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On a planet, a species uses magnets (i.e. float in the air) instead of wheels for transportation. Roads are easy enough, just slap some magnets in it, at it works.
But I was wondering about the feasibility of having transportation which uses the magnetic field of the planet to hover above the ground, or off-road transportation. Is it even remotely possible, and if so, how strong will the magnets need to be? The magnetic field is the same strength as Earth's.
The easy answer is yes. We use one object that moves by the Earth's magnetic field all the time: a compass.
The hard answer is that we don't have the technology to move anything heavier than a pin stuck in a cork floating on water ... much less levitate a truck load of dirt! BUT, don't let that stop you. The concept has been dreamt about for a very long time and it's only a matter of time before we figure out how to make a strong enough counteractive magnetic field that will work.
Your limitations are control. The Earth's magnetic field is constantly in flux and shifting, which is a lot like blowing a small hover craft around with wind. Have you ever steered a hover craft or watched one being steered? They don't turn very readily, and neither would a magnetically levitated craft. Obviously, with enough Clarkian Magic we could take advantage of magnetic shear to navigate. It'll just take us some time to get there.
As for how strong the magnet would need to be. It would not need to be as strong as the Earth's. If you think about it, an object the size of a VW Bug emitting a magnetic flux equal to the Earth's would have catastrophic effects on navigation on the other side of the planet. It would need to be a fraction of that strength.
Think of it this way, Wiki reports that the Earth's magnetosphere is around 50,000 nT (nanoTeslas) while a refrigerator magnet is about 10,000,000 nT. That sounds like it makes me a liar, but remember the Earth's radius is about 3,959 miles and the radius of the refrigerator magnet is maybe 0.375 inches. If you shrunk the Earth's magnetic field to the area of the magnet it would be about 3.5x106 stronger than the refrigerator magnet (which explains why the refrigerator magnet can't levitate).
However, Don't worry about the details. There's enough plausibility to make for a good story.
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.
(the bolded sentences above show why the fact that the Earth's field cause a compass needle to rotate, which JBH's answer pointed to, is not sufficient to show it could equally well cause a magnet to levitate, or exert any net force on it causing its center of mass to accelerate in some direction for travel)
The short answer is no, this is a practical impossibility, but not only because of the earth's magnetic field per se; it's because of the Earth's gravity. Let me break this question down into a few key areas to discuss the matter in detail;
Gravity creates Friction
The reason why wheels are so efficient in terms of propulsion is that they get an assist from gravity. The weight of a car (for example) pushes the tyres down onto the road, resulting in tension, which results in friction. The reason why we use a rubber tyre and grease around the axle is that we want to minimise the friction between the axle and its casing and transfer all the angular force to the wheel. The rubber on the tyre creates so much friction that the only way the angular force can apply is for the wheel to roll rather than spin. This converts the angular force to forward force, pushing the vehicle forward.
This means that the car uses the energy it creates very efficiently to move from its current position to where you want it to go, and gravity is actually helping this rather than the enemy. This is also why cars spin their wheels when bogged in slippery mud or when the energy being applied to the wheels overwhelms the friction applied by the tyre. This in essence reflects the whole debate around power to weight ratio in cars, and also explains why many professional racing cars have such wide tyres (to maximise friction).
But (I hear you say) that only benefits lateral movement, right? It's not true for rising above the earth's surface, right?
Wrong. This is the whole debate around space elevators. Right now, we use rockets to get things out of the Earth's gravitational pull. The problem with that is that the gravitational pull is constant, meaning that thanks to something called the rocket equation, you need massive amounts of fuel to get your payload into space because you don't only have to lift your payload, but the fuel to lift it as well. On the other hand, by gripping some form of very tall, super strong rope tightly, you can use friction to climb out of the gravity well. If you use a counterweight on your elevator car, you can even do so with minimal energy cost for the climb. Either way, if we had the right material science advancements, space elevators can use friction to get stuff out of the gravity well at a far lower cost than getting it to 'float' upwards at a great rate of knots.
Orientation & Lateral Movement
As has already been discussed in the compass example, magnetic fields are good for alignment. That is to say, if you already have something floating freely, the earth's magnetic field will automatically align the magnet to said magnetic field. This means that to steer, you literally need to dynamically adjust the alignment of your magnetic polarity to the direction you want to go. This could be done through friction as well, using a de facto steering wheel to turn a strong magnet under your vehicle, so that fixed thrusters can move you along.
This of course brings us to sunny point number 2 - lateral movement. You still need some way to get your vehicle moving in a given direction, without contact to the ground. On an aircraft or hovercraft, this is usually done with a fan style thruster. Even most modern jet engines are really turbofan engines, capable of great amounts of thrust by pushing air backwards very fast.
It's also why you can't use a hoverboard over water, as we all learned in Back to the Future II. This is not exactly true of course; if you had an oar, you could move along very quickly because the oar maximises the friction between itself and the water, and with NO friction on the hoverboard and the water, in theory you move forward. I say in theory because in a strong headwind, you might be fighting a losing battle with forward movement. But, I digress.
Magnetic Intensity over Distance
Maglev trains (for example) can float a train above the magnetic rail because the magnets are strong, and they're also close together. The earth's magnetic field is strong to be sure, but we're actually reasonably far away from the 'magnet', being the inner core of the Earth. The intensity of the magnetic field at distance is inversely proportional to the square of the distance, meaning that the intensity reduces greatly the further you move away from it. That means to get that same train to float off the track, you need magnets in it that are so intense they can counteract gravity against the earth's inner core, rather than the earth's surface. If you design such a magnetic field, you could use it to melt pidgeons or exsanguinate cows by ripping the haemoglobin directly out them if they stray close enough.
Bottom line is you're better with two magnets interacting in close proximity to maximise the efficiency of your levitation.
Failing Safely
We have to assume here that you're using electromagnets to do all this. Given that gravity is always on, so too do the electromagnets have to be. If they aren't, your vehicle fails badly by collapsing back to the ground. Your electromagnets have to constantly expend energy to counteract gravity.
A car on the other hand simply stops. This is because it's spending all its energy moving forward, not counteracting gravity. Not having to do that, actually using gravity to maximise friction, makes a wheel based vehicle a far more efficient user of kinetic energy than a magnetically levitating vehicle for that reason alone, but it also means that it is by far the safest solution in terms of what happens during an engineering or power failure.
Conclusion
Mark Twain once said 'Thunder is loud. Thunder is impressive. But it's lightning that does all the real work.' Well, mag-lev transportation looks very impressive to be sure but it's friction that is going to do all the real work in terms of transport systems in a real world application. Wheels may sound very 'old tech' but the reason we're still using them is they are still the best and most efficient energy solution for moving people and stuff around. The biggest problems that you have with the model you describe are that you're constantly fighting gravity by reacting to a distant magnet, and you still need mechanisms to propel yourself through the air laterally if you get your craft floating.
Star wars may have worked very hard to make the wheel look unfashionable, but I'm still of the view that it's a very impressive tool.
The Lorentz Force Equation shows how the Earth's magnetic field can be used for transportation $$F = q\vec{E}+q\vec{v} \times \vec{B}$$ where q is electrical charge in coulombs E is the vector of an Electric field (if present) v is the velocity of the charge carriers B is the vector of the Magnetic field
The electric field (or difference in potential aka Voltage) can be considered 0 in the atmosphere of a planet -- unless there is a lightning storm -- so assume the first time drops out leaving only the second term.
To understand the physical significance of the second term look at the image below
In physics, the relationship between the force created by moving charge carriers in a magnetic field follows the right-hand rule. The graphic shows the necessary vectors of $\vec{B}$ and $q\vec{v}$ to produce a force $\vec{F}$ out of the page
The Earth's magnetic field runs north to south parallel to the surface. To understand the proper orientation of components to levitate an object rotate the thumb in the first visual or the palm in the second 90 degrees so they face the top of the page.
The curl of the fingers shows that you need to create a spinning or rotating current to produce a lifting force to counteract the gravitational force. And, since these are vector quantities, forces in the horizontal plane can be created by pitching and rolling the vector $\vec{v}$ a little bit -- most of the force is lift while some is in the direction of travel (like a helicopter)
Consider a more pragmatic form of the Lorentz force equation for your case. $$F = I\int d\ell \times \vec{B}$$ where I is the current (in Amps) traveling over length $d\ell$ in the magnetic field $\vec{B}$
This tells us we need I=200 KAmps per unit length to counteract the 9.8 $\frac{kg*m}{sec^2}$ the force of Earths Gravity.
So one can imagine a pancake looking creature that circulates positively and negatively charged fluids around the interior of its body with great velocity. Positive and negative ions will create forces in opposite directions if they move in the same direction. So moving them in opposite directions means you only have to move half the current. The I term can be a small number of ions moving unbelievably fast or it can be an unbelievable number of ions moving slowly or it can be a lot of ions moving very very fast.
Not a researcher of any note, but suppose the inhabitants of the planet had no ferrous metal in their bodies? This could allow them to use obscenely powerful magnets of any sort, which would be in existence by the electromagnetic technology that we can't bother going into without significant danger.
Or suppose their bodies are largely ferrous and lightweight, with an appearance reminiscent of a jellyfish and capable of blanketing themselves over a large area to fly with biologically-charged electromagnets.
