Refraction and reflection

• Law of reflection:  θ_i = θ_r
• Snell's law or law of refraction:  n_isinθ_i = n_tsinθ_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 n₁ and n₂, the angle of incidence θ_i, and the polarization of the incident light.
We distinguish between p-polarization and s-polarization.

Consider, for example, an air-glass 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 p-polarization by

R_p = ((tan(θ_i - θ_t)/tan(θ_i + θ_t))²,

and for s-polarization by

R_s = ((sin(θ_i - θ_t)/sin(θ_i + θ_t))².

If θ₁ + θ₂ = π/2, then tan(θ₁ + θ₂) = infinite  and R_p = 0.  If light is reflected, it will have s-polarization.  The incident angle at which this happens is called the Brewster angle θ_B.  We then have

n₁sinθ_B = n₂sin((π/2) -θ_B) = n₂cosθ_B.
tanθ_B = n₂/n₁.

Explore using this spreadsheet.  Vary n1 and n2 and observe the changes.

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 on-line simulation from the University of Colorado PhET group to explore the bending of light.
Link to the simulation: https://phet.colorado.edu/en/simulations/bending-light

• Tools and objects can be dragged out of the tool box and then returned.
• The objects in the Prisms tab can be rotated by dragging the handle.
• In the Prisms 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.

• Plot R versus theta.  Paste your graph into your log.  Compare to the graph above.
• Discuss your result. 
  • Is the laser light p-polarized, s-polarized, or unpolarized.  What do your results suggest?
• Calculate sinθ_i and sinθ_t.  Remember that Excel functions require the angles to be in radians.
  The first row if the spreadsheet already contains the formulas.
• 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 color-dependent?  How can you tell?
• Which shapes split the white light into different colors the best?  Did you find a set-up that demonstrates this well?
• Try to arrange a situation so that the light light forms a rainbow.  What shape did you choose?

Reflection

Exploration 2:

Explore image formation with spherical mirrors.  Use this interactive simulation.

• 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 and the height of the image h_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.

• 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.

Experiment:

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 air-glass boundaries and use these angles to determine the index of refraction of crown glass.  Each student will perform the experiment.

• 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 P₁ on the θ_air = 20^o line close to the intersection with the normal on side 1.  Place a second pin P₂ at least 5 cm away on the θ_air = 20^o line.  Both pins should be as vertical as possible.  The line P₁P₂ defines an incident ray.
• Place your head, so that you can look into the glass block from side 2.  View the image of pins P₁ and P₂ and line up both images.  Place a pin P₃ close to the block on side 2, where your line of sight, which lines up P₁ and P₂, enters the block.  P₃ must be lined up with the images of both P₁ and P₂.  (A common error is to line up P₃ with the image of only one of the pins, P₁ or P₂.)  Now place fourth pin P₄ at least 5 cm from P₃ 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.

• 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),

  or, applying Snell's law and trigonometric relations,

  d = w sinθ_air[1 - cosθ_air/(n_glass² - sin²θ_air)^½].

   

• 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?
• Take a picture of a diagram produced by one of your group members and paste it into your Word document.

Convert your log into a lab report.  See the grading scheme for all lab reports.

Name:
E-mail address:

Laboratory 7 Report

• In one or two sentences, state the goal of this lab.
• Make sure you completed the entire lab and answered all parts.  Make sure you show your work and inserted and properly labeled relevant tables and plots.
• Add a reflection at the end of your report in a short essay format.

Save your Word document (your name_lab7.docx), go to Canvas, Assignments, Lab 7, and submit your document.