A few months ago I designed a science project to present it to a contest. I did not win, but I think it turn out quite good, so I thought it would be a good idea to share it here. Although this post explains more some of the calculations, I recommend watching the youtube video I made or having a look at my paper too, because the formulas are a lot easier to understand, since they are not written as simple text and I know that some of the formulas in this post can be dificult to understand because of the way they are written.
Step 1: Concept
Rocket launching is complicated and very expensive, so in this project I try to present a different approach, that uses electromagnetic forces to accelerate a capsule. In theory, this shuttle could easily and cheaply put objects into Low Earth Orbit (LEO), reducing costs and pollution.
Step 2: Functioning
The capsule is made out of Neodymium magnets N52, which have a magnetic dipole moment of approximately 7,35 A*m² per each m³. The structure, as shown in the picture above, consists of a big magnet, which moment is aligned with the external field, and two Hallbach Arrays on each side, with magnetic moments perpendicular to the external field. The magnets are arranged in a Hallbach Array because it maximizes the magnetic field in one direction, almost cancelling it on the other. If you want to know more about them you can read the Wikipidea article about the subject: https://en.wikipedia.org/wiki/Halbach_array (opens in a new tab).
The capsule will be surrounded by a horizontal tube made out of aluminium, and as it moves, a variable flux will go through the aluminium, inducing a current. Once the capsule is moving quickly enough, the forces between the magnetic field of the Halbach Arrays and the induced one will make it levitate. I will be referencing some papers about it at the end, as it is not the focus of this study. The tube has solenoids on it that cover its surface whit 0.5 meters between them, and they will make the capsule accelerate. The idea behind that is that the capsule will be attracted to the centre of every solenoid, and once it reaches it, the solenoid will turn off to avoid the capsule getting stuck on it. Using many solenoids instead of only one maximizes the force, as it will be explained in the study section.
Step 3: Study
The magnetic force over a magnet is given by the expression
F⃗ =∇ ( m·B⃗ )
being m the magnetic dipole moment of the magnet and B the external magnetic field, both vectors (I don't know how to write the little arrow above the vectors here, if you know tell me in the comments). From that expression we get
F=∇ (m∗B∗cos θ)
and having in mind that θ=0º and m=cte, we get
With this we can calculate the average force per solenoid quite easily.
F̄= m*ΔB/x → F̄=m*(B2-B1)/x
B2 is the magnetic field at the centre of the solenoid, and B1 the field at the solenoid's edge, and we know (experimentally) that the field at the edges of the solenoid is half of that in the centre of the solenoid.
L is the lenght of the solenoid.
Magnetic field inside of a solenoid is given by:
B = (μIN)/L
μ: magnetic permeability; I: current that goes through the solenoid; N/L: number of turns per lenght, that I will be writting b.
Writting the force using the expression for the magnetic field inside the solenoid we get:
Looking at this expression we see that the force will be bigger if the current or the number of turns per unit of length increase, and smaller if the total length increases. This is the reason why it is better to use many small solenoids instead of only a big one.
*note: it is important to remember that in these equations, m is magnetic dipole moment, not mass, but in the launch speed study it will represent mass as usual.
Step 4: Launch Speed
If we want to put the capsule on LEO (Low Earth Orbit), on a height of 1000 Km, then the launching speed is calculated:
Em1 = Em2 (because gravitational field is conservative)
Ek1 + Ep1 = Ek2 + Ep2 (mechanical energy is equal to kinetic energy plus potential energy)
(m*v0²)/2 - GMm/r0 = m*vh² - GMm/(r0+h)
Here we write the expression of kinetic energy and potential energy. v0 is the velocity that we give to the shuttle; G is the gravitational constant;r0is Earth radius; M is Earth mass; vh is the velocity in an orbit at height h over Earth surface.
v0²/2 - GM/r0 = vh² - GM/(r0+h) → v0²/2 - GM/r0 = GM/2(r0+h)-GM/(r0+h)
Here we used the orbit speed formula: v = sqrt(GM/(r0+h))
v0²/2 = Gm/r0 - GM/2(r0+h) → v0² = GM(2/r0-1/(r0+h)) → v0 = sqrt(GM(2/r0 - 1/(r0+h)))
putting the values for r0, M, h and G, and doing the math:
v0 = 8.43*10³ m/s
Step 5: Shuttle Specifications
In this section, length of the shuttle, current and other values will be defined. The shuttle should not be too long, it should be less than 150 Km long, so we will assign the values having that in mind.
F = (mμIb)/L
assigning the values:
I = 4000 A
L = 1 m
b = 100 turns/m
μ = 4π*10⁻⁷ NA⁻²
Vacceleration magnet = 1 m³
Vtotal = 3 m³
ρNeodymium = 7.4* 10³ Kg/m³ →mass = 2.22* 10⁴ Kg
Neodymium dipole moment = 7.35*10⁶ Am²V → m = 7.35*10⁶
We can now calculate the lenght of the shuttle
F=(7.35*10⁶*4π*10⁻⁷*4000*100)/1→F = 3.70*10⁶ N
acceleration = Force/mass →a = 3.70*10⁶/2..22*10⁴ →a = 166.67 m/s²
x= x0 + v0*t + (a*t²)/2→x=(166.67*50.58²)/2→x = 85.28 Km
As each solenoid is separated from the others 0.5 m, Xtotal = 127.92 Km
Step 6: Conclusion
I hope that you find this interesting, and I apologize if my English is not perfect, as it is not my first language. If you have any question, piece of advice or correction, do not doubt to write it in the comments.
Step 7: Bibliography and References
·Post, Richard F.; Dyutov, Dmitri D.; Lawrence Livermore National Laboratory, Livermore, CA (March 2000). "TheInductrack: A Simpler Approach to Magnetic Levitation" (PDF). IEEE Transactions on Applied Superconductivity. Volume 10 (n. 1): 901-904. doi:10.1109/77.828377.
·Fernández Fernández, Vicente. “Física” Baía Edicións. 2009. A Coruña. (in Galician) ISBN: 978-84-92630-11-0. ·Catala de Alemany, Joaquín. “Física general” Fundación García Muñoz. Valencia. Isbn: 84-400-3538-1 (in Spanish)