Magnetic levitated tower - magnetic fields and tensile strengths of materials

Question

I was considering a system made from a stack of electromagnetically levitated platforms and wondering how you would go about comparing its tensile (and or compressive) strength to that of a solid material.

There is a similar question and answer here but it is scant on details:

Tensile strength of a chain of electromagnets

I'm interested in whether it would be 'practical' to make a structure with a tensile strength similar or better than say steel in this way? or a strength to density ratio better than a conventional material.

I think this is the same question as asking what strength of magnetic field would be required to tear a material apart.

On that basis. For a given platform size what stength of field is would be required to create a 'joint' at least as strong as if a solid steel bar was used instead.

What current would be needed to power the coil? I'm presuming it would have a 'base' level and increase current draw as strain increased up to some maximum beyond which something fails - the coil burns out or we cut the power drop the platform.

Is such a field readily achievable? I think we can generate a few 10s of Telsas in NMR scanners.

Assuming our best current efficiency how much power would such a device consume per platform at maximum stress?

My thinking so far

So for this thought experiment consider a tensometer (or other device used to evaluate the tensile strength of a material (as in https://en.wikipedia.org/wiki/Tensile_testing) but which does so using electromagnetic repulsion rather than a more common mechanical or hydraulic systrem.

For the tensometer the magnetic field could be distributed in the area around the sample so I think it is a workable design if not efficient.

Returning to the stack of platforms the same force would have to be concentrated into the area of the sample. I wonder if it might need to be so strong it would tear apart the materials that would be used to construct the platforms.

The power requirements would presumably be horrendous too.

I'd like to see an answer of this quality https://engineering.stackexchange.com/a/37923/19065

I wonder also if you usefully make an existing structure weaker or stronger using such a mechanism. Think "polarize the hull plating".

I am assuming that each platform is an active component. They either generate power themselves or have power fed to them by wire or induction from the fields being used to support them. They stabilize the platform above dynamically as in any number of examples on the net (e.g https://www.youtube.com/results?search_query=magnetic+levitation)

Note: the mechanism is allowed to use diamagnetism (ie. superconductors). It need not rely entirely on ferrous electromagnets.

Further I'm assuming the design aims to constrain the fields as far as possible so that energy is focused on levitation via shielding and coupling effects.

Shielding is also necessary so that anything at the top or in an elevator to reach it (e.g. a person) would not be fried.

Thinking this way would it be possible to construct something that has a greater strength to weight ratio than the equivalent solid material would have.

Thinking about practical (though not necessarily sensible) applications consider:

• A magnetic levitating telescopic ladder
• A magnetic levitating car jack

These of course do not need the tensile strength of steel they only need to support the weight of a person or a car so getting a bit more sci-fi consider:

• A tripod
• A cloud city
• A space elevator - taking this idea to the extreme (I believe the current estimate is we would need a tensile strength of around 150GPa for such a structure)

I'm thinking that we need to derive something like magnetic pressure to match the units used for tensile strengths of materials. That page has one equation for force an a coil with resistance in the denominator. For a superconductor that would tend to zero, so I assume that must break down somewhere or be otherwise inapplicable. The other equation giving the pressure as proportional to the square of the field sounds more applicable.

A major practical obstacle is that magnetic fields follow a cube law rather than a square law so even more power is required to produce a strong one.

This question is relevant:

This How strong can an electromagnet be?

The answers reference the strongest eletromagnets so far produced as being from bitter electromagnets and the highest man made field being 45T at enormous cost (liquid helium) and 30MW of power. Noting that the wikipedia page links to some very old archived pages I googled and things have improved slightly a much smaller superconducting device gets 45.5T at only 18MW.

Answer 1

What you want to consider is magnetic pressure. At 1 T the magnetic pressure is approximately 4 atmospheres, and it grows with the square of the field.

Your question makes it hard to see exactly what design you are considering, but if you have a tower of mass $M$ resting on top of a coil repelling it successfully it needs to exert a force of $F=Mg$, or pressure $MG/A$ where $A$ is the cross-section area. Borrowing the formula for electromagnet interaction $F=\mu_0 m_1 m_2/4\pi r^2$ where $m=nIA/L$ is the pole strength (and $I$ is current, $n$ number of windings, $L$ the length). Assuming identical coils, putting it together we get $$M = \frac{\mu_0 n^2 I^2 A^2}{ 4 \pi g L^2 r^2}.$$ Looks great, we can just boost $I$ or $n$ to lift any $M$.

Except for the magnetic pressure. Radius $R$ coils will experience a force due to their own current outwards of $$F=\frac{I^2}{c^2 R}\left[\ln\left(\frac{8R}{a}\right)-1+Y\right]$$ where $Y$ is internal conductance of the could (0 to 0.25) and $a$ the cross-section of the wires (this is per 1 winding; multiply by $n$ for the full force). The thing in the brackets is roughly order unity for our purposes. So when $I$ increases the force pushing the coil outwards grow with the square of the current. Eventually it will become stronger than anything you can make.

What is the ultimate tensile strength we can get? Steel goes up to about 2500 MPa, carbon fibre 7000 MPa, diamond 2800 MPa (80-90 GPa if nanoscale perfect). Instead of trying to patch together all equations, let's use the top approximation: 1 m tall cylinders of steel can handle about 78.5 T, carbon fibres 131 T, and diamond at the extreme 457 T (but in practice much less). Conversely, that pressure can hold up 254600 ton/m$^2$, 713000 ton/m$^3$ or 8.6 million tons...

... if everything is perfect, which it will not be. Note that any break will suddenly make the tower have zero resistance to collapse, and if the coil gives a bit $R$ increases, the force goes down and there are current surges that are likely to heat things up badly. An electromagnetic tower is a magnificent disaster waiting to happen if it is heavily loaded.