PHYS272_Lab7

=Physics for Scientists and Engineers- Lab 7 PreLab =  Assignment: Perform the calculations described below. This assignment is due at the start of the laboratory period. These concepts will be used to complete the lab.

Index of Refraction of Glass
 You shine a laser beam on the surface of a square of glass, collecting the following information: n 2 = 1, p = 10.0 ± 0.2 cm, q = -6.6 ± 0.2 cm. Answer the following questions. 1. Calculate the index of refraction of the glass n 1 given equation 4 below. 2. Calculate the uncertainty in the index of refraction of the glass. Use the fact that if y=f(x 1, x 2 ) then math \displaystyle \Delta y = \sqrt{(\partial f / \partial x_1)^2 \Delta x_1^2 + (\partial f / \partial x_2)^2 \Delta x_2^2}. math

Index of Refraction of Water
 You shine a laser beam on a flat surface of a container of water, measuring 5 different incident angles θ i and corresponding transmitted angles θ t. Since the index of refraction of air is 1.00, your measurements should follow sin θ i = n w sin θ t. You plan to fit your data to a line with y = sin θ i and x = sin θ t so that from the slope of the line you can determine the index of refraction of water. The error in the y measurements will be given by

math \displaystyle \Delta y = \sqrt{(\partial y / \partial \theta_i)^2 (\Delta \theta_i)^2} = \cos( \theta_i ) \Delta \theta_i. math

Since the error is different for the different measurements, we will want to use weighted least squares, which is described on pg 184-185 of “Experimentation”.

Put together a spreadsheet or computer program to do the calculations below. You will then be able to use it to do your lab error analysis.

1. Fill in the following table. Remember to use Δθ i in radians.
 * ~ θ i ||~ θ t ||~ y = sin θ i ||~ x = sin θ t ||~ Δ y ||~ w = 1/( Δ y) 2 ||
 * 10° ± 1° ||> 8° ± 1° ||  ||   ||   ||   ||
 * 25° ± 1° || 18° ± 1° ||  ||   ||   ||   ||
 * 40° ± 1° || 29° ± 1° ||  ||   ||   ||   ||
 * 50° ± 1° || 35° ± 1° ||  ||   ||   ||   ||
 * 60° ± 1° || 41° ± 1° ||  ||   ||   ||   ||

2. Find the weighted slope and y-intercept of the data. These are given by

math \displaystyle m = \frac{\sum w_i \sum w_i x_i y_i - \sum w_i x_i \sum w_i y_i}{\sum w_i \sum w_i x_i^2 - (\sum w_i x_i)^2} math

math \displaystyle b = \frac{\sum w_i y_i \sum w_i x_i^2 - \sum w_i x_i \sum w_i x_i y_i}{\sum w_i \sum w_i x_i^2 - (\sum w_i x_i)^2} math

3. Calculate the deviations of the data points from your straight line fit δ i = y i - (m x i + b) and the standard deviation of those deviations

math \displaystyle S_y = \sqrt{\frac{\sum w_i \delta_i^2}{N-2}} math

where N is the number of data points.

4. Finally, calculate the uncertainty in the slope (S m ) via

math \displaystyle \bar{x} = \frac{\sum w_i x_i}{\sum w_i} math

math W = \sum(w_i(x_i - \bar{x})^2) math

math S_m^2 = S_y^2/W math

=Physics for Scientists and Engineers Lab 7 =

Objectives:

 * To test the laws of reflection and refraction.
 * To measure the indices of refraction for water and glass.
 * To find the position of the image produced by a plane mirror.

Equipment:

 * Plane mirrors and wooden holders
 * Crown glass rectangular plate
 * Plastic semi-disk container
 * Protractor, ruler
 * Craftsman Laser Trac™ leveling tool
 * Desk lamp
 * Graphical Analysis software

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">Physical Principles:
<span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> When light crosses from one material to another, its straight line path will bend by an amount determined by the speed of light in both materials. This “bending” of light is called refraction. The interface between two materials also causes reflection of light. See figure 1. The law of reflection states that the incident angle θ i is equal to the reflected angle θ r :

math \theta_i=\theta_r \ \ (1) math

The law of refraction, referred to as Snell’s Law, states the following relationship between the incident angle θ i and the refracted (transmitted) angle θ t :

math n_1 \sin \theta_i= n_2 \sin \theta_t \ \ (2) math

The parameter n represents the index of refraction, defined as the ratio of speed of light in a vacuum, c, to the speed of light in the material, v.

math \displaystyle n=\frac{c}{v} \ \ (3) math

For example, the index of refraction for air is 1.00, for pure water 1.33 and for crown glass it is 1.52.

Light coming from an object at a distance p from a refracting surface will create an image at a distance q from the surface. If the surface is flat, the relationship between p and q is given by

math \displaystyle \frac{n_1}{p} = - \frac{n_2}{q} \ \ (4) math

where n 1 and n 2 are the indices of refraction on either side of the refractive surface and light travels from n 1 to n 2.

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">A. Law of Reflection and Image Formation by a Plane Mirror
<span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> <span style="background-color: #ffffff; color: #333333; font-family: 'Apple Color Emoji','Segoe UI Emoji',NotoColorEmoji,'Segoe UI Symbol','Android Emoji',EmojiSymbols; font-size: 33.6px;">❗//<span style="color: #333333; font-family: Arial,Helvetica,sans-serif; font-size: 14px;">Note: Pointing a laser purposely at anybody’s face is considered a serious offense and could result in a failing score on this lab!!! //❗
 * [[image:https://www.andrews.edu/phys/wiki/PhysLab/lib/exe/fetch.php?w=240&tok=2b24f8&media=142.lab.9.fig.2.jpg width="240" link="https://www.andrews.edu/phys/wiki/PhysLab/lib/exe/detail.php?id=lab-7&media=142.lab.9.fig.2.jpg"]] || [[image:https://www.andrews.edu/phys/wiki/PhysLab/lib/exe/fetch.php?w=240&tok=b4c357&media=272.lab7.fig.3.jpg width="240" link="https://www.andrews.edu/phys/wiki/PhysLab/lib/exe/detail.php?id=lab-7&media=272.lab7.fig.3.jpg"]] ||
 * 1) Take a sheet of 8.5 × 11 inch paper and draw a set of perpendicular lines in the middle.
 * 2) Support the mirror with at least one wooden slotted block and place along the 8.5 long inch line. Be sure not to move the mirror during this part of the experiment or you will have to begin again.
 * 3) Draw a dot approximately 12 cm in front of the mirror.
 * 4) Shine the laser through the dot, towards the mirror.
 * 5) Make marks where the laser reflects off of the mirror and along the exiting beam.
 * 6) Connect the points to draw the path of the incident and reflected beams.
 * 7) Do this three more times, pointing the laser across the dot at different angles.
 * 8) Select one of the light paths, measure the incident and reflected angle and find the difference. Does equation (1) hold?
 * 9) Extend the reflected beams backwards to find the position of the image. Find the percent error between the object and image distances since for a plane mirror p = q.

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">B. Index of Refraction of Glass
<span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;">
 * [[image:https://www.andrews.edu/phys/wiki/PhysLab/lib/exe/fetch.php?w=250&tok=552941&media=142.lab.9.fig.4.jpg width="250" link="https://www.andrews.edu/phys/wiki/PhysLab/lib/exe/detail.php?id=lab-7&media=142.lab.9.fig.4.jpg"]] || [[image:https://www.andrews.edu/phys/wiki/PhysLab/lib/exe/fetch.php?w=250&tok=5ad4de&media=272.lab7.fig.4.jpg width="250" link="https://www.andrews.edu/phys/wiki/PhysLab/lib/exe/detail.php?id=lab-7&media=272.lab7.fig.4.jpg"]] ||
 * 1) Draw a set of perpendicular lines on an 8.5 × 11 inch sheet of paper.
 * 2) Place the rectangular glass plate so that the clear face is centered along the 8.5 inch line. Trace the outline and measure the width, p, of the plate.
 * 3) Direct the laser beam at an angle less than 20 degrees with respect to the normal of the glass surface, toward the center of the clear side of the glass plate.
 * 4) Mark two points along the exiting beam.
 * NOTE: make sure that it is the beam traveling through the glass, not over. To do this put your hand over the top of the glass, to block any stray light.
 * 1) Repeat this for a different angle on the same side of the 11 inch line and twice more on the other side of the line.
 * 2) Connect the points from the exiting beam and extend them back to where the lines converge and measure the distance q.
 * 3) Using the values of p and q, calculate the index of refraction of the glass and the uncertainty in the index of refraction and compare with the theoretical value of 1.52.
 * 4) Use Snell's Law and one pair of incident and refracted rays to calculate the index of refraction of the glass as well.

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">C. Index of Refraction of Water
<span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;">
 * [[image:https://www.andrews.edu/phys/wiki/PhysLab/lib/exe/fetch.php?w=250&tok=f55402&media=lab9.fig.6.jpg width="250" link="https://www.andrews.edu/phys/wiki/PhysLab/lib/exe/detail.php?id=lab-7&media=lab9.fig.6.jpg"]] || [[image:https://www.andrews.edu/phys/wiki/PhysLab/lib/exe/fetch.php?w=250&tok=4f43bc&media=272.lab7.fig.5.jpg width="250" link="https://www.andrews.edu/phys/wiki/PhysLab/lib/exe/detail.php?id=lab-7&media=272.lab7.fig.5.jpg"]] ||
 * 1) Draw a set of perpendicular lines on an 8.5 × 11 inch sheet of paper. Measure and draw five lines on the paper, representing incident rays with angles 10°, 25°, 40°, 50° and 60° so that they all meet at the intersection of the perpendicular lines.
 * 2) Fill the plastic semi-disk container with water and place it so that the flat side is centered along the 8.5 inch line.
 * 3) Direct the laser beam toward the flat side of the semi-cylinder along the 10° line.
 * 4) Mark the outgoing beam of light that exits the semi-cylinder on the curved side. Be sure to mark the portion of the beam that actually traveled through the water.
 * 5) Repeat this process for each incident angle.
 * 6) Remove the container and draw lines through these points to the center of the paper (where the perpendicular lines intersect). These are the refracted light beams.
 * 7) Measure the angles of refraction (θ t ) and complete a table of the data in your journal.
 * 8) Using the Graphical Analysis software plot a graph of sin(θ i ) versus sin(θ t ). Include error bars and error in the slope on your plot. Find the slope of the line and compare it with the accepted value of the index of refraction of water.
 * 9) Use the procedure given in the prelab for performing the weighted least squares calculation of the slope and uncertainty in the slope. Does this interval contain the accepted value for the index of refraction of water?

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">D. Critical Angle – Total Internal Reflection
<span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;">
 * [[image:https://www.andrews.edu/phys/wiki/PhysLab/lib/exe/fetch.php?w=250&tok=546922&media=lab9.fig.8.jpg width="250" link="https://www.andrews.edu/phys/wiki/PhysLab/lib/exe/detail.php?id=lab-7&media=lab9.fig.8.jpg"]] || [[image:CriticalAngle.png width="560" height="315"]] ||
 * 1) Using the same setup, turn the plastic semi-disk so that the curved side is facing the laser.
 * 2) Place the container so that the 11 inch line is passing through the center and the flat side is along the 8.5 inch line.
 * 3) Aim the laser beam at the center of the container so that it exits on the 11 inch line. Point the laser at different angles until the beam reaches the critical angle and is completely reflected inside the container.
 * 4) Mark the incoming and outgoing paths of the beam.
 * 5) Measure the critical angle, θ C, and calculate the index of refraction for water. Compare this with the accepted value


 * Further Investigations (2 points each)**
 * 1) Repeat part B above using the rectangular container filled with water.
 * 2) Repeat part C above using the semi-circular solid plastic.
 * 3) Repeat part D above using the semi-circular solid plastic.