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WJP, PHY382 (2015) Wabash Journal of Physics v3.3, p.1 Magnetic Field Mapping of Coiled Current-Carrying Wires A. Camacho, B. Hayhurst, K. Sullivan, and M. J. Madsen Department of Physics, Wabash College, Crawfordsville, IN 47933 (Dated: March 17, 2015) The purpose of this experiment is to create a three dimensional vector map of the magnetic field of current-carrying coils of wire. We have measured the magnetic field produced by a coiled, current-carrying wire at predetermined points around the coil and superimposed each vector onto a plot to create a map of the magnetic field. The ”maps” (plots) created by our apparatus agree very closely with the predicted model. Future work on this experiment could include measuring the structure of the magnetic field of a persistent current in a superconducting loop. WJP, PHY382 (2015) Wabash Journal of Physics v3.3, p.2 Magnetic field mapping can be used as a quality control assessment tool in production of complex multipole magnets or complex assemblies such as loudspeakers or photocopier rollers. By accurately knowing the behavior of a magnetic field, we can test the effectiveness and efficiency of different ways to produce a magnetic field, [1] [2]. The purpose of this experiment was to develop a method to accurately map out the magnetic field produced by a coil with approximately constant current running through it [5]. I. MODEL To Achieve the goal of our experiment, we have created a system for mapping the magnetic field in three-dimensional space in a reliable and repeatable fashion. To do this we used knowledge about how current-carrying wires and moving chargea produce magnetic fields. Following from Jackson [5], we consider a circular loop of radius a, lying in the x − y plane, centered at the origin, and carrying a constant current I. From Maxwell’s differential laws of magnetostatics, ∇ × B = µ0 J (1) ∇ · B = 0, (2) and we see that if ∇ · B = 0 everywhere, B must be the curl of some vector A(x), called the vector potential, B(x) = ∇ × A(x). (3) Thus, substituting Eq. (3) into Eq. (1) yields ∇ × (∇ × A) = µ0 J (4) ∇(∇ · A) − ∇2 A = µ0 J. (5) or By making use of the Coulomb gauge, ∇ · A = 0, equation 5 reduces to Poisson’s equation ∇2 A = µ0 J, (6) WJP, PHY382 (2015) Wabash Journal of Physics and the general solution for A has the familiar form Z µ0 J‘ A(x) = d3 x‘ . 4π |x − x‘ | v3.3, p.3 (7) The current density J has only a component in the φ direction, thus Jφ = I sin θ‘ δ(cosθ‘ ) δ(r‘ − a) . a (8) The delta function restricts the current flow to a ring of radius a. Because of the symmetry of our loop configuration, the x component of the current has no effect on the total magnetic field. This leaves only a y component, which is Aφ Z cosφ‘ dφ‘ µ0 I a 2π Aφ (r, θ) = . 4π 0 (a2 + r2 + 2arsinθcosφ‘ )1/2 (9) At this point we switched from Jackson’s use of spherical polar coordinates to cylindrical polar coordinates. The reason behind this is that our solenoid could be modeled after a cylnder and not a sphere. Our new formula for Aφ is then Z cosφ‘ dφ‘ µ0 I a π . Aφ (r, θ) = 4π −π (a2 + s2 + z 2 − 2ascosφ‘ )1/2 (10) The above integral expressed in terms of the complete elliptic integrals is Aφ (r, θ) = − µ0 I [(a + s)2 + z 2 E(k) − (a2 + s2 + z 2 )K(k)] p , 2πs (a + s)2 + z 2 (11) where k= 4as . (a + s)2 + z 2 (12) Eq. (11) along with Eq. (12) is the vector potential for a single loop of wire. Along with this, the components of magnetic induction in cylindrical coordinates are Bs = − Bz = ∂Aφ ∂z 1 ∂ (sAφ ) s ∂s Bφ = 0. (13) (14) (15) All we need to do now is add up all the loops that make up our solenoid while offseting the z-position of the coils accordingly 9 Atot φ = 2 X i=− 29 − µ0 I [(a + s)2 + (z − zi )2 E(k) − (a2 + s2 + (z − zi )2 )K(k)] p 2πs (a + s)2 + (z − zi )2 (16) WJP, PHY382 (2015) Wabash Journal of Physics v3.3, p.4 4as . + (z − zi )2 (17) with k= (a + s)2 Bstot is then obtained by substituting the result of Eq. (16) into Eq. (13), and similarly, Bztot is obtained by substituting the result of Eq. (16) into Eq. (14). The model for our toilet-seat coil followed a similar method, with the exception that 5 turns of the coil were in one direction while the other 5 turns were oriented in the opposite direction. X Y magnometer # Z coil FIG. 1: The above figure shows a simplified schematic of our setup. The magnetometer is mounted on the end of a Lego wand that can actuate in each of the three directions shown. A 10 turn coil of wire connected to a constant current power supply provides the model magnetic field for the sensor to map. The magnetometer outputs data to the arduino where the data is then processed and passed into a computer running National Instruments LabView where the data is collected for further analysis. WJP, PHY382 (2015) Wabash Journal of Physics II. v3.3, p.5 SET UP In an effort to repeat measurements with a good degree of reliability, we engineered an automated mobile sensor support structure. The support structure moves in three dimensions between predetermined points and is automated using programs coded using National Instruments LabView that control Lego Mindstorms servo motors. This entire aparatus moves a Honeywell Magnetometer between positions which gives the magnetic field vector as a function of positon. This measurement allows us to create a map of the field. This setup also allows for a reliable and repeatable measurement technique that also has a very acceptable amount of accuracy. In order to measure a magnetic field, we placed a Honeywell HMC5883L 3-Axis Magnetometer into the magnetic field using a 3-axis automated control mechanism. The data collected from the sensor at each predetermined point gives the magnitude and direction of the magnetic field at that point. Using a collection of points inside of the magnetic field yields field map of the magnetic field. To mitigate the effects of outside/background magnetic fields, we built the support structure and test bed from non-magnetic materials. The Legos, being plastic, and the aluminum frame and test bed allow for the sensor to measure only the magnetic field caused by the coil (a method for canceling out the Earths magnetic field is used but is described elsewhere in this paper). III. DATA ANALYSIS The Honeywell magnetometer collects a magnitude of electric field in the x, y, and z directions; To ensure the data collected from the magnetometer correlated to our theories, we compared the theoretical model field map to the data collected by the magnetometer. To plot the field and compare the fields, we plotted the magnetic field vector vs the position of the sensor. As we can see from Figure 3, the map created by the apparatus is in strong agreement with the predicted field. We can say, then, that the apparatus is a good tool for measuring and mapping magnetic fields. Further calibration could increase the accuracy of the apparatus, but as it is now, the apparatus works in a reliable fashion. This same sort of comparison was done for multiple types of fields in an effort to ensure that the known fields WJP, PHY382 (2015) Wabash Journal of Physics v3.3, p.6 è èèè è è èèè è è è Data èèè è è è Fit èèè è è 100 èèè è è èèè è è è èèè è è 50 èèè è è èèè è è è èèè è èèè 0 0 200 400 600 800 1000 1200 1400 raw position y HmmL 150 FIG. 2: Above is the calibration of the y-axis of our field mapping apparatus. Both the x-axis and z-axis were 0.10601 ± 0.00028 mm/tick (95% CI gaussian pdf) with negligible slop. The y-axis had an up slope of 0.10657 ± 0.00028 mm/tick (95% CI, gaussian pdf) and a down slope of 0.10684 ± 0.00027 mm/tick (95% CI, gaussian pdf). This gives a slop of approximately 1.14mm between up and down. were mapped correctly by the apparatus. IV. CONCLUSION Having compared the expected model of magnetic fields to our measured maps of the same specification, we can conclude that this apparatus is a reliable and reusable tool for measuring magnetic fields in three-dimensional space. Though the obvious uses include measuring the strength and field shape of different solid magnets or wire coil structures, future applications extend to measuring the magnetic field induced by the persistent current in a high temperature superconducting loop. [1] D. Rosa and D. Stenton. “Electric and Magnetic Field Mapping.” The Physics Teacher 35, 136 (1997). [2] C. Christodes, I. Panagiotopoulos, D. Niarchos, G. Jones. “Fast Magnetic Field Mapping of WJP, PHY382 (2015) Wabash Journal of Physics v3.3, p.7 50 50 0 zHmmL x HmmL -50 0 50 0 zHmmL -50 -50 50 0 -50 y HmmL (a) 0 20 40 60 r HmmL 80 100 120 (b) FIG. 3: a) 3D representation of our magnetic field produced by the 15-turn coil shown in gold. b) 2D representation of our magnetic field. The x and y coordinates have been ~ is represented by the length of ~r. integrated into the vector ~r. The magnitude of B Permanent Magnets with GMR bridge and Hall-probe Sensors.” Redcliffe Magnatronics Ltd. 20 Clothier road. Bristol BS4 5PS UK. [3] E. R. Dietz and R. W. Keith “Inexpensive Magnetic Field Mapping Probe.” The Physics Teacher 35 112 (1997). [4] J. R. Merill. “Some Uses of Computers in Introductory Electric and Magnetic Field Mapping.” American Journal of Physics 39, 791 (1971). [5] J. D. Jackson “Classical Electrodynamics” Pg. 180-183. John Wiley & Sons Ltd. 1962 WJP, PHY382 (2015) Wabash Journal of Physics 50 zHmmL zHmmL 50 0 0 -50 -50 0 v3.3, p.8 20 40 60 80 100 0 20 40 60 80 100 r HmmL r HmmL (a) (b) FIG. 4: a) shows our data, as vectors, superimposed with our model as field lines for our 15 turn coil. The black rectangle is our source or coil b) The above figure shows our data, as vectors, superimposed with our model as field lines for our toilet seat coil. The black rectangle is our source or coil. WJP, PHY382 (2015) Wabash Journal of Physics à 2500. à æ æ æ æ à æ BzHmGL 2000. æ æ 1500. à æ æ à à à à à à 1000. à 0 à æ æ à -40 127 à à à -20 255 à æ à 500. TS Model TS Data à æ à æ æ æ UC Model UC Data à v3.3, p.9 0 z HmmL æ æ æ 20 æ æ æ à æ æ æ æ æ -127 à à à à à -255 40 FIG. 5: The above figure shows and overlay of the magnetic field data and model for a z-axis scan through the center of the coils.