Flying vehicles powered by superconducting magnets pushing off the Earth's magnetic field

Question

Maglev trains are fun, but who wants to be stuck on a track? Dr. Acme had the brilliant¹ idea of ditching the track and simply using the Earth's magnetic field as the base that would repel his vehicle, allowing free-flight across the earth! As his head assistant, he has naturally left it to me do the easy part and work out the details, now that he did the hard part and came up with the idea.

Of course the Earth's magnetic field is tiny, so the magnets in our vehicle need to be dialed up to 12. Presumably a superconducting electromagnet will be required to reach the necessary magnetic field strengths. The initial plan involved one large, spin stabilized, superconducting electro-magnet in the center of a saucer-like vehicle. The lack of redundancy of such an arrangement made me wary, but further research has suggested that spin stabilization would not be an option at arbitrary points on the Earth's surface anyway, so I'm working on a new design.

The only other option I can think of is using servomechanics, aka active balancing via sensors and automatic computer adjustment. Dr. Acme assured me that servomechanics will do the job. As a result I'm just left figuring out if the necessary field strengths are even physically plausible. Presumably this will require lots of power. I obviously can't carry around nuclear reactors, but we have some handy Acme^TM Power Plants that can be miniaturized and provide ridiculous amounts of energy as needed.

Of course, this is all pointless if the required magnetic field strength is so high as to be impossible to generate with modern technology, or would even require such unbelievably high strengths as to be effectively impossible to generate for other reasons (requires a magnet so strong it rips its own atoms apart?). To help give some specific numbers, I'm looking to build a typical "car"-sized vehicle: let's say I need to lift 4 tons total, and I'm assuming the weight of the magnets/power plant will be at most half of that.

Since the Earth's magnetic field is a dipole, I presume it grows weaker with distance from Earth faster than gravity, which means arbitrary altitudes may not be possible. Being able to operate up to thousands of feet above sea-level is a requirement, since there are plenty of places in the world where the ground is higher than that. Being able to operate up to or past LEO would be better (ignoring issues related to the need for maintaining a survivable atmosphere). To recap:

Ignoring energy needs, would it be possible to use powerful electromagnets to self-levitate a 4 ton vehicle using the Earth's magnetic field as a base?

Breaking that question down a bit farther:

1. Are the required magnetic field strengths even remotely possible with today's technology?
2. If not, are the magnetic field strengths even physically possible?
3. If such magnets could be built, would it be feasible to use servomechanics/active adjustment to stabilize the system?

¹As brilliant as all his other ideas, anyway.

Answer 1

Magnetic levitation depends not just on the value of the external magnetic field B at a given point, but on the gradient of the magnetic field ∇B, which tells you how quickly the field changes as you move in space. This page from a lab at Radboud University gives the equation for when magnetic levitation can happen:

Whether an object will or will not levitate in a magnetic field B is defined by the balance between the magnetic force F = M∇B and gravity mg = ρV g where ρ is the material density, V is the volume and g = 9.8m/s^2. The magnetic moment M = (χ/µ_0)VB so that F = (χ/µ_0)BV∇B = (χ/2µ_0)V∇B^2. Therefore, the vertical field gradient ∇B^2 required for levitation has to be larger than 2µ0ρg/χ. Molecular susceptibilities χ are typically 10^-5 for diamagnetics and 10^-3 for paramagnetic materials and, since ρ is most often a few g/cm^3, their magnetic levitation requires field gradients ~1000 and 10 T^2/m, respectively.

This lab is concerned with diagmetic levitation of ordinary materials (they show the levitation of a small frog on their site), but the external field gradient doesn't have to be as large if you are levitating a superconductor, which has its magnetic susceptibility χ=1, the highest possible for a diamagnet (diamagnetic materials are repelled by the field of an external magnet, unlike paramagnetic and ferromagnetic materials, so diamagnets are the ones you want for magnetic levitation).

But even with a superconductor, the Earth's magnetic field is so large and therefore changes so little over ordinary human-scale distances that it wouldn't work for magnetic levitation, you would need a huge magnet of similar size to the Earth itself to levitate from the Earth's field. This is explained by a physicist on this page:

The force on an object is related to the change in the energy of a system (not including the kinetic or thermal energy of the object) when the object is moved. We write

F = (change in Energy)/(change in position)

For static fields. The change in position has a direction, and so the force does too (you need some vector algebra with a dot product to express this exactly).

Two small magnets placed together with like poles close to each other feel a repulsive force because of the energy stored in the magnetic field. The energy density in space is proportional to the magnetic field squared, and when the close-by poles are the same, their fields add in more places than they subtract, and so the total energy is higher for this case than when opposite poles are closer, where the field is smaller in more places.

There are two things about the Earth’s magnetic field which makes this effect much smaller. For one, the field is very weak at the surface (about a gauss or less). The more important reason is that because the field extends over such a large space and because we on the surface are far away from the center of the Earth’s dipole, the Earth’s magnetic field strength is very uniform if you look at it over a region of space that is reasonable in size (like the size of the magnet you propose to use).

If you put these two pieces together, you find that the force on a magnet due to the Earth’s field is very small -- if you move the magnet from one place to another, its field adds to the Earth’s field in almost the same way because the Earth’s field is very little different from one place to another, and the total magnetic energy changes by a very very tiny amount. In fact, the total magnetic force on a magnet in a uniform magnetic field is exactly zero, and the forces we normally associate with magnets repelling or attracting are proportional to the rate of change of the field strength with position.

This isn’t the end of the story, however, because the magnetic energy of the system depends on which way the magnet is pointing, relative to the Earth’s field. If it points along the field, the fields add, for a higher energy. If it points the other way, the fields subtract, for a lower energy, and so the magnet prefers to turn to point in this way. Magnets in uniform fields feel torques which make them turn around if they are not pointing in the right direction, but there is no net force making the magnet want to levitate.

That having been said, if you had a really really big magnet, whose field extended over such a large region that the Earth’s field changes noticeably over that region (you might need another Earth-sized bar magnet), then yes, a noticeable force can be produced.

As for actual levitation, that can only happen with materials whose magnetic moment actually points the wrong way, increasing the energy in a magnetic field. These are called diamagnets. Diamagnetism is purely a quantum mechanical effect, with no classical explanation. By far the most intense diamagnets are superconductors. You may have seen superconductors levitating over magnets, or vice-versa. The Earth’s magnetic field does not change rapidly enough from place to place to levitate even a superconductor.

Answer 2

No.

The problem is, the strength of the magnet on the vehicle isn't the limit. Magnetic levitation operates by excluding a magnetic field.

https://en.wikipedia.org/wiki/Magnetic_levitation

The magnetic pressure is proportional to the square of the magnetic field. The Earths' field is 25 to 65 micro-Tesla.

https://en.wikipedia.org/wiki/Earth%27s_magnetic_field

So even with superconductors on your vehicle you would only be able to get a tiny force per square meter. You could never lift the superconductors, quite apart from any vehicle.