Magnetic Field Strength Determination on a Solenoid and Magnetic Field Mapping on Bar Magnets

Magnetic Field Strength Determination on a Solenoid and Magnetic Field Mapping on Bar Magnets
Sims, Jon Xyrus
College of Engineering, University of the Philippines, Diliman, Quezon City

Abstract Magnetic fields are important in a vast array of applications, as well as its natural occurrence in many different phenomena. This experiment aims to determine the dependence of its strength on both a changing current, and for a solenoid, changing number of turns and determine through linear regression the experimental value of permeability. Also, this experiment aims to tackle the similarity and difference between a magnetic field pattern produced by a solenoid and a bar magnet, as well as the difference between the field pattern of a NS-NS and NS-SN configuration, in which the former is known to attract and the latter repel. The author has found the relationship between field strength, current and number of turns per unit length to be fairly linear, with linearity coefficients of 0.9999, and 0.9985, respectively. Calculated permeability for the latter turned out to be further from the true value (RSD = 15.13%), as opposed to the former. With regards to magnetic field line patterns, solenoid patterns turned out to be similar to a bar magnet’s, while SN-SN configuration produced a constructive field line pattern, which had allowed it to act as a single magnet, and attract each other, while the SN-NS experienced the contrary.
1. Introduction
Magnetic fields are created by moving charges and/or currents. Therefore, a solenoid, or a wire coiled several times with a running current, can be treated no differently from a bar magnet. Thus, through Ampere’s law, it is possible to create an integration path across the length of the solenoid and relate magnetic field to both the current and number of turns.[1] Therefore, after integration:

(1)
Where B = magnetic field magnitude, N = number of coil turns per unit length (n = N/L), I = current through solenoid, and = permeability of free space (magnetic constant). It is justified to use the permeability of free space even if the solenoid is definitely made of a metal due to the existence of a hollow space inside it where the magnetic field forms. Magnetic fields, due to their properties have been extremely useful. In the case of imaging, it has shown the capability to align subatomic particles such as protons by their axes in a single direction, thereby allowing a scanner to produce a clear image of the area examined. This process is also known as Nuclear Magnetic Resonance (NMR) or Magnetic Resonance Imaging (MRI). This is harnessed most especially for medical applications, taking advantage of the human body’s own magnetic properties to scan for certain brain fluctuations, or damage control. [2] It is not only humans that have harnessed magnetic fields. Currently, a study conducted by Czech scientists of the University of Duisburg-Essen in Essen, Germany postulates that foxes may use the Earth’s magnetic field in order to hunt due to the fact that unless prey is seen, attacks are always directed to the north-east, in which there is a higher probability of success. The authors suggest that these foxes possess a ring of detection at the retina which is darkest at magnetic north which allows them to calculate distance. [3] Knowing these facts makes the study of the sources of these magnetic fields important. This study explores the effect of both changing current and changing number of terms per unit length across a solenoid. Because the relationships between magnetic field and these two factors (as seen in Equation 1) are linear, the author wishes to confirm this linearity. As an addition to this, the author also wishes to determine the experimental value of permeability of free space. Also, with regards to magnetic field lines, the author wills to determine a sketch of magnetic field lines of both a slinky solenoid and various bar magnet configurations through inductive and visual methods, such as compass and field pattern window application.

2. Methodology

A. Magnetic Field in Slinky, Varying Current
With a displacement of 1m, a slinky was stretched in order to fit a Vernier Magnetic Field Sensor at half the length. With duration of collection set at 10s and collection setting at HIGH, the sensor was connected to LabQuest (see Figure 1). Also, the right end of the slinky was clamped with the positive terminal of a power source, while a negative end was placed on the left. The power supply was turned on, adjusted to a current of I = 2.0A, and the sensor was adjusted according to the direction of the highest magnetic field produced by said current. In order to collect the mean magnetic field produced, LabQuest was made to collect data of magnetic field versus time for ten seconds, while leaving the power supply on only during five seconds of the said ten. This region was then analyzed for the mean, and the procedure was repeated for currents of I = 2.5, 1.5, 1.0 and 0.5 A. Seen in Figure 2 is the plot between magnetic field and current.

B. Magnetic Field in Slinky, Varying Length/Turn Spacing
The procedure above was repeated, with modifications. First, the current was set to a steady and constant I = 0.5 A, while reading replicates were instead taken for varying length: L = 0.25, 0.50, 0.75, 1.0, 1.25. The number of turns, however, was inadvertently taken constant still at 84 turns. The plot between the Magnetic Field magnitude and number of turns per unit length is seen in Figure 3.

Figure 1. Parts A and B Setup. The magnetic field sensor is placed at the centre of the stretched slinky, whilst connecting the two ends to a power source

C. Magnetic Field Sketching of Slinky and Bar Magnet
The slinky, set at a length L = 0.5 m, and I = 1.5 A, was again examined for the magnetic field. However, this time, a compass was used to determine where the magnetic field pointed. The compass was moved across the length of the slinky, both inside and outside, while the direction of where its north pole pointed was mapped.
For the bar magnet, a field pattern window was used to determine the magnetic field pattern by copying the patterns formed by the iron filings in the window, as the magnet is placed on top.

3. Results and Discussion
The permeability of free space, , is experimentally determinable through plotting the magnetic field strength determined against the product of both the current through the solenoid and number of turns per unit length. In figures 2 and 3, the magnetic field strength was found to be relatively linear, further confirming the relationship between magnetic field strength and current, and the same with turns per length, respectively.

Figure 2. Linear Regression Between B (milliTesla) and I (Amperes). Shown is the linearity as well as the regression equation, following the model of

The slope of the line in figure 2 allows for the calculation of by dividing the said slope by the constant number of turns per unit length, which was at 84/m. This yields =. Furthermore, an error of 0.0002 is systematic, reducible to the presence of excess current due to the wiring system of the instrument having a close proximity to the setup. This is logical, as the intercept error assumes zero current, and a presence of a positive intercept may indicate unwanted current. Also, this may be caused by fluctuations at the zeroing of the instrument.

Figure 3. Linear Regression Between B (milliTesla) and n (per meter). Shown is the linearity as well as the regression equation, following the model of

In figure 3, it is possible to deduce the same information. In this case, there is a higher intercept error (0.0115), but the plot is still relatively linear. Calculating yields Lesser linearity and a higher error indicate that a factor independent of the number of turns is at work. If the number of turns is set to be zero, it is the intercept that the magnetic field strength will be equal to. This error, therefore, may represent a magnetic field produced by the wire itself – independent of the number of turns. The data is summarized in Table 1, together with their deviations from the true value.

Table 1.Theoretical and Experimental Permeability from Procedures A and B, With their Standard Deviations

Varying Current
Varying Number of Turns/Length

6.05%
15.13%

As expected, the relative deviation of the permeability found at varying number of turns per length is much greater than with varying current, because of reasons explained above.

In magnetic field sketching, the magnetic field line sketches of both slinky solenoid and bar magnet are found in Figure 4 below.

Figure 4. Magnetic Field Pattern of Solenoid and Bar Magnet

The solenoid has a leftward straight magnetic field at the center, while the lines return to the right end outside the solenoid. This implies that the positive and negative terminals of the power source are at the right and left end, respectively. Since current flows from positive to negative, there is a counter-clockwise current through the solenoid. Through the right hand rule, this current can be interpreted as inward, and an inward curl produces a leftward magnetic field, due to magnetic field having the same direction as magnetic moment.[1] Mathematically:

(2)

where A is the current area vector. It is also for this reason that it is observed that the magnetic field lines produced by both solenoid and bar magnet are similar.

Figure 5. NS-SN and NS-NS Bar Magnet Configuration Patterns The two bar magnet configurations, NS-NS and NS-SN were also examined. From the field pattern window, the patterns were sketched in Figure 5. For the NS-NS configuration, the two magnets seemed to create a constructive and additive magnetic field pattern. Not only were field lines entering the south of one magnet from its north, but some lines also entered the south pole of a magnet from the north of another. Together, two magnets in this configuration can therefore be considered a single, coherent magnet – which also explains while the magnets stick together and are generally attracted. However, a different character is found for the NS-SN configuration. The field lines entering both south poles seem to push each other away, as they cannot intersect. This may explain the repulsion between these two poles.

4. Conclusion
Equation 1 has been confirmed by this experiment, as a linear relationship has been found between magnetic field strength, and both current and number of turns per unit length. Also, values for permeability of free space were deduced from plots derived from both B vs. I and B vs. n. These values deviated by 6.05 and 15.13%, respectively. The deviation found from number of turns per unit length signifies a greater error due to a solenoid being a single-loop conductor even without having turns, which is a more involved property rather than having simple systematic errors caused by instrument wires.

Magnetic field sketches produced from the same solenoid closely resemble such patterns for a bar magnet, which has been explained as a consequence of the right hand rule, and a higher concentration of magnetic field inside the coil. Also, from investigating the interaction between NS-NS and NS-SN bar magnet systems, it was determined that the constructive nature of field lines in the NS-NS system is diametrically opposed to NS-SN, which was found to have a voiding nature, thereby having a quality of repulsion between the two poles.

Acknowledgements The author would like to thank, and curse the gift of the World Wide Web. It has made humans depend on it, and though this would have been difficult without it, it still can cause problems of overdependence, even for communication purposes. The author also gives thanks to his groupmates, who have not left him for the birds. Thanks go out to Sir James Vance, and his expertise on the field. Special thanks to the Filipino masses, as well.

References
1. H.D. Young, R.A. Freedman, A.L.Ford, Sears and Zemansky’s University Physics, Chapter 28, Addison-Wesley, 2012.
2. A. Berger, “Magnetic Resonance Imaging,” BMJ 35(324), p. 1, 2002
3. J. Červený , S. Begall , P. Koubek , P. Nováková , H. Burda, “Directional preference may enhance hunting accuracy in foraging foxes,” Biology Letters 11(3), p.2, 2015.

References: 1. H.D. Young, R.A. Freedman, A.L.Ford, Sears and Zemansky’s University Physics, Chapter 28, Addison-Wesley, 2012. 2. A. Berger, “Magnetic Resonance Imaging,” BMJ 35(324), p. 1, 2002 3. J. Červený , S. Begall , P. Koubek , P. Nováková , H. Burda, “Directional preference may enhance hunting accuracy in foraging foxes,” Biology Letters 11(3), p.2, 2015.