Pcs130 Magnetic Fields 5 o1k3h

Ryerson University - PCS 130

Magnetic Fields Introduction In this experiment, we study magnetic fields of several electrical configurations and their dependence on variables such as position, 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: ~ × rˆ) µ0 I(dL (1) 4π r2 Measuring magnetic fields can be achieved by using 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 power supply used in this lab can generate electric currents above 100 mA 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. ~ = dB

Apparatus • Power supply

• Electric cables

• Ruler

• Vernier LabPro + Logger Pro software

• Retort stand + ruler holder • Magnetic field coil [200 turns, 10.5 cm radius] (2) + base

• Vernier magnetic field sensor • Clamps • Elastic Bands

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Ryerson University - PCS 130

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: ~ wire (I, r) = µ0 I 1 θˆ B 2π r

(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: R2 ~ coil (I, z) = µ0 N I B zˆ 2 (z 2 + R2 )3/2

(3)

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: NI ~ helmholtz,max (I) = 8µ √0 zˆ B 125

(4)

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Ryerson University - PCS 130

Part I -Analysis of a Magnetic Field of a Long Straight Conductor This part of the lab will help familiarize oneself with the LoggerPro interface and how to extract useful information from plots. 1.) Locate (on the website) 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.) The table on the left contains data taken of the magnetic field of a long straight conductor at the specified distances away from the conductor. The data is taken in the direction of the magnetic field. 3.) Based on the data presented, was the probe facing magnetic north or magnetic south? 4.) Calculate the inverse distance

1 r

using the provided distances from the conductor r.

5.) Apply a linear fit to the graph by selecting Analyze → Linear Fit. 6.) What does the slope of the linear fitted line represent? How does it relate to Eqn. (2); the equation for a long straight conductor? 7.) Using the results, calculate the permeability of vacuum. Are we expecting to measure the permeability of vacuum? What might cause the result to be different from expected? Assume that the experiment was done in a reasonable environment. 8.) Briefly comment on the y-intercept value and what you expect it to be.

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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. 2.) With the power supply off, connect a single coil using the white plugs to the power supply. 3.) Using Right Hand Rule, determine the direction of the magnetic field generated by the coil. Hint: Observe the wiring diagram near the plugs of the coil. 4.) Ensure that the sensor is set to the 6.4mT range via the toggle switch on the body of the probe. When changing ranges, LoggerPro will prompt an indication that the sensor settings have changed; Click Use Sensor Settings to continue. 5.) To the best of your ability, position the probe (which should be attached to the end of the meter stick) so it is perpendicular to, and along the central axis of the coil. 6.) Position the probe so that is in the same plane as the coil. The probe will stay in this position for this part of the lab. 7.) With the power supply Off, remove the effect of any local magnetic sources that may contribute to the measurement by zeroing the sensor. This is done by clicking the ∅ button on the LoggerPro interface. 8.) Turn on the power supply and set the electric current to 0.4A. Note that prolonged energization can and will affect the electric current value due to self-heating of the wires. To avoid this, do not leave the system energized (power supply on) for longer than necessary. 9.) Take a measurement by clicking the I button on the LoggerPro interface. Data will collect over 3 seconds. 10.) In the LoggerPro interface, the top graph displays the magnetic field vs. time. 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. 11.) With the top graph selected, select Analyze → Statistics to obtain a time averaged magnetic field and corresponding standard deviation of the selected (or full) region. 12.) Record the magnetic field results, standard deviation, as well as the electrical current in the table adjacent to the graphs. 13.) Measure the magnetic field every 0.4 A up to a maximum of 2 A. Record the results and electric current, I, each time in the table adjacent to the graphs. Page 4 of 8

Ryerson University - PCS 130 14.) Determine the relationship between the magnetic field of a single coil, Bcoil , and electric current, I by selecting the magnetic field vs. electric current graph (bottom) and selecting Analyze → Linear Fit. Analysis II 1.) Compare the relationship between I and BCoil with the expected value. 2.) Check your measurement accuracy by comparing your result to the permeability of vacuum and calculating the percent error. 3.) Comment on the precision of your measurement, whether or not the result is within an acceptable range (ie within the uncertainty) and discuss possible sources of error. 4.) What would you expect to happen if you performed the same experiment slightly out of the plane of the coil?

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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 the power supply Off, zero the magnetic field sensor while it is still at the centre of the coil. 3.) With a single coil still connected, set the power supply so that the current is 2 A when it is on. 4.) With the power supply on, move the probe and ruler (as a single entity) away from the coil until it reaches approximately 20% of the maximum magnetic field strength. Aside: Where would you expect the maximum magnetic strength value to be? 5.) Starting from the position determined above, take and record measurements of the magnetic field strength and position along the central axis in increments of 3 cm until you reach approximtely the same magnetic field strength on the opposite side of the coil. , in general, the more data points obtained, the greater the fit will likely be when fitting the data. 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 (spatially) before the magnetic field magnitude is halved.

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Ryerson University - PCS 130

Part IV - Magnetic Field of a Two Coils Procedure IV 1.) 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. If the coils are loose from the base, tighten them using the thumb screw underneath. 3.) Connect two coils in series using the white plugs such that the magnetic fields of each coil point in the same direction. that the coils are correctly configured to one another by right hand rule for coils. 4.) Set the power supply to output 1 A of current. 5.) In a similar fashion to Part III, measure the magnetic field across the Helmholtz coil in steps of 3 cm 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. 3. 2.) 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.

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Ryerson University - PCS 130

Last Few Steps 1.) 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 LoggerPro file with an easily identifiable name such as PCS130 Magnetic Fields YOURNAME.cmbl. 3.) You can view your LoggerPro file at a later time using the software. You can a copy here. Note: You can only these while on a Ryerson network. 4.) Alternatively, you can save your data (tables) by going to File → Export As and selecting a preferred format (.csv is recommended). The file will contain all the tables made during this experiment. 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.

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