PCS130 Lab 1- Magnetic Fields

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Kaur, 1 Investigating the Effect of Electric Currents on Magnetic Fields Simran Kaur 501029094 Ryerson University Dr. V. Toronov February 4th, 2021 PCS130-212 Gabrielle Lee

Kaur, 2 Introduction Magnetic fields can be explained in a variety of ways, but one of the most important ways being is that magnetic fields are invisible fields that exert a certain force on electric currents, moving charges, and any materials that are affected by magnetism. (Williams, 2016). Magnetic fields are always present any time that magnetism comes into play, however the strength of the field can become limited when certain factors are applied. The magnetic field is said to be stronger when a higher electric current is applied and said to be weaker when the distance between coils is increased. The purpose of this experiment is to delve into and explain how electric currents and the distance between coils can have an impact on the magnetic field. This experiment will aim to support these statements and show how the strength of the field varies with different factors. Theory In order to measure the strength of the magnetic field, this experiment will be largely based upon the equation derived by Jean Baptiste Biot and Felix Savart, known as the Biot-Savart Law: B = NμI 2 R . For this particular experiment, the magnetic field of a near a coil will be measured, which is given by a combination of the Biot-Savart Law as well as by taking a similar integral. This relationship is represented by the following equation: z 2 + R 2 ¿ 3 2 ¿ B ( coil )= N μI 2 R 2 ¿ . N represents the total number of turns the coil makes, μ 0 represents the permeability constant being x 10 4 4 -7 Tm/A, I representing the electric current in amperes, R represents the radius or distance in metres, and lastly with z being the position of the plane. This relationship for the magnetic field of a coil will be used for the entirety of this experiment. By examining and analyzing the equation, it can be seen that if the “z” value was to increase, the value of

Kaur, 3 the magnetic field, B, would decrease. This is also the case with the value of R. However, by taking a look at the values of N or I, it can be seen that by increasing them, one would in turn, achieve an increase in the strength of the magnetic field. Procedure Part 1: Magnetic Field at the Centre of a Single Coil 1. The computer application “Logger Pro” was installed and the “Magnetic Fields” file was downloaded. The “Magnetic Field Simulation” was opened. 2. Using the simulation, the values for the radius and number of turns were recorded, represented by the variables R and N , respectively. 3. The electric current value, denoted by I 1 , was set to zero. This change was made by using the Coil 1 slider. 4. The electric current was increased by 0.5 intervals until eventually reaching 5 A. The magnetic field value was recorded at each interval, which was found in the z-direction of the magnetic field coordinates, measured in milliteslas(mT). The values were all recorded in the Page 1 Single Coil Current Logger Pro table. 5. The data was automatically graphed by the application. The slope was determined by selecting the Linear Fit option, found underneath the Analyze heading. Part 2: Magnetic Field along the Central Axis of a Single Coil 1. Using the simulation, the electric current value was set to 3 A. 2. The z position slider, the z value was set to -2.00 m. This was increased by 0.2 increments until reaching 2.00 m. The magnetic field value was recorded at each increment and recorded in the Page 2 Single Coil Distance Logger Pro table. 3. Using the simulation, the electric current value was set to 4 A. 4. Repeat step 2

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Kaur, 4 5. Using the simulation, the electric current value was set to 5 A. 6. Repeat step 2 7. The data was automatically graphed. The curve of best fit for each of the three resulting curves was found by selecting the Curve Fit option, followed by the General Equation and Single Coil options, found underneath the Analyze heading. Results and Observations Part 1: Magnetic Field at the Centre of a Single Coil Figure 1: The linear relationship between the magnetic field for a single coil and the electric current applied. These are measured in the units milliteslas(mT) and amperes(A), respectively. The radius of the coil is measured to be 0.74 m, while undergoing 2867 turns. Referring to the left side of Figure 1, the values of the current and the magnetic field are seen to have a directly proportional relationship, meaning that as one value increases, so will the other. This is shown through the graph on the right side, which displays a linear trend. Using this information, as well as the Right Hand Rule, it is possible to determine the direction of current flow in the coil, which is the clockwise direction. This can be seen as pointing one’s right thumb straight outwards, would lead to their fingers curling in a clockwise direction. This is also true as positive charges have a velocity in the positive z-direction, which is clockwise. Using the equation derived by the application, the experimental slope is observed to be

Kaur, 5 0.002445 T/A. This can be compared to the theoretical slope calculated using the Biot-Savart Law, which is shown below. The percent error can also be found to observe the accuracy. R = 0.74 m N = 2867 turns μ 0 = 4π x 10 -7 Tm/A B coil /I = [(μ 0 N)/2 ] x [(R 2 /(R 2 + Z 2 ) 3/2 ] =[(4π x 10 -7 Tm/A)(2867 turns)/2] x [(0.74 2 m / (0.74 2 m + 0 2 ) 3/2 ] =0.002434 T/A ∴The theoretical slope is 0.002434 T/A y = 2.44 + 0.002273 5? * 1 mT/A = 0.001 T/A * 2.445 mT/A = 0.002445 T/A ∴The experimental slope is 0.002445 T/A % Error = [( | experimental - theoretical | ) / theoretical ] x 100% = [( | 0.002445 T/A - 0.002434 T/A| ) / 0.002434 T/A] x 100% = 0.45 % ∴The percent error is found to be 0.45% Part 2: Magnetic Field along the Central Axis of a Single Coil Figure 2: The relationship between the magnetic field for a single coil and the distance in the z-direction. These are measured in the units milliteslas(mT) and metres(m), respectively. Shows three different sets of data each taken at different values for the current, denoted by I . When taking a look at Figure 2, it can be seen that unlike magnetic field vs current, the magnetic

Kaur, 6 field vs distance actually yields an inversely proportional relationship. This means that as one value increases, the other decreases, therefore resulting in the parabolic shape seen above. Along with what was previously stated, it can be seen that by increasing the electric current, the magnetic field would also increase, which explains why the maximum point for I 3 is much higher than I 1 . Part 3: Exploring More Complex Systems In order to find an equation for the magnetic field of a two coil system, the Biot-Savart Law can be used. z 2 + R 2 ¿ 3 2 ¿ B ( coil )= N μI 2 R 2 ¿ z 2 + R 2 ¿ 3 2 ¿ z 2 + R 2 ¿ 3 2 ¿ ¿ B ( tot )= N μ I 2 R 2 ¿ R 2 ¿ 3 2 ¿ − R 2 ¿ 3 2 ¿ ¿ B ( tot )= N μ I 2 R 2 ¿ 5 R 2 4 ¿ 3 2 ( ¿ ) 2 ¿ B ( tot )= N μ I R 2 2 × ¿

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Kaur, 7 B ( tot )= 0.72 ¿ μ R This final equation, which is a variation of the Biot-Savart Law, can be used to find the magnetic field for a two coil system, much like the one in the Magnetic Field Simulation. Discussion and Conclusion Part 1: Magnetic Field at the Centre of a Single Coil Through the entirety of this experiment, only positive values were used for the electric current, however negative values could also be used. As previously shown in Figure 1, the magnetic field and electric current have a directly proportional relationship. This means that if the electric current was set to a negative number, the magnetic field, in turn, would also yield a negative number. The value of the magnetic field represents the strength of the field, meaning that if it was below zero, it would be much weaker than those above zero. The direction of the magnetic field as a vector would also change, being counterclockwise instead of clockwise. This is also true as negative charges have a velocity in the negative z-direction, being counterclockwise. Looking back to what was previously calculated, it can be seen that the percent error for the experimental slope is only 0.45%. Due to unforeseen circumstances, the lab had to be conducted online using computer programs, which are known for being extremely accurate. This can explain why the percent error is very small, being under one percent. Since the experimental and theoretical slopes are very close in value, it can be concluded that the results do, in fact, support the equation originally derived by theory. The equation was derived to calculate the magnetic field as a function of the electric current, represented by B coil /I. If the magnetic field was measured slightly out of the plane of the coil, meaning that “z” does not equal to zero anymore, the results would differ. If “z” was changed to either a negative or positive value, the strength of the magnetic field would decrease. The higher the increase of “z” in either direction, would lead to a larger decrease in the magnetic field. Part 2: Magnetic Field along the Central Axis of a Single Coil Taking a look back at the results for the relationship between the magnetic field and the distance, it is seen that the magnetic field values never become negative, even if the “z” values are negative. Instead, the field has the same value at both the positive and negative “z” values. This can be explained as

Kaur, 8 per the Biot-Savart Law, the “z” would be put to a power of two, therefore resulting in a positive value, regardless of the original sign. This relationship is what leads to the parabolic shape of the graph. It is also seen that the magnetic field is at its strongest point, the closer that “z” is to zero. This can be related back to part 1, where it was found that if the experiment was conducted with measuring the field outside of the plane, there would be a much smaller outcome in value. The three curves shown in the graph represent essentially the same data, with just the electric current being increased at intervals of 1 A each time. The curves each, all have three fit parameters, given as A, B, and C, and these would represent the current, the radius, and the length. Looking at Figure 2, it can be seen that it is in fact, possible to achieve similar results from part one of the experiment. By ignoring about the second half of the curve and solely focusing on the points leading up to the maximum point, those being the negative values for “z”, it is seen that they have a positive trend, meaning that as one increases, so does the other. This relates back to Figure 1, where it was found that the magnetic field and electric current shared a directly proportional relationship. The only difference as to why Figure 1 is a linear graph, whereas the first half of Figure 2 has a slight curve to it, is because in the latter figure, the magnetic field was measured slightly out of the plane, with the “z” values constantly changing. All in all, the negative values for “z” leading up to the maximum point as well as the graph for magnetic field vs electric current, are essentially giving very similar data, only differing in the position of the z-plane. Part 3: Exploring More Complex Systems Throughout the experiment, the magnetic field was studied for a single coil. However, there is still much to be said about a two-coil system. In the Magnetic Field Simulation, by activating the second coil and setting I 2 to the negative value of I 1 , the magnetic field at that value can be observed. It is seen to be a negative value, even though I 2 is set to a positive value. This is because now that two currents are simultaneously flowing, each with an opposite sign, it would lead to one heading in the clockwise direction, with the other in the counterclockwise direction. The results would prove to be the same even if I 1 was positive and I 2 was negative. This is because as long as one of the coils has a negative electric current value and flows in the counterclockwise direction, then the magnetic field will also result in a negative number. As the coils come closer together, it is observed that the strength of the magnetic field will increase. On the other hand, if the coils were to move further apart, the strength of the magnetic field would be seen to decrease. Moving on to the equation derived for the magnetic field of a two coil system, the Biot-Savart Law was initially applied. Since there are two coils as opposed to just one, the law had to be added to itself to represent the presence of two coils. The only difference in the added version of the law was that the distance, R , had to be negative as the equation is meant to determine the field of two

Kaur, 9 coils, with one coil being negative and one positive. Much like the theoretical slope calculation previously done, the “z” value is zero, so it was taken out of the equation. After simplifying the equation further, the final result came out to be B ( tot )= 0.72 ¿ μ R . This does support what was observed earlier because by using the equation, it can be seen that the closer the coils are put together, the stronger the magnetic field will be and vice versa. This is due to the inverse relationship between distance and field that shows if distance increases, meaning the coils are further apart, the field strength will decrease. Along with if distance decreases, meaning the coils are closer together, the field strength will increase. All in all, all three parts of this experiment aimed to support the hypothesis that the magnetic field would decrease when distance increases, as well as increase when electric current increases. After conducting and analyzing the data, it was concluded that the hypothesis was in fact supported and the expected results were what were seen. These patterns and trends help demonstrate the magnetic field and the impact that certain factors can have on its strength. References Knight, R. D. (2017). Physics for Scientists and Engineers: A Strategic Approach . Boston?: Pearson. Ling, S. (2020, November 5). 12.2: The biot-savart law . Physics LibreTexts. https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physic s_(OpenStax)/Map%3A_University_Physics_II_-_Thermodynamics_Electricity_and_Ma gnetism_(OpenStax)/12%3A_Sources_of_Magnetic_Fields/12.02%3A_The_Biot-Savart _Law Lucas, J. (2015, July 29). What is magnetism? | Magnetic fields & magnetic force . livescience.com. https://www.livescience.com/38059-magnetism.html Vedantu. (2021, January 13). Biot Savart law - Statement, derivation and FAQ . Online Tuition

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Kaur, 10 with Best Teachers for CBSE, ICSE, IIT JEE & NEET Prep. https://www.vedantu.com/physics/biot-savart-law Wagner, D. (n.d.). Introduction to magnetism and induced currents . Rensselaer Polytechnic Institute (RPI) :: Architecture, Business, Engineering, Humanities, IT & Web Science, Science. https://www.rpi.edu/dept/phys/ScIT/InformationStorage/faraday/magnetism_a.html Williams, M. (2016, October 23). What is a Magnetic Field? Universe Today. https://www.universetoday.com/76515/magnetic-field/

PCS130 Lab 1- Magnetic Fields

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