Hoverboard Math

Disclaimer: Everything here might be wrong. Feel free to correct me if you believe it so.

Let’s say you wanted to create a solenoid strong enough to levitate at the poles of the Earth’s magnetic field, and thus, the first skate truck for a true hoverboard. How practical would that be?

First, let’s use Lenz’s Law as an approximation for the solenoid. We are able to treat our solenoid as a long wire, as the magnetic field is going to be uniform in X, Y, and Z for the recommended use case of a hoverboard.

Force = Current * Length x B-Field

Assuming we want to levitate a 1kg, 100m long solenoid coil in the uniform earth magnetic field…

F = 9.8 Newtons
L = 100 meters
B = 50 uTesla

(9.8 N) = I * (100 m) * (0.00005 T)

I = 1960 Amperes

Thus we need a current of 1960A running in our solenoid to generate this field.

Naturally, this is impossible with copper wire. However, plasma is extremely conductive, so perhaps we could create a long glass tube using a fiber-optic manufacturing process. Then we must ask, what are the resistive losses in a coil of 1keV plasma?

Let’s use a simplified version of Spitzer Resistivity to find an approximate specific resistance:

n = 5.15 * 10^(-5) * (0.993 * 10) / ((1000)^(3/2))

n = 1.617 * 10^-8 ohm/m

Then use ohm’s law to find the coil’s losses:

P = I^2 * R

P = (1960 A)^2 * (1.617 * 10^-8 ohm/m) * (100 m)

P = 6.212 W

Despite the high current, our resisitve losses for this plasma are relatively low at only 6.2 Watts!

Now, how much plasma really is this? We can get an approximation by assuming a unit charge per molecule. A current of one Ampere is equal to 1 coulomb per second, which is approximately 6.25 x 10^18 elemental charges passing by per second. We can use Avogadro’s number to make sense of this.

(1960 A) * (6.25 x 10^18  charges/coulomb) = (1.55 * 10^22 charges/s)

1.55 * 10^22 / Avagadro’s Number = 0.02034 mols

Assuming our plasma is Xenon with a molar mass of 131 grams we find, that’s only 2.6 grams of xenon.

2.6 g == 0.02034 mol * 131 g / mol

At standard temperature and pressure, that’s less than 1 liter of xenon.

Seems like not too much?

Questions for the reader:

What other losses are not taken into account?

What is the power needed to maintain the 1000eV plasma temperature in the 100 meter tube?

What would the mass of a 100 meter glass tube be, to maintain the structural integrity needed to contain the plasma?

Responses are welcome :-)