Lab1 (1)

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Investigation of Magnetic Fields PCS130 - Physics II

Introduction A basic method of creating magnetic fields is through the movement of electrons or an electric current. Through a conductive medium, the current path is chosen which in turn affects the magnetic field created. Through different equations, the magnetic field of different configurations can be determined. In this experiment, a loop will be investigated. The strength of the magnetic field with different currents, distances, and two loops instead of one. Theory To begin understanding how the experiment works from a theoretical level, the physics behind straight wire currents and their magnetic fields must be understood. Say there is a straight, infinitely long, wire carrying a current. Surrounding this wire is a magnetic field, distance r away, rotating counterclockwise from the direction of the current (right-hand rule ). The farther away from the current, the weaker the magnetic field. This can be proven by the Biot-Savart Law: dB= (μ 0 /4π)(I(dL ✕ r)/r^2) (1) The outcome of the equation for the same wire changes based on the radius. Where, I, L, and μ 0 remain constant, r is changing. The larger r become, the smaller the values become, proving the magnetic field decreases with distance. Moving onto magnetic fields near a loop of wire, the magnetic field changes shape, therefore changing the behaviour of the field. In the centre of the loop, all the field lines pointing the same direction creates the “north” pole where the magnetic lines “leave” from. The visualization of a straight wire’s magnetic fields and a loop’s magnetic fields are shown here: Figure 1: Current lines illustration It could be easier to imagine this as the wire’s magnetic field turns to form a ‘doughnut’ shape. In the middle of the loop will be one direction of magnetic fields, and outside of the loop will be the opposite direction. To determine which way the ‘north’ pole of the loop is, use the right-hand

rule where the current flows in the direction of the curved fingers, and the thumb is pointing north. The formula to quantify the strength of a loop is: B loop (I, z ) = (μ 0 I/2)(R^2/( z ^2 + R^2)^(3/2)) z (2) To increase the magnitude of the field, the current should be increased (with proper precautions), and more turns of wire around the coil to compile the magnetic fields of each loop. With these modifications, the formula shown below can be used to determine the field strength. B coil (I, z , N) = NB loop (I, z ) (3) Procedure Part 1 Before beginning the experiment, we ensured that the Magnetic Field Sensor was connected to CH-1 of the LabPro software on the computer by checking the top left corner of the screen for an icon. The sensor was set to 6.4mT range. Once the connection has been confirmed, the sensor was zeroed (Experiment -> Zero) while the coils were off, and separated from potential external magnetic fields. Keeping the power supply off for safety, the first coil was connected. On the bottom of the coils that were used, there is a diagram showing the direction of the current. The right-hand rule was applied to determine the north pole of the magnetic field. The sensor was attached to a ruler suspended on an adjustable clamp. An attempt was made to position the sensor in the central axis of the coil where the magnetic field should be the strongest. This is where the sensor remained for the rest of part 1. Before starting to take any measurements, a button with this symbol Ø was clicked on the LoggerPro interface to zero the sensor again. The power supply was turned on and the current was set to 0.4A. By clicking on the green triangle button, measurements ( current, magnetic field, error ) were recorded and then those values were inputted into an excel spreadsheet. This process was repeated, but the current was increased by 0.2A each time until reaching 2A. The power supply was turned off after. Once all the measurements were recorded, a graph was created showing the linear relationship between current and magnetic field strength.

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Part 2 Once again with a single coil, the power was turned off and the sensor was zeroed while still in the centre from the part 1 setup. The power supply was once again turned back on and the current was set to 2A. Moving just the ruler (to maintain central position) away from its initial position so it is approximately receiving 20% of the maximum magnetic field strength. Measurements were taken similarly to before, and all values were recorded on another spreadsheet page. After each measurement, the ruler was moved closer to the centre position in 2cm increments. This process was repeated until the ruler passed beyond the centre and is reaching the approximate 20% area on the opposite side of the loop. The power supply was turned off after. Once all the measurements were recorded, a graph was created showing the relationship between distance and magnetic field strength. Part 3 With the power supply still off, connect the other coil in series with the first loop used. The two coils were positioned so that they are parallel to each other, and the magnetic fields of each coil pointed in the same direction. Before performing the experiment, information such as coil radius, number of turns, and distance between the two coils were recorded. The current was set to 1A. Similarly to part 2, the sensor was set a distance away from the coils and moved by increments of 2cm towards and through the two coils. At each interval, measurements were recorded on another excel sheet, and a graph showing the relationship between distance and magnetic field strength was created. Results and Calculations Part 1 The expected outcome of this experiment is to see a linear relationship between current strength and magnetic field strength. According to formula 2 in the theory section, when μ 0 and R remain constant, the magnetic field is dependent on the current strength, I. The theoretical values were calculated by using formula 3, without changing the z value. Example Calculation( 1A measurement): B loop = (μ 0 I/2)(R^2/( z ^2 + R^2)^(3/2)) = (4π ⋅ 10^(-7) ⋅ 1)/2 (0^2 + 0.0525^2)^(3/2) = 1.197 mT

Figure 2: Relationship between current and magnetic field. Blue is experimental, and orange is expected For the error bars, the current ( x-axis ) error bars were taken from the LoggerPro statistical analysis function while recording values throughout the experiment. The magnetic field reading error bars ( y-axis ) was obtained from the Vernier website stating the expected errors when using the sensor device. As stated on the website, “the standard deviation of the readings…when you do not move anything typically [are] 0.009” (1). There are vertical error bars in the graph, but they are very small. Part 2 The expected curve should appear as such, where the maximum strength is approximately 2.4 mT. Example calculation( at the centre, z=0 ): B = (μ 0 I/2)(R^2/( z ^2 + R^2)^(3/2)) = (4π ⋅ 10^(-7) ⋅ 2)/2 (0^2 + 0.0525^2)^(3/2) = 2.393 ⋅ 10^(-2) T = 2.393 mT Figure 3: y= 1.26 ⋅ (0.525^2 / ( (x^2 + 0.525^2)^(3/2) )

Figure 4: Relationship between distance and magnetic field. Blue is experimental, and orange is expected For the error bars, the current ( x-axis ) error bars were taken from the LoggerPro statistical analysis function while recording values throughout the experiment. The magnetic field reading error bars ( y-axis ) was obtained from the Vernier website, using 0.009mT again. There are vertical error bars in the graph, but they are very small. It is to be noted that the expected and collected values are nearly identical for x=0, there is a slight overlap between the dots. Part 3 Figure 5: Relationship between distance and magnetic field. Blue is experimental, and orange is ‘expected’ The error bars are there, just small again, and follow the same sources as part 2’s bars.

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Discussion and Conclusion Part 1 This part of the experiment was successful at showing the linear relationship between current and magnetic field strength. The expected values and the collected data values were very close to each other, proving the accuracy of how the experiment was conducted. Only a few points were out of the error range, but overall, accurate. The precision of the sensor is very small, so the vertical error should not be contributing much to the cushion space for error. Had the experiment been conducted slightly out of plane, the results would have shown a slightly lower trendline, ideally parallel to the expected value line. Part 2 The goal of part 2 was to determine the distance where the field’s strength is 50% of its maximum value. Through data and graphical analysis, the halfway point from the max (2.384mT) would be around 8cm to the left or right (1.087mT). By calculating the true maximum strength(2.4mT), it is possible to determine the distance where the 50% strength lies. That would be at 4cm away(1.2mT). Where these deviations came from will be discussed further down when sources of error are addressed. Part 3 Despite using the Helmholtz double coil equation, the graph did not represent how the expected values should appear. The theoretical result should have shown a graph where two single-loop curves overlap each other and create a flat line for a certain interval. Ignoring the incorrect ‘expected’ series line, the experimental values show how the graph should look more accurately. The objective of part 3 is to also determine the distance where 50% of the field's strength is, and according to the values, it lies 12cm away from the centre. At the centre, the field strength is 1.3 mT, and half of that lies around 12cm, 0.7027mT. If the coils were moved closer together, the graph should show a curve similar to the one in part 2, where the overlapping regions of each coil’s magnetic field add onto each other. If they were moved further apart, a graph similar to the orange line would appear, where there is a dip. This ‘dip’ in the graph would be due to the fields getting weaker farther from the source, and the distance between each coil is not close enough to make up for this lost strength.

Sources of error Magnetic fields are subject to change when there are other fields present. In a room with 10 other experiments being conducted, plus electronic devices that are on, there is definitely enough excess magnetic fields interfering with the sensor. As the experiments went on, the power source began to fluctuate its hold on a steady current. For future experiments, it should be stressed to allow time in between for it to cool down so the wires do not overheat and other parts of the generator don't malfunction. Finally, it is important to ensure that the sensor does not move while measurements are being taken. Its position in the centre is already being estimated, so having it move while values are being recorded creates a higher error than necessary. If for the next time, an apparatus that is able to centre and maintain the sensor's position is available, it is advised to use that. It should be noted that there are human and environmental errors throughout the lab. Although error can be calculated and accounted for, to have results match expected values, the experiment should be done in a specialized room specifically for these measurements.

Citations 1. Vernier. (-1, November 30). What is the accuracy of the magnetic field sensor? Technical Information Library. Retrieved January 26, 2023, from https://www.vernier.com/til/1574

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Lab1 (1)

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