Beyond the magnetic field of a finite wire: a teaching approach using the superposition principle
BBeyond the magnetic field of a finite wire: a teaching approachusing the superposition principle
J.E. Garc´ıa-Farieta ∗ Departamento de F´ısica, Universidad Nacional de Colombia - Sede Bogot´a, 11001
A. Hurtado-M´arquez † Facultad de Ciencias y Educaci´on, Proyecto Curricular deLicenciatura en f´ısica - Grupo de Investigaci´on FISINFOR,Universidad Distrital Francisco Jos´e de Caldas, Bogot´a, 11021-110231588
I. INTRODUCTION
The traditional lecture-based scheme of teaching physics have shown to be ineffectivein several aspects [1–7]. It results in a lack of motivation, with low impact on students,to address more complex physical systems than the ones presented in textbooks. A clearexample in university-level education is the electromagnetics course, in particular when themagnetic field concept is introduced [8–10].Empirical laws of magnetism are usually discussed after a phenomenological descriptionof the magnetic interaction and its relation with electric currents as magnetic sources. Thetextbooks show in that section how to apply the Biot-Savart law to evaluate the magneticfield in any point in space due to a little element of a current-carrying wire of arbitraryshape. Then, it is exemplified considering a straight finite wire and evaluating the magneticfield B at a specific point, however, it involves several assumptions, which make the problemeasy to solve, but which are not discussed in detail, such as: i) choosing a symmetry pointto evaluate B ; ii) applying the superposition principle to simplify vector operations; and iii)reducing the number of dimensions of the system, from 3D to 2D or 1D. Even if it can bejustified, it makes difficult for students to solve some of the exercises proposed at the end ofthe textbook section, that aim to explore the magnetic field of more complex systems like an ∗ [email protected]; permanent address: Carrera 45 No. 26-85, Bogot´a, Colombia † [email protected] a r X i v : . [ phy s i c s . e d - ph ] J a n infinite wire, square loop, polygons of n sides, circular loop among others. Omitting thesekey points to evaluate the magnetic field in such systems, leads students to use incorrectassumptions, while they try to adapt the methodology previously used in the simple case ofa finite wire, resulting in algebra tricks or unreasonable results.In this article, the expression of the magnetic field for a finite wire is used in a ‘induc-tive’ way to obtain the magnetic field created by several configurations. This procedure iseasy generalized from a computational approach, to evaluate the field in any point of spaceby using the analytic expression of the finite wire n -times and exploiting the superpositionprinciple in those configurations. This hybrid methodology allows students to improve theiranalytic skills by extending the result of simple cases, as long as the assumptions made havebeen discussed. It also provides a motivation to create their own models and use program-ming tools to visualize the patterns of the magnetic field lines in different configurations,that are not always possible to perform in a laboratory demonstration. 𝜃 𝜃 𝜌 O P x i z 𝒓 𝒓 𝑟 𝐴 𝑟 𝐵 𝑑𝑙 𝜃 B A L 𝒓 𝒓 𝐁(𝒓)
FIG. 1: Finite straight wire carrying a steady current along z -axis. The magnetic field isevaluated at the point P . II. A FINITE STRAIGHT WIRE
The Biot-Savart law describes the magnetic field d B at any point in space due to anelement d l of a current-carrying wire as follows: d B = µ I π d l × ˆr r , (1)where µ is the vacuum permeability and ˆ r is an unit vector along r . For a thin and straightwire carrying a current I placed along the z -axis, as shown in Fig. 1, the magnetic field atan arbitrary point P is given by: d B = µ I π ρdz ( ρ + z ) / ˆe ϕ , (2)where cylindrical coordinates have been used for simplicity to map the three-dimensionalspace, taking advantage of the fact that d l and r are located in the same plain, causingthat the magnetic field is always oriented along ˆe ϕ for each point P , i.e, following concentriccircle trajectories regardless of the segment size. By integrating Eq. (2), the total magneticfield produced by the wire can be expressed as: B = µ I πρ (cos θ − cos θ ) ˆe ϕ where cos θ i = z − z i (cid:112) ρ + ( z − z i ) and i = A, B. (3)This equation gives a general description of the magnetic field for a current-carrying wireof finite length L , however, for the computational implementation it is useful to express itin Cartesian coordinates considering that the wire can be oriented in any direction. Usingthe law of cosines and vector decomposition, the magnitude of the magnetic field for anarbitrary located wire can be re-written as [11] B = µ I π ( r + r ) ( L − r − r + 2 r r ) r r (cid:112) r r + 2 r L + 2 r L − r − r − L . (4)Fig. 2 shows the function written in python language, whose inputs are: i) the coordi-nates ( x, y ) where B is evaluated, ii) the position of the wire ( r A , r B ), and iii) the value ofthe electric current I . This short function allows students to explore how affects each oneof the parameters in Eq. (4) and to visualize different configurations, taking advantage thatpython language is very friendly and has a wide set of tools that can be used without a deepprogramming knowledge.FIG. 2: Python routine to compute the magnitude of the magnetic field produced by afinite straight current-carrying wire arbitrarily oriented in space.Fig. 3 shows the plot of evaluating B in a meshgrid for a wire aligned along the z -axis.Subfigure a) displays B as a function of the distance ρ for points at different heights asshown by the colorbar. Inside the wire, the magnitude of the field increases linearly withdistance until reaching the wire radius, then outside the wire, it decreases inversely withthe distance. The field at points within the radius of the wire but above/below it tends tozero as expected. Subfigure b) shows the intensity map of the total magnetic field in a crosssection view of the wire ( upper ) and in the z − ρ plane ( lower ). The direction of the arrowsindicate that the current is coming out from the plane following the right-hand rule. III. APPLICATION OF THE SUPERPOSITION PRINCIPLE
The superposition principle states that the field created by different sources, e.g., by twoor more currents, simply added together as vectors. Applying the superposition of magneticfields before any calculations makes a transition to problem-solving easier. This approach isrepresented in Fig. 4. Starting from the analytic expression for a finite wire, the magneticfield of a composite system can be expressed as the one created by an arrangement of n -wires, being simple to compute numerically by iterating Eq. (4), or equivalent the pythonroutine (see Fig. 2).To illustrate this idea, let’s consider a n -side regular polygon carrying a current I . Eachside corresponds to a wire, then Eq. (3) can be used to find B at any point of space. Byinstance, the field at the center of the polygon can be obtained using Fig. 5, where AB is (a) (b) FIG. 3: a) magnitude of the magnetic field created by a current-carrying wire as a functionof the distance ρ ; b) intensity map of the total magnetic field in a cross section view( upper ) and in a parallel plane to the wire ( lower ).FIG. 4: Basic idea to obtain the magnetic field of a system using the superpositionprinciple.one of the sides and ρ is the radius of the inscribed circle. The angles θ and θ are derivedfrom the geometrical construction, taking into account that the inner angle of the polygonis given by ∠ AOB = 2 π/n , thus ∠ BOC = π/n . Considering the n wires and using angleproperties, the magnitude of the magnetic field is given by: B = n µ i πρ (cid:104) cos (cid:16) π − πn (cid:17) − cos (cid:16) π πn (cid:17)(cid:105) , = n µ i πρ sin (cid:16) πn (cid:17) . (5)As example, let’s evaluate this expression in two particular cases. For a square loop of side 𝜃 𝜃 𝜌 𝜋𝑛 𝑟 𝑟 A B C O P x 𝐁(𝑟)
FIG. 5: Systems considered for several configurations. l : n = 4 and ρ = l/
2, thus Eq. (5) becomes B = 2 √ µ iπl , (6)whereas for a circular loop of radius r : n → ∞ and ρ = r , then the magnetic field at itscenter is given by B = lim n →∞ n µ i πr sin (cid:16) πn (cid:17) , = µ i r lim n →∞ sin (cid:0) πn (cid:1)(cid:0) πn (cid:1) = µ i r . (7) IV. NUMERICAL RESULTS
Using the python code for a wire, it is simple to extend the result to more elaborategeometric distributions. Thus the student can obtain the magnetic field at any point inspace, visualize and characterize it, even without deriving the analytic expression for thesystem. Fig. 6 shows the results for four different configurations: a) two finite parallel wires;b) square loop, c) circular loop, and d) an irregular arrangement with a fish shape.FIG. 6: Magnetic field distribution of different composite configurations evaluated with thesuperposition principle by using the expression of a finite wire.The systems can be described in terms of the field intensity and identifying if there areregions where the field is canceled. The students can also change the input parameters andexplore its impact graphically, by example varying the electric current (value and direction),the wire lengths and also recover the configurations proposed in textbooks to compare withthe analytic solution.As an exercise, one can suggest to create their own models in more than two dimensionsfor specific distributions, and explore what happens changing the parameters. This inductiveprocess leads to a interesting activity, since an arbitrary system is defined only in terms of thecoordinates of the wires, thus an arraignment in 3D can show even more complex, and at thesame time more realistic, configurations. For example, it can be used to find the magneticfield of solenoids, loops configurations like Helmholtz coils or even a toroidal system, beforeperforming exhausting theoretical calculations.
V. CONCLUSION
We have presented a practical methodology to use the superposition principle of magneticfields starting from a finite current-currying wire. A short python routine was designed toillustrated how to apply the superposition principle to obtain the magnetic field of complexconfigurations in any point in space, without need of an explicit analytic expression. It canbe used as a complement of classroom experiments, motivating the discussion on magneticphenomena and providing a better understanding of the Biot-Savart law. This approachallows students to explore systems with different level of complexity, from regular polygonsto very irregular distributions as illustrated in Fig. 6, combining basic analytic skills withcomputational tools. [1] D. Sokoloff and R. Thornton, “Using interactive lecture demonstrations to create an activelearning environment”.
Phys. Teach . , 340 (Sept. 1997)[2] L. Mcdermott, “Millikan Lecture 1990: What we teach and what is learned-Closing the gap”. Am. J. Phys . , 301 (1991)[3] M. Welzel, “Student-centred instruction and learning processes in physics”. Research In Sci-ence Education . , 383 (1997)[4] R. Hake, “Interactive-engagement versus traditional methods: A six-thousand-student surveyof mechanics test data for introductory physics courses”. Am. J. Phys . , 64 (1998)[5] M. Good, A. Mason and C. Singh, “Comparing introductory physics and astronomy students’attitudes and approaches to problem solving”. Eur. J. Phys . (6), 065702 (2018)[6] J. Li, and C. Singh, “Investigating and improving introductory physics students’ understand-ing of electric field and the superposition principle: The case of a continuous charge distribu-tion”. Phys. Rev. Phys. Educ. Res. , 010116 (2019)[7] M. Dancy and C. Henderson, “Framework for articulating instructional practices and concep-tions”. Phys. Rev. St Phys. Educ. Res. , 010103 (2007)[8] A. Tasoglu and B. Mustafa, “The effect of problem based learning approach on conceptualunderstanding in teaching of magnetism topics”. Eurasian J. Phys. & Chem. Educ . (2), 110(2014) [9] F. Herrmann, “Teaching the magnetostatic field: Problems to avoid”. Am. J. Phys . , 447(1991)[10] J. Guisasola, J. Almud´ı and J. Zubimendi, “Difficulties in learning the introductory magneticfield theory in the first years of university”. Science Education . , 443 (2004)[11] Z., Yu, C. Xiao, H. Wang and Y. Zhou, “The calculation of the magnetic field produced byan arbitrary shaped current-carrying wire in its plane”. Advanced Materials Research, TransTech Publications .756