Refraction and reflection
In this session you will explore the behavior of light at the boundary
between two transparent media with different indices of refraction. A
fraction of the incident intensity will be reflected, and the rest of the light
will be transmitted. The direction of propagation of the reflected and
transmitted light is given by the laws of reflection and refraction.

Law of reflection: θ_{i} = θ_{r}
 Snell's law or law of refraction:
n_{i}sinθ_{i }= n_{t}sinθ_{t}.
How much of the light is reflected and how much is transmitted?
The reflectance R is the ratio of the reflected
flux to the incident flux, and the transmittance
T is the ratio of the transmitted flux to the incident flux.
Energy conservation requires that R + T = 1 (if there is no absorption).
R and T depend on the indices of refraction of the two media n_{1}
and n_{2}, the angle of incidence θ_{i},
and the polarization of the incident light.
We distinguish between
ppolarization and spolarization.
Consider, for example, an airglass interface as shown. The plane of incidence contains the normal to the boundary and the
incident ray. The electric field vector E of the incident
wave is perpendicular to the direction of propagation and can have a component in the plane of incidence,
E_{p},
and a component perpendicular to the plane of incidence E_{s}.
We have E = E_{p}+
E_{s}.
The reflectance R depends of the polarization and is given for ppolarization by
R_{p} = ((tan(θ_{i } θ_{t})/tan(θ_{i}+ θ_{t}))^{2},
and for spolarization by
R_{s} = ((sin(θ_{i } θ_{t})/sin(θ_{i}+ θ_{t}))^{2}.
If θ_{1 }+ θ_{2} = π/2, then tan(θ_{1
}+ θ_{2}) = infinite and_{ }R_{p} = 0. If light is reflected, it
will have spolarization. The incident angle at which this happens
is called the Brewster angle θ_{B}. We then have
n_{1}sinθ_{B} = n_{2}sin((π/2) θ_{B}) = n_{2}cosθ_{B}.
tanθ_{B} = n_{2}/n_{1}.
Reflection and refraction can result in
image formation. Spherical
mirrors form images by reflection. The mirror equation tells us
where the image is formed and if it is real or virtual.
 mirror equation: 1/x_{o
}+ 1/x_{i }= 1/f
 magnification: M = h_{i}/h_{o}=
x_{i}/x_{o}
If the magnification is negative, the
image is inverted.
Things that always go together for spherical mirrors:
 real image <> inverted image <> x_{i} is positive
<> M is negative
 virtual image <> upright image <> x_{i} is negative
<> M is positive
x_{o} and x_{i} are positive for locations in front
of the mirror and negative for locations behind the mirror. R and f are positive for concave and negative for convex mirrors,
and f = R/2.
In this session you will explore refraction at a plane
interface and image formation by reflection from spherical surfaces.
Equipment needed:
 Glass block (square)
 White paper
 Thick cardboard backing
 Pins
 Millimeter ruler
 Protractor
Open a Microsoft Word document to keep a log of your procedures,
results and discussions. This log will become your lab report. Address the
points highlighted in blue. Answer all questions.
Refraction
Exploration 1
Use an online
simulation from the University of Colorado PhET group to explore the bending of
light.
Link to the simulation:
https://phet.colorado.edu/en/simulation/legacy/bendinglight
Explore the interface!

Tools and objects can be dragged out of the tool box and then
returned.

The objects in the Prism Break tab can be rotated by dragging the
handle.

In the Prism Break tab, the protractor rotates and the laser translates.

All the tools work in both Ray and Wave mode, but some are easier to use
in Wave mode because the region where the tool can read is larger.
Click the intro tab.
(a) Let red light move from air into water.
For incident angles θ_{i} from to zero 80^{o} in 10 degree
steps measure the angle of refraction θ_{t }and the reflectance R.
Download this spreadsheet and enter your measured
values on sheet 1.
 Plot R versus theta. Paste your graph into
your log. Compare to the graph above.
 Discuss your result.
 Is the laser light ppolarized, spolarized,
or unpolarized.
 What do your results suggest?
 Calculate sinθ_{i} and sinθ_{t}.
 Remember that Excel functions require the angles to be in radians.
 Into cell D2 enter =sin(A2*pi()/180) and into cell E2 enter
=sin(B2*pi()/180).
 Copy these formulas into the other cells of columns D and E.
 Plot sinθ_{i} versus sinθ_{t}.
 What does the plot look like?
 Use the trendline to find the slope.
Paste the graph with trendline into your log.
 What value do you obtain for the slope?
 Given Snell's law, what value do you expect
for the slope? Discuss!
(b) Design experiments to determine the index of refraction of mystery
materials A and B.
 Describe your procedure and discuss why you
decided to proceed this way. What are your results for n_{A}
and n_{B}?
(c) Design and describe a setup that has the
refracted ray bend away from the normal?
 Paste a screen shot of your setup into your log.
(d) Click on the prism break tab. Use red light with a wavelength
of 650 nm. Try to arrange various prisms in such a way, so that the laser
beam after total internal reflections moves parallel to the incident beam but in
the opposite direction. Try to use as few pieces as possible.
 Paste a screen shot of your design into your log.
(e) Now switch to white light and experiment with various prisms to
answer the following questions.
 Are the reflection and refraction of light
colordependent? How can you tell?
 Which shapes split the white light into different
colors the best? Did you find a setup that demonstrates this well?
 Try to arrange a situation so that the light light
forms a rainbow. What shape did you choose?
Experiment 1:
In this experiment you will trace the path of a light ray through a block of
glass. You will determine the angle of incidence and the angle of refraction at
two airglass boundaries and use these angles to determine the index of
refraction of crown glass.
 Place a sheet of paper onto the cardboard backing. Place the glass block
onto the paper and outline its position accurately with a sharp hard pencil. Use the protractor to draw a normal to the block close to one corner. Draw
another line through the intersection making an angle of about θ_{air }= 20^{o} with the normal.
 Make sure the block is at its original position on the paper. Place a
pin P_{1 }on the θ_{air }= 20^{o}
line close to the intersection with the normal on side 1. Place a second pin
P_{2} at least 5 cm away on the θ_{air }= 20^{o}
line. Both pins should be as vertical as possible. The line P_{1}P_{2
}defines an incident ray.
 Place your head, so that you can look into the glass block from side 2. View the image of pins P_{1} and P_{2 }and line up both
images. Place a pin P_{3 }close to the block on side 2, where your
line of sight, which lines up P_{1} and P_{2}, enters the
block. P_{3} must be lined up with the images of both P_{1 }
and P_{2}. (A common error is to line up P_{3 }with the
image of only one of the pins, P_{1 }or P_{2}.) Now place
fourth pin P_{4 }at least 5 cm from P_{3 }along your line of
sight on side 2, so that all four pins appear to be placed along a straight
line.

Remove the glass block and carefully complete the diagram on the sheet
of paper as shown in the figure below.
 Measure the angles θ_{air} and
θ_{glass} with an uncertainty of less
than 0.5^{o}. Measure the width w of the block and the displacement
d of the ray with an uncertainty of less than 0.5 mm. Enter your
measurements into the table on sheet 2 of your spreadsheet.
trial# 
θ_{air} 
θ_{glass} 
n_{measured} 
w 
d_{measured} 
d_{calc} 
difference (%) 
























 Repeat this process two more times for angles θ_{air}
of approximately 45^{o} and 60^{o}. Use fresh sheets of
paper. It becomes more difficult to align the pins when θ_{air} gets larger, but the precision of the measurements
improves.
Data Analysis:
 Use the results of each of your trials to determine the index of
refraction n of crown glass.
 Find the average value.
 Find the percent
difference between this average measured value and the nominal index of
refraction for crown glass, n = 1.52.
 The expected displacement of a ray passing through the glass block is
d = wsin(θ_{air } θ_{glass})/cos(θ_{glass}).
From the figure on the right we see that
d/L = sin(θ_{air } θ_{glass}),
w/L = cos(θ_{glass}),
and therefore
d = wsin(θ_{air } θ_{glass})/cos(θ_{glass}).
 Use your measured values of the width of the block w and of the angles θ_{air} and θ_{glass}
to calculate d. Compare this calculated value with your measured value of d and find the percent difference.
 Insert your table into your log.
 Report the average value of the index of
refraction of crown glass from your measurements and the percent
difference between this average value and the accepted value.
 Comment on
your three diagrams. How does the deviation d vary with θ_{air}?
 Hand in your diagrams to your instructor.
Reflection
Exploration 2:
Explore image formation with spherical mirrors.
Open the simulation at
http://www.shermanlab.com/science/physics/optics/SphericalMirror.php.
Note: If Java is blocked, add http://www.shermanlab.com to the Exception Site List in the Java Control Panel.
 You can choose the radius of curvature of the mirror R (positive or
negative), the object position x_{o} = o, and the height of the object h_{o} = h.
 The simulation calculates the image position x_{i} = i and the
height of the image h_{i} = h'.
Note. the simulation displays the image position i with the wrong sign.
Use x_{i} = i.
 The simulation also draws a ray diagram.
Investigate 4 different situations and fill out the
table on sheet 3 of your spreadsheet. You choose the radius of curvature R
and the object position x_{o}.
 Use a concave mirror to produce a real image which is bigger that
the object.
 Use a concave mirror to produce a real image which is smaller that
the object.
 Use a concave mirror to produce a virtual.
 Use a convex mirror to produce an image.
case 
R 
f 
x_{o} 
x_{i} 
1/x_{0} + 1/x_{i} 
1/f 
M 
image real? 
image upright? 
concave mirror, real image: hi > ho 









concave mirror, real image: hi < ho 









concave mirror, virtual image










convex mirror










 Paste the table into your log.
 Discuss your results.
 Can you think of situations where spherical
mirrors are used to produce the images explored in case 1  4.
Convert your log into a session report, certify with you signature that
you have actively participated, and hand it to your instructor.