PCS130 Lab Report #1 (1)

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Dec 6, 2023

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Faculty of Science Department of Physics Course Number PCS 130 Course Title Physics II Semester/Year Winter 2023 Instructor Tetyana Antimirova TA Name Sukhraj Virdee Lab/Tutorial Report No. Lab Report #1 Report Title Magnetic Fields Section No. 03 Group No. 59 Submission Date 02/02/2023 Due Date 02/02/2022 Student Name Student ID Signature Hodo Wardheer ****58201 H.W. Amber Bittle ****54827 A.B.

Introduction: The objective of this lab was to investigate the magnetic field of electrical coils and how change in various variables (position and current) can affect it. Current ( I ), describes the amount of energy (volume of electrons) flowing down a potential gradient and is measured in amperes (A) ( Zemaitis et al., 2022). The flow of currents are directed along a particular path and direction using conductive materials like such metals. Charge configuration is a significant factor in determining the magnetic field produced by electrical currents. In this lab, two electrical setups were utilised: a single coil and a double Helmholtz coil. A vernier sensor probe was used to detect the magnetic field at the centre of a single coil, throughout the axis of a single coil, and through the centre axis of two coils. As the experiment was performed variables were altered to determine their effect on the magnetic field. LoggerPro software and Microsoft Excel sheets were used to generate graphs and store data. By manipulating the distance of a probe and intensity of the currents, the magnetic field could be observed and their relationship could be further investigated and understood.

Theory: Magnetic Fields are produced by electric currents, which can be macroscopic currents in wires, or microscopic currents associated with electrons in atomic orbits (Hyperphysics, 2022). The magnetic field is often visualised through magnetic field lines, or lines of force, that leave one end of the magnet, arc through space, and re-enter the magnet at the other end (GOC, 2019). The Helmholtz coil is an electromagnetic apparatus consisting of two identical coils in which they are placed at the same distance from each other as their radius and have a region with a nearly uniform magnetic field (Virtuelle, 2022). Image retrieved from: https://virtuelle-experimente.de/en/b-feld/b-feld/helmholtzspulenpaar.php To properly investigate the magnetic fields created by electric current, the Biot-Savart Law was utilised. The Biot-Savart Law is an equation created by scientists Felix Savart and Jean Baptiste Biot. It describes the magnetic field with relation to its direction, length and proximity of the electric current produced and can be used for long straight conductors, or single/Helmholz coil. The direction of the magnetic field can be determined by using the right hand rule. In the equation, dB represents the magnetic field, I represents the electric current, r is the distance away from the conductor, n is the number of turns in a coil, and the constant represents the vacuum µ 0 permeability (classical vacuum definition: μ0 = 4π × 10−7 H/m) . Biot-Savart Law: ?? = µ 0 4π 𝐼(?𝐿 × ?) ? 2 The magnetic field near a long, straight conductor is represented as the equation: ? ?𝑖?? (𝐼, ?) = µ 0 𝐼 2π 1 ? θ

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The magnetic field near a circular loop of wire is represented as the equation: ? ???? (𝐼, ?) = µ 0 𝐼 2 𝑅 2 (? 2 +𝑅 2) 3 ? The magnetic field for the Helmholtz coil is represented as the equation: ? = µ 0 𝑁𝐼 𝑅 8 5 5 Since the strength of the magnetic field can be influenced by variables such as position and current, it can be assumed that they are directly proportional to one another (as one increases/decreases it is reflected in the other). For this experiment, a vernier magnetic field sensor (probe) was used to measure the magnetic field at the centre of a single coil and throughout the central axis of a single and double coil. The position and current were manipulated throughout the experiment to observe the effect of position and current intensity on the magnetic field. Utilising the variations of equations provided, the magnetic fields for each of the experiments were determined and the corresponding graphs were generated to visualise the relationship . Finally, the percentage error was also determined by utilising the equation: Percentage error = 𝐹𝑖?𝑎? − 𝐼?𝑖?𝑖𝑎? 𝑖?𝑖?𝑖𝑎? | | × 100.

Procedure: The following paraphernalia required to perform the laboratory experiment were gathered. a) Power Supply b) Retort stand c) Ruler/Holder d) 2 × Magnetic field coil [200 turns, 10.5cm radius] + base e) Electric cables f) Vernier LabPro g) LoggerPro h) Vernier magnetic field sensor i) Clamps j) Elastic Bands Part 1- Magnetic Field at the Centre of a Single Coil: Firstly, the magnetic field sensor was connected to the LabPro interface. The field sensor was then calibrated using the zero function. Before turning on the power supply, a single coil was connected to it using the white plugs provided. The right hand rule was then used to confirm the direction of the coil's magnetic field. The sensor was set to 6.4 mT in range and the probe was positioned at the end of the metre stick, where it was perpendicular to the coils central axis point. While keeping the power supply off, the sensor was zeroed to remove the effect of any magnetic source. The power supply was turned on and set to 0.4 A. The play button was pressed and results were measured and recorded for a 10 second period. Observing the magnetic field vs. time graph generated by LoggerPro, the function analyse—> statistics was selected. The mean, standard deviation, magnetic field, electric current ( I ) and uncertainty results were recorded in the corresponding excel sheet provided. The process was repeated by increasing the magnetic field by 0.2 A till a maximum of 2 A. The relationship between the B coil (magnetic field of a single coil) and I (electric current) were plotted and a linear fit was applied for future analysis and reference. Part 2 - Magnetic Field along the Central Axis of a Single Coil: The power supply was turned off. The magnetic field sensor was zeroed while maintaining its position in the centre of the coil. The power supply was turned on and the current was set to 2 A. The metre stick and probe were moved away from the coil so that it remained at 20% of its maximum magnetic field (as observed in Part 1). Beginning at that position, the LoggerPro software was used to observe the magnetic field and position along the coils central axis. The

ruler and probe were moved 2 cm at a time until the initial magnetic field strength was reached on the opposite side of the coil. The results as well as the uncertainty value for the magnetic sensor were recorded in the excel sheet provided for future analysis. Part 3 - Magnetic Field of a Two Coils: The power supply was turned off and the magnetic field sensor was zeroed. The two coils were positioned parallel to one another and screwed in tightly to avoid movement. The plugs were connected to the other coil in correspondence with the right hand rule. The power supply was turned on and set to output a current of 1 A and an excel sheet named “Helmholtz Coil '' was created to record the results. The metre stick and probe were moved away from the coil so that it remained at 20% of its maximum magnetic field. Beginning at that position, the LoggerPro software was used to observe the magnetic field and position along the two coils central axis. The ruler and probe were moved 2 cm at a time until the initial magnetic field strength was reached on the opposite side of the two coil system. The equation for the magnetic field of the two coils was created and recorded for future use. The results as well as the uncertainty value for the magnetic sensor position were recorded in the excel sheet provided for future analysis. Part 4- Saving Data: The radius of the loops, number of turns, distance between the loops, and the excel files with recorded data were saved and stored for further analysis.

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Results and Calculations: Part 1 - Magnetic Field at Center of a Single Coil Table 1: Magnetic Field at the Center of a Single Coil Increases with Current Current (A) B coil (mT) Standard. Deviation 0.4 0.395 0.04757 0.6 0.675 0.05569 0.8 0.857 0.05911 1 1.145 0.06215 1.2 1.415 0.06528 1.4 1.627 0.06679 1.6 1.923 0.06652 1.8 2.189 0.06279 2 2.405 0.06432 Using a Sensor and Vernier Lab Pro, the magnetic field at the centre of a single coil was recorded at increasing currents. The current was first set to 0.4A and then increased in 0.2A increments until 2.0A was reached. The corresponding magnetic field strength for each current was then used to construct a plot of magnetic field as a function of current to determine the relationship between the two.

Figure 1: The Strength of a Magnetic Field is Relative to the Current Generating it. The figure above illustrates the relationship between the strength of a current passing through a single coil and the magnetic field strength in the centre of the coil. The plot was constructed using the data from Table 1 And illustrated that as the current increases, the strength of the magnetic fields increases as well. Calculations 1. Expected relationship between I and B coil Expected slope: /A ?2−?1 ?2−?1 = 2.405−0.395 2−0.4 = 1. 256?𝑇 Slope from graph: 1.185mT/A % Error = x100 = 5.65% 1.185−1.256 1.256 | | | | 2. Measurement Accuracy Permeability of Vacuum (experimental) B coil = µ𝑁𝐼 2𝑅 0 = µ 2𝑅 𝑁 ???𝑖? 𝐼 0 = = 1.244x10 -6 Tm/A µ 2(0.105?) (200 ?????) 1. 185?𝑇/? Permeability of Vacuum (expected)

0 = 1.257x10 -6 Tm/A µ % error = x 100% ?????𝑖????𝑎? ?𝑎??? − ???????? ?𝑎??? ???????? ?𝑎??? | | | | % error = x 100% 1.244?10 −6 𝑇?/? −1.257?10 −6 𝑇?/? 1.257?10 −6 𝑇?/? | | | | | | % error = 1.03% Part 2 - Magnetic Field along the Central Axis of a Single Coil Table 2. Magnetic Field Strength at Different Positions Along the Central Axis. Position, Z (m) B (I=2.0A) mT 0.63 (z=12cm) 0.48 0.61 0.6649 0.59 0.868 0.57 1.145 0.55 1.57 0.53 2 0.51 (z=0) 2.297 0.49 2.414 0.47 2.278 0.45 1.989 0.43 1.505 0.41 1.153 0.39 0.862 0.37 0.649 0.35 (z=-16cm) 0.479 A sensor was used to detect the strength of the magnetic field along the central axis of a coil with a current of 2.0A. The position 0.51m corresponds to z=0 and is the centre of the coil. The sensor

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was slid in 0.02m increments from one end of the central axis, through the centre of the coil, to the opposite end and magnetic field data was recorded and is displayed in the table above. Figure 2. The Magnetic Field of a Coil is Strongest at the Center (z=0). The figure above shows data points found in Table 2 with the expected values using fitting coefficients laid overtop. Calculations 1. Fitting Coefficients A = µ 0 𝐼𝑁 2 A = (4Π?10 −7 𝑇?/?)(2?)(200 ?????) 2 A = 0.251 B = R ∴ B = 0.105m C should be small enough to be negligible. Percent Errors % Error for A = x 100% ?????𝑖????𝑎? ?𝑎??? − ???????? ?𝑎??? ???????? ?𝑎??? | | | | = x100% 0.241 − 0.251 0.251 = 3.98%

% Error for B = x 100% ?????𝑖????𝑎? ?𝑎??? − ???????? ?𝑎??? ???????? ?𝑎??? | | | | = x100% 0.099−0.105 0.105 = 5.71% 2. Full Width Half Maximum B coil max = 2.43mT Half of B coil max = 2.43/2 = 1.215mT The x-values corresponding to according to the equation with fitting parameters are ??𝑎? 2 0.414m and 0.566m FWHM = x2-x1 = 0.566m - 0.414m = 0.152m Part III - Magnetic Field of Two Coils Table 4. Magnetic Field of Helmholtz Coil Z (m) B_z (I=1A) mT 0.63 0.161 0.61 0.195 0.59 0.262 0.57 0.375 0.55 0.479 0.53 0.625 0.51 0.771 0.49 0.835 0.47 0.883 0.45 (z=0) 0.897 0.43 0.889 0.41 0.878

0.39 0.804 0.37 0.671 0.35 0.56 0.33 0.476 0.31 0.355 0.29 0.247 0.27 0.18 0.25 0.157 To measure the magnetic field strength along the central axis of a helmholtz coil, the current for both of the coils was set to 2.0A and the probe was slid along the axis in 0.02m increments. The magnetic field at each position on the axis was recorded by vernier pro and then used to construct Figure 3. Figure 3: Magnetic Field of Centre of Helmholtz Coil is Constant. The data from Table 3 was used to construct the figure above, which illustrates the magnetic field (in mT) of two coils, both with a current of 1A, as a function of the position of the probe

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along the central axis. The probe measured the magnetic field of the coil in steps of 0.02m all the way through the system, and the magnetic field detected by the probe at each distance along the central axis was recorded in Table 3 and then plotted in the figure above. Calculations 1. Fitting Coefficients Equation for the Magnetic Field at a distance z from the Centre of two coils r distance apart: ? ???𝑎? = µ0𝑁𝐼? 2 2 ( 1 [? 2 +(?+ ? 2 ) 2 ] 3/2 + ( 1 [? 2 +(?− ? 2 ) 2 ] 3/2 ) Where, N = 200 for both coils R = 0.105m for both coils I = 1A for both coils Note: the coils are r distance apart Equation using Fitting Coefficients A,B,C,and D ? ???𝑎? = ? ? ( 1 [? 2 +(?+?) 2 ] 3/2 + ( 1 [? 2 +(?+𝐷) 2 ] 3/2 ) Where, A = µ0𝑁𝐼? 2 2 B = r C = + ? 2 D = - ? 2 2. Full Width Half Maximum B coil max =0.897mT Half of B coil max = 0.897/2 = 0.448mT The x-values corresponding to would be approximately 0.32m and 0.56m ??𝑎? 2 FWHM = x2-x1 = 0.56m-0.32m = 0.24m.

Discussion and Conclusion: Part I The goal of part I was to understand how the magnetic field at the centre of a coil, B coil , and the current, I, are related. I and B coil have a positive linear relationship, with a slope of 1.185mT/A, as indicated in Figure 1. As the current (I) increases, the strength of the magnetic field at the centre of the coil (B coil ) will increase as well. The expected slope (1.256mT/A)and experimental slope (1.185mT/A) of the graph were fairly similar, with a % error of around 5.65%. Using the data from Figure 1, the experimental value for the permeability of vacuum, μ 0 , was found to be 1.244x10 -6 Tm/A, which was used to check the measurement accuracy. Comparing the experimental and expected permeability of vacuum, found that the measurement accuracy was satisfactory, as the % error between the μ 0 experimental and μ 0 expected was only around 1.03%. Based on the % error between the expected and experimental permeability of vacuum values, and the results being within an acceptable range based on their standard deviations, it would be reasonable to say that the measurements are relatively precise. Possible sources of error include the sensor not being exactly on the central axis of the coil, and interference between the sensor and the coil. If the same experiment was conducted, only the sensor was slightly out of the plane of the coil the magnetic field sensor picked up by the sensor would be smaller in magnitude, as the magnetic field is strongest at the centre of the coil. Part II The second experiment was to determine how the magnitude of the magnetic field changes as the position of the senor changes. After using the data from Table 2 to construct Figure 2, a set of fitting coefficients were used to create a fit equation for the data. These coefficients, A, B, and C, were fairly similar to their expected values based on the % errors between the coefficients for the fit equation and the expected values. The % error for A, and B were 3.98% and 5.71% respectively. Due to the fact that C is very small compared to the position, z, it was considered negligible and no % error was calculated. Even though the percent errors are low for the fitting coefficients, A and B, the experimental values (A=0.241 and B=0.099)are still not exactly the same as the expected values (A=0.251 and B=0.105). Potential sources of error include the sensor not being exactly on the central axis of the coil, or the presence of some other object such as a magnet interfering with the sensor.

The full half width maximum (FHWM) was found to be 0.152m, and was determined by finding the x-values that correspond to half of the maximum magnetic field (2.43mT/2 = 1.215mT) based on the equation with the fitting parameters, these x-values were determined to be x1=0.414m and x2=0.566m. The FHWD is equal to the difference between these two values and is therefore 0.152m (x2-x1=FHWM). Part III The final experiment was to determine the magnetic field of Helmholtz coils as a function of position along the central axis. The data is Table 3 was used to construct Figure 3, which shows that the magnetic field is rather constant in between the two coils that are r distance apart. The figure also illustrated that the magnetic field is highest in the centre of the Helmholtz coil and decreases as the position moves further away from z=0. An equation for the magnetic field at a distance z (m) from the centre of the Helmholtz coil can be found in the Results and Calculations section under part III. This equation was then adjusted to fit the data using the following fitting coefficients: A, B, C, and D. The coefficient A is equal to the product of the vacuum permeability constant, number of turns, current, and radius squared all divided by 2 (A= ), B is equal to the radius of the coils (B=0.105m), C is the radius µ 0 𝑁𝐼? 2 2 divided by two (C = + ), and D the negative quotient of the radius divided by two (D = . ? 2 − ? 2 ) Sources of error for the fitting coefficients likely involved the experimental setup, if the coils were not exactly one radius apart or if the sensor was not directly aligned with the central axis of the helmholtz coil. The FWHM of the helmholtz coil was estimated to be around 0.24m, based on the half maximum magnetic field being 0.448mT and the estimated x-coordinates corresponding to that 0.448mT in Figure 3. The FWHM of the helmholtz coil was larger than that of the FWHM of the single coil from part II, which was only 0.152m. The result of moving the coil closer together or farther apart would be an increased or decreased magnetic field strength. Coils that have less distance between then will have a stronger magnetic field than coils with a much larger distance between them. This is supported by the fact that the maximum magnetic field for the single coil is around 2.41mT/A whereas the maximum magnetic field of the helmholtz coil is around 0.987mT/A, though it is worth noting that these two magnetic fields had a different current in the coil producing them, which does make the comparison less reliable. If the experiment were to be conducted in the same way, only the coils had equal (in magnitude) but opposite currents (I 2 = -I 1 ), there would be no magnetic field due to the currents cancelling

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each other out. For example, if the current in coil 1 is 1.0A and the current in coil 2 is -1.0A, the magnetic field at a point equidistant between the two coils would be nonexistent because the currents would cancel eachother out. This series of experiments allowed for the understanding of a few significant takeaways relating to current and magnetic fields. The first part of the experiments proved that the relationship between the current and magnetic field at the centre of a coil is positively related - if the current increases the magnetic field will as well. The relationship between position and magnetic field was also explored in this lab. The magnetic field is strongest at the centre of the coil, and will decrease the further from the centre of the coil the measurement is taken at. Finally, the Helmholtz coil was introduced and explored. It was determined that the magnetic field in the centre of a helmholtz coil is relatively constant, and decreases as the position of the measurement is further away from the area between the two coils, where the field is at a maximum strength.

References: 1. GOC, N. R. C. (2019, March 1). Government of Canada / gouvernement du Canada . Government of Canada, Natural Resources Canada, Canadian Hazards Information Service. Retrieved February 2, 2023, from https://www.geomag.nrcan.gc.ca/mag_fld/default-en.php 2. Hyperphysics.(2022). Magnetic field. Retrieved February 2, 2023, from http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfie.html 3. Virtuelle. (2022.). Magnetic field of two Helmholtz coils. Retrieved February 2, 2023, from https://virtuelle-experimente.de/en/b-feld/b-feld/helmholtzspulenpaar.php 4. Toronto Metropolitan University PCS-130- Magnetic Fields. Retrieved from: https://courses.torontomu.ca/d2l/le/content/707020/viewContent/4970943/View 5. Zemaitis MR, Foris LA, Lopez RA, et al. Electrical Injuries. [Updated 2022 Sep 9]. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2022 Jan-. Available from: https://www.ncbi.nlm.nih.gov/books/NBK448087/?report=classic

PCS130 Lab Report #1 (1)

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