in the
string of double bonds, so in the ground state all energy levels up to n=11 are filled.
Therefore, the first energy level that in the "ground state" is not occupied by electrons is n=12.
A. The first excited state of the molecule is defined as the state that is closest to the ground state in
energy. It involves bringing an electron into the lowest free energy level of n=12. From which energy
state should the electron be excited if we want to excite the lowest energy state above the ground state?
B. According to the data shown at the top, this chromophore (beta carotene) does a good job absorbing
photons of wavelength lambda = 450 nm. What is the energy of this photon in eV?
C. Use the vibrating string model of the electrons in the chromophore to estimate the size of the
chromophore, L. The actual length is about 3 nm. Is the model giving you a reasonable result? 4. The Microscope
A microscope is a device where a first lens makes a magnified real image of an object and that real
image is then viewed with a second lens used as a magnifying glass to make a virtual image that is
magnified further.
A. In a copy of the figure below (not to scale, but arranged so that the image formation works properly)
draw a ray diagram showing the formation of the real and virtual images. (The "x" marks are the focal
points belonging to the left lens. The "o" marks are the focal points belonging to the right lens.) B. If the focal length of the left lens is 1 cm, the focal length of the right lens is 8 cm, and the distance
between the two lenses is 15 cm, what will the magnification of the system be (i.e., How much bigger will
the final image be than the object is?) when the object is at a distance of 1.125 cm from the left lens?

 

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