Lab 4

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Physics

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Feb 20, 2024

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Lab 4: Magnetic Field of a Solenoid Rex Kilpatrick Group Partners: Cooper Wooten John Reina Performed October 4 th , 2023

Objective The goal of this experiment is to observe the patterns of magnetic fields that are created by solenoids. This experiment also aims to quantify the strength of magnetic fields at the center of a solenoid and to experimentally determine the permeability of free space. Theory Solenoids are coiled conductors that create controlled magnetic fields. Magnetic fields are characterized by lines that leave the north end of a solenoid, travel to the south end of the solenoid on a path outside of the solenoid, and then loop back through the solenoid to emerge from the north end of the solenoid once again. Most magnetic field lines emerging from a solenoid form an elliptical loop shape that extends outside the solenoid. However, a straight, non-looping line forms along the central axis of the solenoid which the coils of the solenoid wrap around. Unlike the other magnetic field lines which travel north, then south, and then back north, this line only travels from the southern to northern end of the solenoid. The magnetic field generated around a solenoid has a strength that can be modeled by the equation [1]: B = N L μ 0 i Eq. 1 In this equation, B is the strength of the magnetic field in teslas (T), N is the number of turns or coils within the length of the solenoid, L is the length of the solenoid in meters (m), i is the current flowing through the solenoid in amperes (A), and μ 0 is the permeability of free space constant, which is equivalent to 4π ∙ 10 -7 Tm/A. Materials/Equipment The materials and equipment used for this lab were a Vernier LabQuest Stream interface with LoggerPro software, a laptop, an adjustable DC power supply, a 400-turn solenoid, a piece of paper, a pencil, a Vernier magnetic field sensor, a small compass, banana cables, alligator cables, and a slinky. Procedure (Procedure A) For the first part of this experiment, the magnetic field that forms from a current- carrying solenoid was visually depicted. In order to accomplish this, first a circuit with a solenoid was set up. This was done by wiring the 400-turn solenoid in series with the DC power supply. The current output of the power supply was then set to 1.5 A. After setting up the circuit, a small compass was used to depict the lines formed by the solenoid. This was done by first

determining which end of the solenoid caused the compass to orient in a manner where north pointed away from the solenoid. After determining which end of the solenoid was the north end, the compass was placed at the north end of the solenoid, and a small dot was drawn on the paper at the point where the north arrow of the compass pointed. The compass was then moved so that the south end of the compass aligned with the dot, and a new dot was drawn where the north end pointed. This process was repeated until a full loop to the south end of the solenoid was formed, and this process was repeated for multiple field lines until the general form of the magnetic field could be seen. (Procedure B) For Procedure B, the circuit was reconfigured so that a slinky would be used for the solenoid instead of the previously used 400-turn solenoid. This was done by first stretching and taping the slinky to the table so that there was about 1 cm of space between the coils of the slinky. Next, the slinky was wired in series with the power supply using alligator clips. With the slinky wired in, the length of the stretched slinky in meters and the number of turns within that length were measured and recorded. After setting up the circuit and collecting solenoid specifications, the magnetic field sensor was set up halfway along the slinky and with its probe resting on the central axis of the slinky. After setting up the circuit and the probe, the probe’s setting was set to “high sensitivity x100,” and it was connected to the LabQuest. The LabQuest was then connected to the laptop, and it was verified that LoggerPro was collecting data from the magnetic field sensor. Next, the power supply was turned on, and the current was set to 0.5 A. Once the current was set to 0.5 A, the power supply was turned off, and the magnetic probe was zeroed in LoggerPro. Then, data collection in LoggerPro was started while the power supply was still off. After being off for a few seconds, the power supply was turned on, and the magnetic field data was recorded for a few seconds. After being on for a few seconds, the power supply was turned off again, and data collection continued a few more seconds before being stopped. After collecting this data set, analytics tools within LoggerPro were used to find the average B- field, and this was compared with the theoretical B-field. (Procedure C) Procedure C was conducted using the same set up and process as Procedure B. However, Procedure C collected the magnetic field readings at 0.2 A, 0.4 A, 0.6 A, 0.8 A, and 1.0 A. So, the magnetic field produced by each current was measured via the same process outlined in Procedure B as was used for measuring the field generated by the 0.5 A current.

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Data (Figure 1) In this image, a few magnetic field lines generated by a 1.5 A current flowing through a 400-turn solenoid are depicted. Note that the arrowheads point in the direction of north as is shown on the compass. Therefore, the northern end of the solenoid is the end closer to the bottom of the image. (Graph 1) The graph above shows the B-field data collected for the magnetic field as generated in Procedure B.

(Graph 2) The graph above shows the B-field data collected for the magnetic fields generated in Procedure C. Note that the line generated by 0.2 A is at the bottom, by 0.4 A is the first above the 0.2 A line, by 0.6 A is the first above the 0.4 A line, by 0.8 A is the first above the 0.6 A line, and by 1.0 A is the top line. 10 20 30 40 50 60 70 80 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 f(x) = 0 x + 0 R² = 1 B-field vs (N/L)i (N/L)i (A/m) B-field (mT) (Graph 3) The graph above shows the relationship between the B-field in mT and the product of the number of turns in the slinky N , the inverse of the length of the slinky L , and the current flowing through the slinky i in A/m. Please note that the number of turns was 86 and the length of the slinky was 1.183 m.

Analysis (Procedure A) The magnetic field lines as depicted in Figure 1 form as the theory section predicted them to do so. Therefore, the theory that magnetic field lines form loops with north- south polarities is supported. (Procedure B) As can be seen in Graph 1, the experimentally determined B-field of the slinky solenoid is 0.04821 mT. Knowing the length of the slinky L (1.183 m), the number of turns N (86 turns), and the current running through the slinky i (1.5 A), the theoretical B-field of the slinky can be determined using Eq. 1; this value is found to be approximately 0.04568 mT. The experimental B-field errs from the theoretical B-field by approximately 5.5%. The small percent error indicates that the measurement for Procedure B is fairly accurate. (Table 1) The table below shows the data collected for Procedure C. Please note that like in Procedure B, the length of the slinky was 1.183 m, and the number of turns was 86. Run i (A) Avg B-field (mT) σ B (mT) µ 0 = B/(Ni/L) (Tm/A) (N/L)i 1 0.2 0.01922 0.001344 1.32193E-06 14.5393 1 2 0.4 0.03786 0.001583 1.30199E-06 29.0786 1 3 0.6 0.05627 0.001276 1.29007E-06 43.6179 2 4 0.8 0.07547 0.001528 1.29769E-06 58.1572 3 5 1.0 0.09401 0.001486 1.29318E-06 72.6965 3 (Procedure C) As can be seen in Graph 3, the experimentally determined permeability of free space, which is the slope of the trendline, is approximately 0.0013 mTm/A, which is equivalent to 0.0000013 Tm/A. This experimentally determined value errs from the theoretical value by 3.3%. It is also worth noting that the standard deviation of the slope as calculated using the method outlined in Appendix C-II [1] is approximately 4.5 ∙ 10 -9 Tm/A. Since the standard deviation is such a low value, it indicates that the collection of the data was very precise. However, the accuracy of the experimental value is not very high, for it is more than 7 standard deviations away from the theoretical value. This indicates that some systematic error is occurring within the experiment. Conclusion In conclusion, the objective of this lab has been met. Through Procedure A, the magnetic field generated by a current-carrying solenoid was visualized. Through Procedure B, the strength of the center of a magnetic field was successfully measured and analyzed. Through Procedure C,

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precise data points were able to be used to calculate an experimental value for the permeability of free space. As modeled in Procedure A, the magnetic field lines around a current-carrying solenoid form loops that progress from north to south to north, except for a central magnetic field line which only progresses from south to north. In Procedure B, the strength of the center of the magnetic field was found experimentally to be 0.04821 mT, which errs from the theoretical strength value of 0.04568 mT by 5.5%. A few sources of error could have caused the difference in the values. One possible source could be using a meterstick to measure the active length of the slinky, for a meterstick can only be so precise in its measurement. A second source of error could be from not having the current at an exact value. While the current displayed on the digital readout of the power supply as 0.50 A, since the current is set by a continuous knob, the actual current output could have been different by less than a one-hundredth of an amp. In Procedure C, the experimental permeability of free space was found to be approximately 1.3 ∙ 10 -6 Tm/A, which erred from the theoretical permeability of free space by approximately 3.3%. The standard deviation of the slope was determined to be approximately 4.5 ∙ 10 -9 Tm/A. Such a low value is indicative of precise data. However, since the experimental value for the permeability of free space is over seven standard deviations away from the theoretical value and is not within one or two, the accuracy of the experimental permeability of free space value is poor. This indicates that some systematic error is occurring during data collection. One possible source of that error could be from interactions of the magnetic field with metal in the table. Another possible source of error could be from deformations within the structure of the slinky that results in imperfect coils. Other sources of error could be the ones listed above that could have occurred with Procedure B. References [1] Physics 262 . Fifth ed., Van-Griner Learning, 2024.

Lab 4

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