Check your schedule! This is a reminder that Experiment 2 does not necessarily follow Experiment 1. You need to login to your course homepage on Sakai and check your lab schedule to determine the experiment rotation that you are to follow. My Lab dates: Exp.2:... Exp.3:... Exp.4:... Exp.5... Note: The Lab Instructor will verify that you are attending on the correct date and have prepared for the scheduled Experiment; if the lab date or Experiment number do not match your schedule, or the review questions are not completed, you will be required to leave the lab and you will miss the opportunity to perform the experiment. This could result in a grade of Zero for the missed Experiment. To summarize: There are five Experiments to be performed during this course, Experiment 1, 2, 3, 4, 5. Everyone does the first experiment on their first scheduled lab session. The next four experiments are scheduled concurrently on any given lab date. To distribute the students evenly among the scheduled experiments, each student is assigned to one of four groups, by the Physics Department. Your schedule is shown in your course homepage on Sakai. 11
(ta initials) first name (print) last name (print) brock id (ab17cd) (lab date) Experiment 2 Faraday rotation In this Experiment you will learn that a beam of laser light can be affected by a magnetic field; how a polarizer interacts with a plane polarized incident beam of light; how to determine the limits of validity for an experimental result; to extend your data analysis capabilities with a computer-based fitting program; to apply different methods of error analysis to experimental results. Prelab preparation Print a copy of this Experiment to bring to your scheduled lab session. The data, observations and notes entered on these pages will be needed when you write your lab report and as reference material during your final exam. Compile these printouts to create a lab book for the course. To perform this Experiment and the Webwork Prelab Test successfully you need to be familiar with the content of this document and that of the following FLAP modules (www.physics.brocku.ca/pplato). Begin by trying the fast-track quiz to gauge your understanding of the topic and if necessary review the module in depth, then try the exit test. Check off the box when a module is completed. FLAP PHYS 1-1: Introducing measurement FLAP PHYS 1-2: Errors and uncertainty FLAP PHYS 1-3: Graphs and measurements FLAP MATH 1-1: Arithmetic and algebra FLAP MATH 1-2: Numbers, units and physical quantities WEBWORK: the Prelab Faraday Test must be completed before the lab session! Important! Bring a printout of your Webwork test results and your lab schedule for review by the TAs before the lab session begins. You will not be allowed to perform this Experiment unless the required Webwork module has been completed and you are scheduled to perform the lab on that day.! Important! Be sure to have every page of this printout signed by a TA before you leave at the end of the lab session. All your work needs to be kept for review by the instructor, if so requested. CONGRATULATIONS! YOU ARE NOW READY TO PROCEED WITH THE EXPERIMENT! 12
13 The Faraday effect The Faraday effect, discovered by Michael Faraday in 1845, was the first experimental evidence that light and electromagnetism are related. This effect occurs in most optically transparent dielectric materials (including liquids) when they are subjected to strong magnetic fields. Light, and in general, electromagnetic radiation (EMR) takes the form of self-propagating waves in vacuum or in matter. These waves consist of alternating magnetic and electric field components that oscillate perpendicular to one another and to the direction of motion of the wave. By convention, the electric field vector E defines the polarization angle of the wave at any instant of time. A beam is said to be unpolarized when the E orientation of the component waves is a random mixture of all possible angles. Figure 2.1 depicts some electric field E oscillations striking a polarizer grid with a vertical polarization axis. A polarizer selectively transmits only the component of E that is parallel to the polarization axis of the polarizer, in this case E y. Recalling that E = E x + E y, then the vertical wave is transmitted fully ( E y = E), the horizontal wave is attenuated fully ( E x = 0), and the diagonal waves transmit only their E y component, although this is not shown in the diagram. The transmitted beam is said to be planepolarizedbecauseallthe E y pointinthesamedirec- Figure 2.1: Polarization of light tion, as shown by the arrow on the viewing screen. The Faraday effect or Faraday rotation is a magneto-optical phenomenon, or an interaction between light and the magnetic field in a dielectric, or non-conducting, medium. A magnetic field induces a rotation of the atomic magnetic dipoles in the dielectric, making it dielectrically polarized. This causes a beam of EMR entering the material to split into two beams by the effect of double refraction, or circular birefringence. These beams propagate throught the material at different speeds so that upon emerging from the material, they recombine with a phase shift that is expressed as a rotation in the polarization angle of the beam, as shown in Figure 2.2. The rotation angle β of the plane of polarization is proportional to the intensity of the component of the magnetic field B in the direction of the beam of light, as well as the length l of the sample: β = νbl (2.1) The Verdet constant ν is an optical parameter that describes the strength of the Faraday effect for a particular material; it varies with the temperature of the sample and the wavelength of the incident light. Figure 2.2: Faraday rotation of plane-polarized wave by angle β.
14 EXPERIMENT 2. FARADAY ROTATION Malus Law of polarization In 1809, Etienne-Louis Malus (1775-1812) observed that when a polarizer is placed in front of a beam of plane polarized incident light of intensity I 0, the intensity I of the plane polarized transmitted beam is given by I = I 0 cos 2 β, (2.2) where β is the angle between the light s initial polarization E and the polarization axis of the polarizer. From Equation 2.2 it is apparent that when β = 0,I = I 0 and the light is fully transmitted, when β = 90,I = 0 and the light is fully blocked, and when β = 45,I = I 0 /2. Equation 2.2 can be easily derived from the previous discussion of E components and by recalling that the intensity of a wave of amplitude A is I = A 2. Procedure The Faraday rotation apparatus consists of four basic components: the light source, the solenoid and power supply, the analyzer polariod and the optical detector. 1: The light source The rectangular enclosure on the right side of the Faraday apparatus contains the light source, a red laser pointer of 650 nm wavelength. The laser light exits the enclosure through an integral polarizing filter so that the output of the light source is a 95% plane polarized wave. By manipulating the four nylon thumb screws on the laser mount, the laser beam can be adjusted to traverse the apparatus along its central axis and properly arrive at the optical detector. The beam should be adjusted to yield a maximum meter reading. 2: The Solenoid The solenoid is a multilayer coil of wire 150 mm long that surroundsa sample of dielectric material, a l = 100 mm long rod of SF-59 high-index glass. When a current i flows throught the coil, a magnetic field develops around the coil. Inside the coil, this field vector B points along the axis of the coil, in the direction of the analyzer polaroid, as shown in Figure 2.2. While the magnetic field does vary along the coil axis, this variation is not significant for samples shorter than and properly centered in the solenoid. The calibration constant for the solenoid is: where i is in Amperes (A) and B in Tesla (T). B = 0.0111i (2.3) To generate a magnetic field, set a voltage on the external power supply, then press the pushbutton on the Faraday apparatus to energize the solenoid briefly. The current flowing throught the solenoid is displayed on the power supply current meter. The coil resistance is R 2.6 Ω so that according to Ohm s Law, the coil current is i = V/R. Avoid prolonged current flow through the solenoid; the coil will heat up and increase the temperature of the sample, altering your results. 3: The analyzer polaroid This component is a polaroid film that can be rotated 360 in a calibrated mount graduated at 5 intervals. A set screw is used to lock the protractor at a specific angle. Hold the protractor flat against the mount to prevent the angle, or meter reading, from changing as you gently tighten the set screw. Do not overtighten the assembly.
15 4: The Detector The intensity of the transmitted beam is measured with a photodiode detector. The detector is sensitive to the visible as well as some of the infrared spectrum. The output of the detector is a current i d directly proportional to the input intensity I. The current flows through a resistor R, generating a voltage V = i d R that is displayed in units of 0.1 mv on the digital readout. A gain switch on the detector and gain adjustment knob on the front panel are used to set R and scale this output to the 0 1999 range of the four-digit 7-segment display, the region for which the detector output is linear. Since only relative intensity (I/I 0 ) measurements will be made, calibration of the beam intensity is not required. Part 1: Verification of Malus Law The task is to verify that Equation 2.2 is valid for this apparatus. Turn on the bench-top power supply, then switch on the laser. The red laser beam should be visible at the polarizer. If there is no visible beam or the glass rod is protruding from either end of the coil, then the apparatus needs to be re-aligned. See the lab instructor. Allow 5 minutes for the laser to warm up. Turn off the laser. The meter should display a zero reading, but may not.? Why might the detector show a non-zero value with the laser is off? Explain how you would test your hypothesis. Turn on the laser, then for each of the three detector switch settings slowly rotate the analyzer polaroid over 360 and note how the intensity readout varies. Select the switch position that gives a maximum reading that does not exceed the range of the display, 1999. This will yield the best display resolution for the measurement of I. This maximum reading is the unattenuated beam intensity I 0.? How many maxima do you note as you rotate the protractor over 360? Record below an I 0 value and the corresponding protractor angle: I 0 =..., β 0 =...? What is the periodicity, in degrees, of the cos 2 (θ) function? At which angles of θ is the cos 2 (θ) function a maximum? What is the conversion from degrees to radians? In 5 increments over a 180 range that includes the above I 0 angle, record the beam intensity I as a function of the polarizer angle setting. Set the angle carefully; you should not need to tighten the polarizer set screw as you take these measurements. Close any open Physicalab programs, then start a new PhysicaLab session and enter your email address and that of your partner, if any, in the email entry box. Your graphs will be sent there for later inclusion in your online lab report. Enter the data pairs (β,i) in the data window. Select scatter plot. Click Draw to plot a graph of your data. Select fit to: y= and enter A*(cos(B*x-C))**2+D in the fitting equation box. Note that the cosine function in the fitting equation expects an argument in radians while your data is in degrees.
16 EXPERIMENT 2. FARADAY ROTATION I (mv) I (mv) I (mv) I (mv) Table 2.1: Intensity as a function of polarization angle? What are the dimensions of the four fit parameters A, B, C, D?? What is the physical meaning of the four fit parameters A, B, C, D? From your graph, estimate and enter values for the fitting parameters A, B, C, D. Click Draw to perform a fit on your data. If the fit fails, you may need to reconsider some of your guesses. Label the axes and enter your name and a description of the data as part of the graph title. Click Send to to email the group a copy of the graph. Record the fit results below: A =...±... B =...±... C =...±... D =...±...? For which value of x = β 0 does the cos 2 (Bx+C) function yield a maximum I? Determine β 0 from the fit results, then compare this value to that obtained from a visual estimate of β 0 from the graph (it may help to check the X grid box) and to the value obtained previously when rotating the polarizer to I 0 : β 0 (fit) =... =... =... β 0 (graph) =... β 0 (I 0 ) =... The angle of interest is not actually β 0, since I does not depend much on β near I 0. The greatest change, hence the best resolution, in intensity I with β occurs when β = β m = β 0 ± 45 and I = I m = I 0 /2.? Set β = β m. How did the intensity change? Is this as you expected?? Does it matter which of the two angles is used? What differences would you note as β is increased?
17 Part 2: Determination of Verdet constant As shown in Equation 2.1, the change in polarization angle β is proportional to the magnetic field B and hence to the solenoid current i. To optimize the measurement of this rotation in the plane of polarization: 1. set the analyzer polaroid to β m, at 45 relative to the incident beam, as follows: (a) rotate the analyzer to get a maximum reading on the display; (b) record the maximum intensity I 0 =..., when β = β 0. (c) rotate the analyzer until I m = I 0 /2 =... Now, β = β m. 2. Record the intensity at β m without a magnetic field, when i = 0 and B = 0; 3. Apply a current i to the solenoid to generate a magnetic field B. The intensity reading will change. 4. Rotate the analyzer polaroid until the intensity reading matches the previous B = 0 value, then estimate the new angle β. The coil current is only readable during the time that the button is pressed and the coil has current flowing through the windings. This makes it difficult to set a specific current value, quickly, so that the coil does not heat up and hence increase the temperature of the glass sample. The Verdet constant varies with temperature as well as with the frequency of light transmitted. As already mentioned, the coil has a resistance R 2.6 Ω and obeys Ohm s Law, V = ir. Hence the equation i = V/2.6 can be used to predict the value of voltage V to set on the power supply so that the coil will be energized with a current i when the button is pressed. Record below β and i for set voltages V corresponding to nominal currents of i 0,1,2,3 A. V (V) i (A) Table 2.2: Rotation data, measured with protractor Enter the four points (i,β), then fit a straight line to your data by entering A+B*x in the fitting equation box. Label the graph, then email a copy. Combining Equations 2.1 and 2.3 yields β = (0.0111νl)i and the slope from your fit can then be used to get a value for the Verdet constant: A =...±... B =...±... B ν = =...=... 0.0111 l ( B ) 2 ( ) l 2 (ν) = ν + =...=... B l ν =...±...
18 EXPERIMENT 2. FARADAY ROTATION Part 3: Determination of Verdet constant, a better method You may have noticed that the rotation angles measured in the previous section are very small for the range of currents available. With the coarse 5 resolution of the analyzer scale, the measured angles exhibit an error of 2.5, large enough to make these results, as well as the estimate for the Verdet constant ν, meaningless. This being said, you can still guess the angle values by interpolating between the scale increments to get a qualitative feel for the relationship between the solenoid current i and the resulting rotation β. You will now apply an indirect method that does not require angle measurements to determine the rotation angle and hence the Verdet constant. Using the Malus Equation 2.2 and solving for β yields: I I = I 0 cos 2 β β = arccos (2.4) I 0 The rotation angle can thus be calculated using only the values of intensity I 0 and I. This approach leads to a substantial improvement in resolution since both I and I 0 are measured precisely with the digital meter. To determine a value for the Verdet constant: Rotate the analyzer to get a maximum I 0 =... reading on the display, as before; rotate the analyzer until I = I 0 /2, then secure it with the thumb screw; Fill Table 2.3 with a series of voltage values 0 < V V max, where as before V max yields a current i 3 A. For each V entry in the table, adjust the power supply to set this output voltage. Press the button to energize the solenoid and note the solenoid current i and the resulting beam intensity I, then release the button and record the data in Table 2.3; Use Equation 2.4 to calculate the rotation angle β. Enter your data in Table 2.3; V (V) i (A) I Table 2.3: Rotation data calculated from intensity ratio Plot your data as (i,β). The graph should approximate a straight line. Select fit to: y= and enter A*x+B in the fitting equation box and perform a linear fit on your data. Email a copy of your graph, then from the slope, determine a value and error estimate for the Verdet constant: A =...±... B =...±... ν = =...=...=... (ν) = =...=...=... ν =...±...
19 Part 4: Verdet constant of SF-59 glass From the Internet or another source, determine the value of Verdet constant ν for SF-59 glass at room temperature and incident light of 650 nm. Include the appropriate units and an estimate of the error in the value, if available: ν(sf 59) =...±... IMPORTANT: BEFORE LEAVING THE LAB, HAVE A T.A. INITIAL YOUR WORKBOOK! Lab report Go to your course homepage on Sakai (Resources, Lab templates) to access the online lab report worksheet for this experiment. The worksheet has to be completed as instructed and sent to Turnitin before the lab report submission deadline, at 11:00pm six days following your scheduled lab session. Turnitin will not accept submissions after the due date. Unsubmitted lab reports are assigned a grade of zero.