Physics Lab Report 1

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1 Physics Lab Report 1 Magnetic Fields Course: PCS130 Section: 09 Teaching Assistant: Ary Sakafish Instructor: Tetyana Antimirova Lab Partners: Ahmed Abdul Lathif, Rafia Kibria January 17, 2023

2 Table of Contents Introduction ……………………………………………… 3 Theory …………………………………………………… 4 Procedure ……………………………………………… .... 5-7 Results and Calculations …………………………………. 8-12 Discussion and Conclusion ………………………………. 13-14 References ………………………………………………… 15

3 Introduction A Magnetic field is a vector field that describes the influence magnetism has on moving electric charges. Electric charges that are in motion create magnetic fields due to the electrical current flow in a conductor. The strength and direction of the magnetic field depend on how much current is flowing and the direction of its flow. The magnetic fields created can be calculated by the Biouts-Savart Law known as = . In this experiment, we studied ?? µ0 4π ?(?𝐿 ×𝑟 ) 𝑟 2 the magnetic fields generated by an electric current passing through the conductor which in this case was a coil of a conductive metal. In discovery, we found the shape and size of the coils to impact the magnetic field that was generated, which showed us the relationship of the magnetic field and the electrical current. We also conducted this experiment to understand the relations the distance and the strength of the magnetic field at those specific points had by using two coils of wires and a 10.5 cm radius. To complete this experiment with success, three parts such as finding the magnetic field in the center of the coil, the magnetic field along the central axis of a single coil, and the magnetic field along the central axis of a helmholtz coil were to be conducted.

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4 Theory The Biot-Savart Law is used to calculate magnetic fields created by the electric current: ?? → = µ 0 4π ?(?𝐿 → 𝑥 𝑟 ^ ) 𝑟 2 According to the equation, a very small magnetic current is generated by a current-carrying ?? → ( ) conductor of a small cross-section at a point distance away from it. The constant is ? ?𝐿 → 𝑟 µ 0 the permeability of vacuum. We can use the Biot-Savart Law to find the magnetic field due to a current-carrying loop of wire. Consider a circular loop that has a radius R, carries a current I, and lies in the xz -plane:

5 Procedure Part (I) Magnetic Field at the center of a single coil 1. Connect the Magnetic Field Sensor to CH-1 of the LabPro computer interface and open logger pro. Don’t forget to calibrate the field sensor. 2. With the power supply off , connect a single coil using the white plugs to the power supply and set it to the 6.4mT range. 3. Position the probe so that it is in the same plane as the coil. The probe will stay in this position for this part of the lab. 4. Turn on the power supply and set the electric current to 0.4A. Do not leave the power supply on for too long of a period to avoid your electric current value being affected. 5. In the LoggerPro interface, the top graph displays the magnetic field vs. time . Select Analyze → For statistics. 6. Record the magnetic field results, as well as the electrical current in the excel sheet and assign an uncertainty to your magnetic field value based on the fluctuations observed. 7. Measure the magnetic field every 0.2 A up to a maximum of 2 A . Record the results and electric current, I, each time. 8. Determine the relationship between the magnetic field of a single coil, Bcoil, and electric current, I by plotting your results and applying a linear fit.

6 Part (II) Magnetic Field along the central axis of a single coil 1. With the power supply Off , zero the magnetic field sensor while it is still at the center of the coil and with a single coil still connected, set the power supply so that the current is 2 A when it is on. 2. With the power supply on, move the probe and ruler away from the coil until it reaches approximately 20% of the maximum magnetic field strength. Switch to the second sheet (B vs. z). 3. Record measurements of the magnetic field strength and position along the central axis in increments of 2 cm until you reach the same magnetic field strength on the opposite side of the coil. 4. You should record an uncertainty value for the magnetic field sensor’s position and test a fit by creating a set of data from the theoretical equation: ? ?𝑜𝑖?(?, 𝑧) = ? ? 2 ((𝑍+?) 2 +?) 2 ) 3/2 𝑧

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7 Part (III) Magnetic Field along the central axis of a Helmholtz coil 1. With the power supply Off, zero the magnetic field sensor and position two coils within the rectangular outlines on the base. Coils must be parallel with one another. 2. Connect two coils in series using the white plugs such that the magnetic fields of each coils point in the same direction. Verify that the coils are correctly configured to one another by right hand rule for coils. 3. Set the power supply to output 1 A of current. 4. Create a new sheet in your Excel file called Helmholtz Coil. 5. Measure and record the magnetic field across the Helmholtz coil in steps of 2 cm all the way through the two coil system. 6. Write the equation for the magnetic field of a two coil system with the coils R distance apart. 7. Fit your Helmholtz data using the equation you found above and 2-3 fitting coefficients. 8. Record relevant values such as the radius of the loops, number of turns, and distance between the two coils 9. Lastly, tidy up your workstation and turn off the power supply (after setting values to 0) for your fellow students in other sections.

8 Results and Calculations Part 1: Magnetic Field at the Center of a Coil For magnetic field at the center of a coil, we can use the following formula: ? = µ 0 𝑁? 2𝑅 Where ● B = magnetic field in Tesla (T) ● I = Current in Amperes (A) ● R = Radius of coil in meters (m) = 10.5 cm = 0.105 m ● N = Number of turns in coil = 200 For the first case, I = 0.4 A B = -0.01227 mT = -0.00001227 T Therefore, µ 0 = 2?𝑅 𝑁? = 2 • −0.00001227 • 0.105 200 • 0.4 = − 3. 22 • 10 −8 ?/? We can do the above calculation for all the other readings, and hence the experimental value of can be found. These values can then be used to determine the relationship between the µ 0 magnetic field B and current I. Below are the values of the magnetic field we measured for the different values of current: Current (I) [A] B_z (mT) Experimental value of µ 0 (H/m) 0.4 -0.01227 -3.22 10 -8 • 0.6 -0.001857 -3.25 10 -9 • 0.8 -0.005424 -7.12 10 -9 • 1 -0.005498 -5.77 10 -9 • 1.2 -0.009436 -8.26 10 -9 • 1.4 -0.007141 -5.36 10 -9 • 1.6 -0.006081 -3.99 10 -9 • 1.8 -0.01936 -1.13 10 -8 • 2 -0.006871 -3.61 10 -9 • Below is the graph obtained for the measurements attained:

9 Part 2: Magnetic Field Along the Central Axis of a Single Coil Since , where: ? = µ 0 𝑁?𝑅 2 2 (𝑅 2 +𝑧 2 ) 3 ● I = current in the coil = 2A ● N = Number of coils = 200 ● R = Radius of coil = 10.5cm = .105m ● B = Magnetic field associated with the coil (mT) ● z = distance from the coil along the central axis For the first case: I = 2A B = 0.4636 ± 0.04272 µ 0 = 2? (𝑅 2 +𝑧 2 ) 3 𝑁?𝑅 2 = 2 • 0.4636 • (0.105 2 +0 2 ) 3 200 • 2 • 0.105 2 = 2. 43 • 10 −2 ?/? Uncertainty calculation: ∆µ 0 µ 0 = ∆? ? ⇒ ∆µ 0 = µ 0 ∆? ? = (2. 43 • 10 −2 ) • 0.04272 0.4636 = 2. 24 • 10 −3 ?/?

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10 We can do the above calculation for all the other readings, and hence the experimental value of can be found. These values can then be used to determine the relationship between the µ 0 magnetic field B and the distance from the coil along the central axis. Below are the values of the magnetic field we measured for the different values of distance from the coil along the central axis: z position (cm) B_z (mT) Experimental Value of µ 0 (H/m) 0 0.4636 ± 0.04272 2.43 • 10 -2 ± 2.24 • 10 -3 2 0.6115 ± 0.05193 3.39 • 10 -2 ± 2.88 • 10 -3 4 0.8029 ± 0.0552 5.17 • 10 -2 ± 3.55 • 10 -3 6 1.076 ± 0.05636 8.63 • 10 -2 ± 4.52 • 10 -3 8 1.473 ± 0.06006 1.54 • 10 -1 ± 6.27 • 10 -3 10 1.874 ± 0.06282 2.59 • 10 -1 ± 8.69 • 10 -3 12 2.226 ± 0.06261 4.09 • 10 -1 ± 1.15 • 10 -2 14 2.5 ± 0.06898 6.08 • 10 -1 ± 1.68 • 10 -2 16 2.48 ± 0.05868 7.88 • 10 -1 ± 1.87 • 10 -2 18 2.167 ± 0.06146 8.89 • 10 -1 ± 2.52 • 10 -2 20 1.682 ± 0.05851 8.79 • 10 -1 ± 3.06 • 10 -2 22 1.251 ± 1.318 8.22 • 10 -1 ± 8.66 • 10 -2 24 0.9423 ± 0.05921 7.68 • 10 -1 ± 4.83 • 10 -2 26 0.7715 ± 0.04806 7.71 • 10 -1 ± 4.81 • 10 -2 28 0.6608 ± 0.0524 8.01 • 10 -1 ± 6.35 • 10 -2 30 0.5348 ± 0.0474 7.79 • 10 -1 ± 6.90 • 10 -1 Below is the graph obtained for the measurements attained:

11 Part 3: Magnetic Field Along the Central Axis of a Helmholtz Coil We use the following equation to calculate the magnetic field of a Helmhotlz coil along its central axis: , where: ? = µ 0 𝑁?𝑅 2 2 (𝑅 2 +𝑧 2 ) 3 ● I = current in the coil = 2A ● N = Number of coils = 200 ● R = Radius of coil = 10.5cm = .105m ● B = Magnetic field associated with the coil (mT) ● z = distance from the coil along the central axis For the first case: I = 2A B = 0.3335 ± 0.03949 µ 0 = 2? (𝑅 2 +𝑧 2 ) 3 𝑁?𝑅 2 = 2 • 0.3335 • (0.105 2 +0 2 ) 3 200 • 2 • 0.105 2 = 1. 75 • 10 −2 ?/? Uncertainty calculation: ∆µ 0 µ 0 = ∆? ? ⇒ ∆µ 0 = µ 0 ∆? ? = (1. 75 • 10 −2 ) • 0.03949 0.3335 = 2. 07 • 10 −3 ?/? We can do the above calculation for all the other readings, and hence the experimental value of can be found. These values can then be used to determine the relationship between the µ 0 magnetic field B and the distance from the Helmholtz coil along the central axis. Below are the values of the magnetic field we measured for the different values of distance from the Helmholtz coil along the central axis: z (cm) B (mT) Experimental Value of µ 0 (H/m) 0 0.3335±0.03949 1.75 • 10 -2 ± 2.07 • 10 -3 2 0.4171±0.0417 2.31 • 10 -2 ± 2.31 • 10 -3 4 0.5655±0.05282 3.64 • 10 -2 ± 3.40 • 10 -3 6 0.6778±0.05455 5.44 • 10 -2 ± 4.38 • 10 -3 8 0.9007±0.05619 9.40 • 10 -2 ± 5.86 • 10 -3 10 0.8997±0.05676 1.24 • 10 -1 ± 7.85 • 10 -3 12 1.225±0.05878 2.25 • 10 -1 ± 1.08 • 10 -2 14 1.23±0.06378 2.99 • 10 -1 ± 1.55 • 10 -2

12 16 1.589±0.6353 5.05 • 10 -1 ± 2.02 • 10 -1 18 1.879±0.06647 7.71 • 10 -1 ± 2.73 • 10 -2 20 1.804±0.06143 9.43 • 10 -1 ± 3.21 • 10 -2 22 1.444±0.0587 9.49 • 10 -1 ± 3.86 • 10 -2 24 1.456±0.05978 1.19 ± 4.87 • 10 -2 26 2.047±0.06467 2.05 ± 6.47 • 10 -2 28 1.522±0.05976 1.85 ± 7.25 • 10 -2 30 1.279±0.06226 1.86 ± 9.07 • 10 -2 32 0.9796±0.05484 1.70 ± 9.50 • 10 -2 34 0.7349±0.057257 1.50 ± 1.17 • 10 -1 36 0.5449±0.479 1.30 ± 1.15 38 0.3543±0.03801 9.85 • 10 -1 ± 1.06 • 10 -1 Below is the graph obtained for the measurements attained:

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13 Discussion and Conclusions In regards to our calculations, first we calculated the magnetic field at the center of the coil by using “ to identify the experimental value of , which was -3.22*10^-8 H/m. ? = µ 0 𝑁? 2𝑅 " µ 0 Once this was found, we could apply the formula to the other readings we obtained and come to an understanding of the relationship between Magnetic field B and Current I. The formula was easy to use since all the values of the unknowns were either given or obtained from the experiment so all that was needed to be done was for it to be plugged in. Since we started from 0.4 amps up t0 2 amps, a total of nine trials were conducted not including the mess ups and with all the pieces of data, a ‘ Magnetic field vs Current’ graph was constructed. From the graph, a general conclusion stating that, as the current increases, the magnetic field increases proportionally can be stated. Furthermore, after performing our second part of the experiment, we calculated the magnetic field along the central axis of a single coil by using a similar but different formula known as “ ” to once again obtain the value of . By using the information gained ? = µ 0 𝑁?𝑅 2 2 (𝑅 2 +𝑧 2 ) 3 µ 0 from the experiment and data already given we were able to plug in the values, solve and get 2.43*10^-2 H/m. For this experiment we executed a total of 16 trials in which this formula could be applied and help us determine the relationship between magnetic field B and the distance from the coil along the central axis. With both the values of the magnetic field and its positions established, a Magnetic field vs Position was created and conclusions drawn from it told us that the magnetic field is inversely proportional to the position, so as position increases the magnetic field decreases and as it increases the field decreases causing the graph to look like a bell curve. Finally, the last part of the experiment was done where once again we found the value of using the formula ” which occurred to be 1.75*10^-2 H/m. We implemented µ 0 "? = µ 0 𝑁?𝑅 2 2 (𝑅 2 +𝑧 2 ) 3

14 this calculation on some, but not all thirty trials to help us determine the relationship between the magnetic field B along the central axis of the Helmholtz coil. Once the final graph of this experiment was created, it was in the shape of a quartic function but we can observe how it stays almost the same until the middle. The conclusion drawn from the graph was that the magnetic field along the central axis of the helmholtz coil produces a uniform magnetic field. To conclude, by experimenting with three different motives surrounding magnetic fields, we were able to understand the concepts of how magnetism has an influence on moving charges and how different factors surrounding coils or the strength and direction of magnetic fields influence different magnetic properties.

15 References Kraft, Franz. “Helmholtz coil.” Wikipedia , https://en.wikipedia.org/wiki/Helmholtz_coil. Accessed 27 January 2023. “What are magnetic fields? (article).” Khan Academy , https://www.khanacademy.org/science/physics/magnetic-forces-and-magnetic-fields/ magnetic-field-current-carrying-wire/a/what-are-magnetic-fields. Accessed 27 January 2023.

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