Magnetic Fields. Introduction. Ryerson University - PCS 130

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1 Ryerson University - PCS 130 Introduction Magnetic Fields In this experiment, we study magnetic fields of several electrical configurations and their dependence variables such as postion, and electric current. We start by observing the magnetic field in a long straight conductor (as a class), how to measure the magnetic field, and how it changes as a function of distance. Here we will use the results evaluate the permeability of air. The magnetic field of a single, then two coil systems will be studied; in particular a specific two coil configuration defined as a Helmholtz coil. This is the condition where two coils of wire form a (spatially) broader magnetic field in the volume between the two coils. The equation for each configuration is derived by integrating the Biot-Savart Law differential: db = µ 0 I(dL ˆr) (1) 4π r 2 Measuring magnetic fields can be achieved by using a sensor comprised of a Hall-effect transducer. The sensor measures only the component of the magnetic field that is perpendicular to the end of the sensor. A positive value indicates that the sensor is pointing to magnetic south. The Hall sensor probe head can be bent 90 degrees as seen in the figure below. Please note that the probe head only bends one way and the wires within are susceptible to breaking when handled roughly. Magnetic field sensor configurations Page 1 of 8

2 Apparatus Power supply Ruler Retort stand + ruler holder Magnetic field coil [200 turns, 10.5cm radius] (2) + base Ryerson University - PCS 130 Electric cables Vernier LabPro + Logger Pro software Vernier magnetic field sensor Clamps Pre-Lab Questions Please complete the following questions prior to coming to lab. Completing these questions will greatly assist with understanding the concepts covered during the lab. 1.) Read through the entire lab writeup before beginning 2.) Starting from Biot-Savart Law (Eqn. 1), show that the function for the magnetic field for a single wire (of infinite length) as a function of distance r (away from the conductor) and electric current I is: B wire (I, r) = µ 0I 2πr ˆθ (2) 3.) Starting from Biot-Savart Law (Eqn. 1), show that the function for the magnetic field for a coil of wire of radius R and N turns along the central (perpendicular) axis z and electric current I is: B coil (I, z) = µ 0NI 2 R 2 ẑ (3) (z 2 + R 2 ) 3/2 4.) Using Eqn. 3, write the function for the magnetic field along the central (perpendicular) axis of two identical coils spaced R apart. Show that the magnetic field at the midpoint between the two coils is: B helmholtz,max (I) = 8µ 0NI 125 ẑ (4) Page 2 of 8

3 Measurement Procedure Ryerson University - PCS 130 In the section, the general procedure of taking a proper measurement is outlined. By following this process, you increase the accuracy of your results. Following this procedure removes the effect of any local magnetic sources that may contribute to the measurement. 1.) Ensure all components are properly connected in the desired configuration. The power supply can generate electric currents above 100mA which can be dangerous to humans. Care must be taken when using the power supply by ensuring the dials are turned to a minimum (counter clockwise) when connecting or disconnecting any configuration of the apparatus. The red (+) and black (-) terminals are to be used. For safety, turn off the power supply when connecting/disconnecting components. 2.) Start by setting the power supply to the correct value indicated in each part of the lab. 3.) Place the sensor to the desired position of measurement. 4.) With the power supply Off, zero the sensor by clicking the button on the LoggerPro interface. 5.) Energize the system by turning the power supply on and waiting a few seconds while the system comes to a steady state. Note that prolonged energiziation can and will change the set electrical current value due to self-heating of the wires. To avoid this, do not leave the system energized (power supply on) for longer than neccessary. 6.) Take a measurement by clicking the button on the LoggerPro interface. Data will collect over 5 seconds. 7.) In the LoggerPro interface, view the magnetic field vs. time graph (top) and select an adequate region on the graph that best represents the measurement. If the whole measurement is adequate, you can skip highlighting a region and use the full measured data set. 8.) With the graph selected, select Analyze Statistics to obtain a time averaged magnetic field and corresponding standard deviation of the selected (or full) region. Page 3 of 8

4 Ryerson University - PCS 130 Part I - Magnetic Field of a Long Straight Conductor This section of the lab will be performed by the TA as a demonstration of how to use the equipment including using certain features of the LoggerPro software. Procedure I 1.) Locate and open the LoggerPro file PCS130 Magnetic Fields.cmbl and ensure that Page 1 Straight Wire Current is selected on the navigation bar at the top. 2.) Connect the long straight conductor to the power supply. 3.) Set the power supply so that the current is 7A when it is on. 4.) Set the sensor to the 0.3mT range and in the bent position. 5.) Starting from the 2cm away from the conductor, measure and record the results with the distance r from conductor in the table adjacent to the graphs. 6.) Repeat taking measurements along the r direction in increments of 1cm for a total of 5 points. 7.) Determine the relationship between the magnetic field of a straight conductor, B wire, and inverse distance, 1/r by selecting the magnetic field vs. distance graph (bottom) and applying a linear fit. Analysis I 1.) Determine the measured permeability of air including its uncertainty. Check your measurement accuracy by comparing your result to the permeability of vacuum and calculating the percent error. Comment on the precision of your measurement, whether or not the result is within an acceptable range (ie within the uncertainty) and discuss sources of error. 2.) Briefly comment on the y-intercept value and what you expect it to be. 3.) Would translating the sensor in a direction parallel to the conductor change the magnetic field? Comment on what you would expect. Page 4 of 8

5 Ryerson University - PCS 130 Part II - Magnetic Field at the Centre of a Single Coil Procedure II 1.) Navigate to Page 2 Single Coil Current on the LoggerPro interface. 2.) With the power supply off, connect a single coil using the white plugs to the power supply. 3.) Set the sensor to the 6.4mT range and in the unbent configuration. 4.) To the best of your ability, position the ruler (using the retort stand) such that it passes through the coil just below the central axis. 5.) Hold the probe ontop the ruler so that it is positioned at the centre of the coil. Keep the sensor in the same place for the following measurements. 6.) Collect magnetic field values for electric current values starting with 2A in decreasing increments of 0.4A. Record the results and electric current each time in the table adjacent to the graphs 7.) Add uncertainty for the electric current I by double clicking the column title, selecting Options, and clicking the Fixed Value, Error Constant +/- radio buttons under Error Bar Calculations. 8.) Determine the relationship between the magnetic field of a single coil, B coil, and electric current, I by selecting the magnetic field vs. electric current graph (bottom) and selecting Analyze Linear Fit. Analysis II 1.) Compare the slope of the graph to the expected value 2.) Compare the result with the expected value. Check your measurement accuracy by comparing your result to the permeability of vacuum and calculating the percent error. Commet on the precision of your measurement, whether or not the result is within an acceptable range (ie within the uncertainty) and discuss sources of error. Page 5 of 8

6 Ryerson University - PCS 130 Part III - Magnetic Field along the Central Axis of a Single Coil Procedure III 1.) Navigate to Page 3 Single Coil Distance on the LoggerPro interface. 2.) With a single coil still connected, set the power supply so that the current is 2A when it is on. 3.) Take measurements of the magnetic field along the central axis in increments of 3cm starting from a position along the ruler with a magnetic field strength of approximately 1th of the maximum value determined in Part II (for 2A). Record your results and the 5 position in the adjacent table. 4.) Continue taking measurements until you reach approximately the same initial value on the opposite side of the coil. Remember, the more data points obtained, the greater the fit will likely be when fitting the data. 5.) Add uncertainty for displacement in a similar manner to Part II. 6.) With the bottom graph (Magnetic Field in a Coil vs. Distance) selected, apply a curve fit by selecting Analyze Curve Fit. Scroll down the list of General Equations and choose Single Coil from the list of functions. Press the Try Fit button to see observe the fit and click OK to finish fitting the data. Analysis III 1.) Comment on what each fitting coefficient means and compare the relevant ones to the expected values. Calculate a percent error for these coefficients and discuss sources of error. 2.) Determine the Full Width Half Maximum (FWHM) using the single coil equation and fitting parameters. The FWHM is the measure of how broad (wide) a signal is before it loses half of its intensity. In this case, its a measure of the distance spanned before the magnetic field magnitude is halved. Page 6 of 8

7 Part IV - Magnetic Field of a Two Coils Procedure IV Ryerson University - PCS ) Navigate to Page 4 Helmholtz Coil Distance on the LoggerPro interface. 2.) Position two coils such that they are within the rectangular outlines on the base. Ensure that the coils are parallel with one another. 3.) Connect two coils in series using the white plugs such that the magnetic fields of each coil point in the same direction. Verify that the coils are correctly configured to one another by right hand rule for coils. 4.) Set the power supply to output 1A of current. 5.) In a similar fashion to Part III, measure the magnetic field across the Helmholtz coil in steps of 3cm all the way through the two coil system. 6.) Similar to Part III, apply a curve fit to the bottom graph (Magnetic Field in a Helmholtz Coil vs. Distance) but selecting Helmholtz Coil in the list of General Equations. Analysis IV 1.) Compare the maximum value measured to the calculated maximum value obtained by using Eqn ) Comment on what each fitting coefficient means and compare the relevant ones to the expected values. Calculate the percent error for these coefficients and discuss sources of error. 3.) Calculate the FWHM for the Helmholtz coil and compare it to the FWHM of the single coil in Part III. 4.) Discuss what would happen if the magnetic coils were moved closer together or further apart. 5.) Describe the magnetic field you would observe if the coils were connected with magnetic fields pointing in opposite directions. Page 7 of 8

8 Last Few Steps Ryerson University - PCS ) Be sure to record relevant values such as the radius of the loops, number of turns, and distance between the two coils (for Part IV) as they will be needed to complete the lab. 2.) Save your data (tables) by going to File Export As and selecting a preferred format (.cvs is recommended). The file will contain all the tables made during this experiment. 3.) Save your LoggerPro file with an easily identifiable name such as PCS130 Magnetic Fields YOURNAME.cmbl. 4.) You can view your LoggerPro file at a later time using the software. You can download a copy from: 5.) Submit the.cmbl file to your group submission folder on D2L. 6.) Once this is complete and are certain that the data is saved, restart the computer when all experiments are completed. 7.) Lastly, tidy up your work station and turn off the power supply (after setting values to 0) for your fellow students in other sections. Page 8 of 8

Sound Waves and Beats

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