## Studio Session 8

### 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:  θi = θr
• Snell's law or law of refraction:  nisinθi = ntsinθt.

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 θi, 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, Ep, and a component perpendicular to the plane of incidence Es.  We have E = Ep+ Es.

The reflectance R depends of the polarization and is given for p-polarization by

Rp = ((tan(θi - θt)/tan(θi+ θt))2,

and for s-polarization by

Rs = ((sin(θi - θt)/sin(θi+ θt))2.

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

n1sinθB = n2sin((π/2) -θB) = n2cosθB.
tanθB = n2/n1.

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/xo + 1/xi = 1/f
• magnification:  M = hi/ho= -xi/xo

If the magnification is negative, the image is inverted.

Things that always go together for spherical mirrors:

• real image <--> inverted image <--> xi is positive <--> M is negative
• virtual image <--> upright image <--> xi is negative <--> M is positive

xo and xi 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.

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 θi from to zero 80o in 10 degree steps measure the angle of refraction θt and the reflectance R.
• 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 sinθi and sinθt.
• 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 sinθi versus sinθt.
• 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 nA and nB?

(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.

(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?

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 air-glass 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 θair = 20o with the normal.
• Make sure the block is at its original position on the paper.  Place a pin P1 on the θair = 20o line close to the intersection with the normal on side 1.  Place a second pin P2 at least 5 cm away on the θair = 20o 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 θair and θglass with an uncertainty of less than 0.5o.  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# θair θglass nmeasured w dmeasured dcalc difference (%)

• Repeat this process two more times for angles θair of approximately 45o and 60o.  Use fresh sheets of paper.  It becomes more difficult to align the pins when θair 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(θair - θglass)/cos(θglass).

From the figure on the right we see that

d/L = sin(θair - θglass),
w/L = cos(θglass),

and therefore

d = wsin(θair - θglass)/cos(θglass).

• Use your measured values of the width of the block w and of the angles θair and θglass 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 θair?
• 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 xo = o, and the height of the object ho = h.
• The simulation calculates the image position xi = -i  and the height of the image hi = h'.
Note. the simulation displays the image position i with the wrong sign.  Use xi = -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 xo.

• 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 xo xi 1/x0 + 1/xi 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.