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.


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