Category Archives: Advanced EM

Advanced Electromagentism (Phys 341)

Project Proposal – Modeling the Magnetic Field of Magnetic Dipoles using Mathematica

For my project, I will work with Tewa Kpulun and Ramy Abbady on modeling vector fields of different geometries using Mathematica software. I plan on modeling the magnetic field $\vec{B}$ of a magnetic dipole with the eventual goal of modeling large scale magnet shapes. I will begin by modeling a single magnetic dipole with a set magnetic moment $\vec{m}$ before modeling more complicated systems. Eventually, I plan to superimpose many magnetic dipoles to create more complex magnetic geometries, such as a bar magnet or classic horseshoe magnet, possibly with regions where different internal currents create individual, unique magnetic dipoles.

Elias Kim Spring Project

Project Proposal:

Electrodynamics provides for some frighteningly fascinating applications. John Loree and I plan on investigating the physics of rail guns, powerful machines that fire slugs at high velocities using electromagnetic forces. We plan to build a functioning gun and analyze its motion with LoggerPro. Furthermore, I hope to use Mathematica to model the physics involved in the circuitry of the gun. This analysis will include finding current as a function of resistance, charging capacitors in the system, and the induced EMF. John and I will combine our results to produce general equations of motion for the rail gun.

Mathematica Script

Here is a link to the Mathematica script that has my graphs and animations used in this blog https://vspace.vassar.edu/brrachmilowitz/Graphs%26Pictures.nb

Why it’s so difficult to model maglev systems:

(Borcherts and Davis, “Force on a Coil Moving over a Conducting Surface Including Edge and Channel Effects, Journal of Applied Physics (1972))

The complexity of equations (like the one seen above) for the magnetic field produced by a real current-carrying loop with finite dimensions is why the perfect dipole approximation had to be made in the course of my research. Potentially, future work would involve using the equations for the “real” case and attempting to model the motion of trains that use these coils to levitate.

Future Considerations

In order to expand upon the work I’ve done with this project thus far, I would consider modeling the actual motion of a maglev train, that is, its motion relative to the height above the track. I would begin by attempting to solve the second-order differential equation associated with this motion, based off of Newton’s Second Law, given by the equation:

(1) $\begin{equation*} M\ddot{z}=-Mg+\frac{3\mu_{0}m^{2}}{32\pi(z)^{4}}\left(1-\frac{\omega}{\sqrt{v^{2}+\omega^{2}}} \right) \end{equation*}$

However, this equation would still only apply under the assumption that the superconducting coils on board the train act as a perfect dipole. Furthermore, I could consider the effects of the magnetic forces that guide the train along the tracks, and how this force affects the magnetic lift and drag forces on the train in that case.

References and Mathematica Code

References are here.

1) Jacobsen, Neil E. NMR Spectroscopy Explained: Simplified Theory, Applications and Examples for Organic Chemistry and Structural Biology. Hoboken, NJ: Wiley-Interscience, 2007. Print.

Mathematica Code and the PowerPoint for other figures. To download the Mathematica notebooks, right click the links and Save As.:

https://vspace.vassar.edu/milueckheide/NMR%20Magnetic%20Field%20Model.nb

https://vspace.vassar.edu/milueckheide/Shielding%20Factor%20Tables.pptx

https://vspace.vassar.edu/milueckheide/Wave%20Plots.nb

References & Mathematica Code

Here are the various references that I consulted while researching the methods of magnetic levitation:

Jayawant, B. V. “Electrodynamic Suspension and Levitation Techniques.” Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 416.1851 (1988): 245-320. Web. 25 Apr. 2012. <http://www.jstor.org/stable/pdfplus/2398062.pdf?acceptTC=true>.

Kraftmahker, Yaakov. “Maglev for Students.” European Journal of Physics. 29. (2008): 663-669. Web. 25 Apr. 2012. <http://www.rose-hulman.edu/~moloney/Ph425/ProjectPDFs/ejp_jul_08_maglev_0143-0807_29_4_001.pdf>.

Reitz, John R. “Forces on Moving Magnets due to Eddy Currents.” Journal of Applied Physics. 41. (1970): 2067-2071. Web. 25 Apr. 2012. <http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=JAPIAU000041000005002067000001&idtype=cvips&doi=10.1063/1.1659166&prog=normal>.

Rossing, Thomas D., and John R. Hull. “Magnetic Levitation.” Physics Teacher. (1991): 552-562. Web. 25 Apr. 2012. <http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=PHTEAH000029000009000552000001&idtype=cvips&doi=10.1119/1.2343425&prog=normal>.

Here is a link to the Mathematica code used to produce the various graphs and plots in my project:

https://vspace.vassar.edu/joandrade/Physics%20Project%20-%20Mathematica%20Code.nb

Conclusion

Initially, I sought out to study the interaction of magnetic fields and moving charges that allows Maglev trains to function as one of the world’s leading competitors in future transportation. Conceptually, their supposed high efficiency was not something unclear or doubtable: through magnetic levitation, the methods involved in EMS and EDS virtually eliminate all ground drag forces that moving vehicles typically face. However, though these machines seem to be “free of all forces,” it now seems that forces – and the delicate balance of interacting forces – is extremely important in determining whether or not a maglev system will work properly and efficiency.

I found that there are many variables that influence how a maglev system will function, and that different types of variables affect different types of magnetic levitation. Although the levitation force in the EMS system is entirely attractive and not very dependent upon the speed of the moving train, the EDS system relies upon repulsive forces and speed drives the levitation. I’ve revisited inductance and its quantitative – rather than qualitative – value beyond the simplicities of Lenz’s Law. In the non-ideal physics world, there are real forces that affect our system that we can’t always “assume away.” Although I did approximate the superconducting coil configuration with that of a magnetic dipole, I would have compared my outcome with that of a real rectangular coil with finite dimensions if an equation for the magnetic field or force was available.

To extend upon this project, I would like to do several things: (i) determine an exact analytical equation for the magnetic force that would allow a superconducting coil with physical dimensions to levitate; (ii) compare my result with similar results, and vary some other values besides speed (current, number of loops, area of loops, etc.); (iii) determine which settings are optimal for a real maglev train to attain maximum efficiency; and (iv) introduce the concept of a guidance force (the force that stabilizes the train in the horizontal direction), and analyze how this additional force impacts the lift, magnetic drag, and the ratio between the two.

Lastly, I think it would be most valuable to conduct experiments on my own, testing the relationships proposed by physicists in the past, and searching to develop a model to describe the exact relationship between the speed of the train (or some scale model) and the levitation force, magnetic drag force, and aerodynamic drag force that result. Ideally, then, I could derive an expression for the ideal height (the height that maximizes $\frac{F_{L}}{F_{D}}$ ) of the maglev train at any given speed.

Results IV – Graph of Forces and Final Analysis

Using the equation for the lifting force on a magnetic dipole by a conducting plane, as well as the relationship between lifting force and drag force (which happens to apply for all coil configurations) given by:

(1) $\begin{equation*} \frac{F_{L}}{F_{D}}=\frac{v}{\omega} \end{equation*}$

(stated by Jayawant, using the same variables as previously)

I produced the following graph in Mathematica, a plot of the ratio of the force and the ideal image force with respect to speed, ignoring the constant term out front:

where the blue line represents the lifting force $F_{L}$ , the red line represents the drag force $F_{D}$ , and the yellow line represents the ratio of the lifting force and the drag force $\frac{F_{L}}{F_{D}}$ .

From this graph, it appears that our theory for the limiting case (as $v>>\omega$ ) holds true, since the lifting force starts off proportional to $v^{2}$ , but then quickly becomes asymptotic toward the ideal image force $F_{I}$ as speed continues to increase. Meanwhile, the drag force is proportional to $v$ at low velocities, and is actually greater than the lifting force within this low speed interval. However, at around $2m/s$ , the lifting force surpasses the drag force, as the lifting force continues to rise while the drag force reaches a peak. Then, as speed increases, the drag force appears to fall off as $\frac{1}{\sqrt{v}}$ . Additionally, for this magnetic dipole approximation, the ratio of lifting force to drag force remains fairly linear and very steep.

If I were able to compare these results with those for a rectangular coil of wire – a configuration more appropriate to maglev trains – I would expect the same patterns to appear, though the lifting force might take longer to approach the limiting value of the ideal image force, since the rectangular coil situation represents a real case rather than an ideal case. Also, the drag peak might vary, thus causing a change in the train speed at which the lifting force surpasses the drag force. Because the ratio of lifting force to drag force is one measure of the efficiency of a maglev system (i.e., because it measured some desired outcome with respect to an undesired outcome, both dependent upon the speed of the train), I would expect the ratio for this perfect dipole case to always be greater than any nonideal case: thus, the slope of the ratio curve should be lower for real EDS maglev systems.

Results III – Calculating Lift and Drag

Considering the lifting magnetic force on the train’s superconducting coils, we know that at extremely low speeds, the magnetic force of repulsion will likely not be great enough to levitate the train. However, as the speed of the train increases, the lifting force should increase proportionally, as the magnetic repulsion force increases. It would be nice to have a compact equation directly relating the speed of the train to the lifting force; regrettably, I could not discover any such equation. There is, however, a method by which one can estimate this force. Just as there was the “Method of Images” to solve problems with conducting planes in electrostatics, there is a similar method for moving sources of magnetic fields. As the superconducting coils on the train pass over the conducting coils in the track, the repulsive force generated by the opposing magnetic fields between the two loops can be thought of as an equal and opposite force being created by an image of the superconducting coil located an equal distance away on the opposite side of (underneath) the track. Then, as the train moves forward some distance, a new image is created underneath the track, and the previous image begins to move away from the track, further into the ground. The speed at which the images recede from the plane is given by $\omega=\frac{2}{\mu_{0}\sigma*T}$ , where $\mu_{0}$ is the permeability of free space, $\sigma$ is the conductivity of the conducting plate (coils) in the track, and $T$ is the thickness of the conducting plate. Though most of the force created by the images is repulsive, there is still the electromagnetic drag force created by the remaining currents in the passed loops, which can be thought of as coming from the previous images. This successive line of images is known as the “wake of images.”

Figure 2. The “wake of images” that appears on the opposite side of a conducting plane, caused by the eddy currents that are established by the moving coil above. (a) When the speed of the superconducting coils is low, the previous images move away from the plane (and the real coils) relatively slowly, allowing the eddy currents to produce a force on the coil in the direction opposite its motion. (b) When the speed of the superconducting coils is high, the images move away relatively quickly, allowing the lifting force to approach the ideal image force. (Reproduced from Rossing.)

As the train increases speed, the distance between successive images appears to increase, until the point at which the lifting force is best approximated by the repulsive force that would be produced if an exact image of the superconducting coils existed an equal distance beneath the tracks. This force is called the ideal image force, and is essentially the limiting value of the lifting force on the moving object.

In order to quantify the lifting force, I attempted to use the following equation, given in Jayawant:

(1) $\begin{equation*} F_{L}=F_{I}\left(1-\left(1-\frac{v^{2}}{\omega^{2}}\right)^{-n}\right)\right) \end{equation*}$

which relates the ratio of the lifting force $F_{L}$ and the ideal image force $F_{I}$ to the speed $v$ of the moving coils (speed of the train) and the recession speed $\omega$ of the coil’s images inside the conducting plane, and $n$ is related to the dimensions of the coil. However, when graphing this function (the ratio of $F_{L}$ to $F_{I}$ in Mathematica, the output did not seem to model the behavior that I predicted; the function did not increase steadily with increasing speed, nor did it asymptotically approach a value of 1.0. Thus, I followed an assumption suggested by Kraftmakher, and replaced this equation for the lifting force on a rectangular coil with the simplified equation for the lifting force on a magnetic dipole, given by Reitz as:

(2) $\begin{equation*} F_{L}=\frac{3\mu_{0}m^{2}}{32\pi(z_{0})^{4}}\left(1-\frac{\omega}{\sqrt{v^{2}+\omega^{2}}}\right) \end{equation*}$

where $m$ is the magnetic dipole moment of the train’s coils, $z_{0}$ is the height of the train’s coils above the conducting plate (track coils), $v$ is the speed of the train, and $\omega$ is the recession speed of the coil’s images. It is clear that in the limiting case where $v>>\omega$ , the lifting force approaches the ideal image force for the dipole configuration, given as the constant out front in the right side of the equation. With this relationship, I will be able to approximate what the lifting and drag forces would be on a coil of superconducting wire inside the moving train.

Modeling and Experimental Tools with Prof. Magnes

Ideal for getting ideas and feedback on upper level physics tools.