Calculating the limits of magnetic acceleration

[Ariphaos]

This is something that's been bothering me for awhile now, as Wikipedia is a bit self-contradictory on the subject, there are a few places where units magically disappear and I'm not really comfortable with that.

So here is my question - given a constant magnetic field of a certain strength (say, 2.5 Tesla), acting on 1 kg of a ferromagnetic material (say, iron), what would the net acceleration be, or if that question is posed completely wrong, what is missing so I can complete it?

I'm trying to calculate the practical limits of a coilgun, if you are curious. If someone's already looked into this just point me to a link and I apologize.

[Enola Straight]

I'd say...just shy of the speed of light.

Particle accellerators use magnetic acceleration to punch up particles to high relativistic velocities all the time.

A coilgun evacuated to high vaccuum would make an excellent mass accelerator.

[Enola Straight]

I'd say...just shy of the speed of light.

Particle accellerators use magnetic acceleration to punch up particles to high relativistic velocities all the time.

A coilgun evacuated to high vaccuum would make an excellent mass accelerator.

Amendment: you didn't say how long your coilgun was.

A long enough gun ( kilometers long) should kick up the 1k iron slug to several times the speed of sound.

[Wyrm]

I'd say...just shy of the speed of light.

Speed ≠ acceleration. Different dimensions.

Particle accellerators use magnetic acceleration to punch up particles to high relativistic velocities all the time.

A coilgun evacuated to high vaccuum would make an excellent mass accelerator.

All true, but doesn't answer the question. After all, if acceleration is unlimited, then no matter what length of the coilgun, you can accelerate the mass to any speed you want.

The answer, I suspect, will have to do with how much current you have available, and the saturation of the slug.

I've never seen any equations relating the strength of a magnet and the force it induces on magnetic objects, or a magnetic potential equation to the same effect. I'm sure someone (gooses Mike) has some figures handy.

[Wyrm]

Obviously, I'm not qualified to say what the consequences of all this is, and I don't know how real particle accelerators do it (other than the fact they are huge rings rather than linear ones), but it might give you another pointer to help in your research.

Since the driving forces on subatomic particles can be made much greater than they can on macroscopic objects for the availible current, acceleration of the beam can be accomplished with a short 'injection' accelerator, with acceleration elements specifically tuned to the particles accelerated. The apparatus works on automatic with simple RF power.

Once you reach nearly the speed of light, the synchronity problem effectively disappears, since added energy doesn't go into making the particle go faster so much as the particle gets heavier. The reason why synchrotrons have such large rings is because bending a beam of many MeV or GeV is crazy hard; the less bending you have to do, the better, and larger rings bend less over a given distance.

[Kanastrous]

Does it matter what the specific material is, that's used as a projectile?

The lack of specificity in the OP led me to wonder. And I remember reading about a USAF test rail that was being used to sling plastic slugs...

[Wyrm]

Does it matter what the specific material is, that's used as a projectile?

Yes. Try to pick up a penny (or other copper coin) with a magnet to see why.

The lack of specificity in the OP led me to wonder. And I remember reading about a USAF test rail that was being used to sling plastic slugs...

The "rail" part here is crucial, as it indicates the test rig is a railgun and not a coilgun. In a railgun, the acceleration is provided by Ampère's force law (the projectile is part of the circuit, which means that the "plastic" slug must have some conductor in it, perhaps in a sabot). For a coilgun, it's the force on a ferromagnetic material.

[starslayer]

Force due to a magnetic field is given by the Lorentz equation for a charged particle, and this formula for our "wire": I(l*B). An iron bar sitting in a uniform B field will not accelerate once it has rotated to point in the direction of the field. With a coilgun, the projectile must be placed outside the coil before the current is switched on, and it must then be quickly switched off so that the projectile does not simply come to rest inside the coil. Remember also that magnetic fields only accelerate charged particles. When a coilgun is switched on, the quick change in magnetic flux induces a current in the bar (Faraday's Law), creating a possibly very large magentic field. These two fields then atrract each other and accelerate the iron bar. The current is then switched off once the bar has reached the coil's midpoint for maximum acceleration. This creates another quick change in magnetic flux through the bar, creating a field that repels the coil. This provides more acceleration, and the iron bar flies out. If this explanation is wrong, someone with some more knowledge please correct me. I hope that helps.

[Ariphaos]

Wyrm and starslayer have the right idea about what I'm trying to ask, thank you.

So the limit of acceleration is actually dependent on the amount of current as well, not just the raw strength of the field? I see that in some equations.

I know that iron reaches magnetic saturation somewhere between 1.6 and 2.2 tesla, depending on who you ask (it actually looks like a logarithmic curve?), but if it's dependent on current, that seems to be a way to cheat, unless something I'm not understanding is limiting the current.

[starslayer]

I should explain the formula I posted above. I is the current through the wire, l is vector with magnitude equal to the length of the portion of the wire in the B-field, and pointing in the direction of the current, and B is the magnetic field itself.

Any current will generate a magnetic field, and any changing magnetic field will generate a current in a conductor. So yes, the acceleration is dependent on the current, because that is what creates the B-field in the first place. For a coilgun, you wouldn't want to run a current through your projectile directly, as this would just make it a railgun, and I believe a hybrid design is unworkable (you would have to be extremely careful about which way your current is flowing, where all your B-fields are pointing, etc.).

I'm going to have correct one of my statements earlier. B-fields do not accelerate charged particles. In fact, B-fields can't do any work. With "magnetic attraction," it's as I explained earlier: E-fields are what is accelerating everything, becuase of induced currents. There also isn't really a "limit" to acceleration as you seem to think; the acceleration of a given object being subjected to B- and E-fields is due to the mass of the object and the strength of those E- and B-fields. There will only be one value for the acceleration for any given system. The maximum practical acceleration for any given coil system is going to be limited by how much current you force through the coil before it melts, and the how low the mass of your projectile is.

[Wyrm]

When a coilgun is switched on, the quick change in magnetic flux induces a current in the bar (Faraday's Law), creating a possibly very large magentic field. These two fields then atrract each other and accelerate the iron bar.

starslayer's description is a little inacurate. The current induced by a changing magnetic field is always such that the magnetic field induced by the current repels the inducing current. (Lenz's Law) This is a result of the conservation of energy.

For the source of the attraction, we must look to the fact that iron is ferromagnetic. When the iron bar is exposed to the coil's magnetic field, the magnetic domains within the bar align with the coil's field, so the bar itself becomes a small magnet. This reaction magnetic field is such that the field lines cross the coil's wires in such a way that there is a force that is towards the magnet. (Ampère's Law) Of course, Newton's third law states that an action force requires a reaction force, which is the force on the iron bar. Attraction.

Now, what if all the magnetic domains are as aligned as they can be with the magnetic field? Then the iron bar has reached magnetic saturation. The bar cannot possibly be magnitized any further. At this point, increasing the field intensity does not increase the magnetic field in the material. The upshot of this is that the attractive force stops increasing with the square of the current (the magnetic field is proportional to the coil current, so far from saturation, there's a double-whammy with respect to coil-current) and becomes only proportional to the current.

I know that iron reaches magnetic saturation somewhere between 1.6 and 2.2 tesla, depending on who you ask (it actually looks like a logarithmic curve?), but if it's dependent on current, that seems to be a way to cheat, unless something I'm not understanding is limiting the current.

The only way to "cheat" here is to make the projectile itself into a whopping powerful (and very magnetically hard) permanent magnet, or powerful electromagnet. But not really. The acceleration is still proportional to the current in these cases, rather than the square of the current.

[starslayer]

Ghetto edit: Wyrm, you will never such equations for magnets, since B-fields don't do work. This is explained in relativity, where B- and E-fields are the same thing viewed in different reference frames, and it turns out that the B-field ends up simply disappearing.

Upon reading up some on hysteresis and magnetic saturation, the maximum practical acceleration will also depend on the material's saturation limit. So, pumping more current into the coil to jump up the B-field won't work for an iron projectile after the field reaches about 2.2 T. Using an NIB projectile would probably give you the maximum possible acceleration for any given system (a quick Google search unfortunately didn't reveal its magnetic saturation).

[Wyrm]

Ghetto edit: Wyrm, you will never such equations for magnets, since B-fields don't do work. This is explained in relativity, where B- and E-fields are the same thing viewed in different reference frames, and it turns out that the B-field ends up simply disappearing.

Yet magnets attract and repel each other. You can't tell me that engineers haven't come up with some rules of thumb for determining how they move.

See, the thing you miss in your absorbing relativity is that it doesn't matter a whit whether B-fields disappear in some frame of reference or another. In the frames of reference where B-fields do appear, they appear with all the physical reality of E-fields.

[Ariphaos]

I've seen some reference to CoFe alloys reaching magnetic saturation at 2.4 Tesla.

[starslayer]

Wyrm: Conceeded on the points of magnetic attraction and relativity. However, you aren't stating Ampere's law correctly. Ampere's law has to do with magnetic potential, not with magnetic force. Are you thinking of the Lorentz force? Like you, I still can't find any equation relating the force between two magnets. I would imagine it would be like Coulomb's law, except phrased like this: F = (constant cluster) B1*B2/r^2. I have no idea what the constants our front would be.

[Wyrm]

Ampere's law has to do with magnetic potential, not with magnetic force. Are you thinking of the Lorentz force?

Funny that it's called Ampère's force law... Although this law is derivable from the Lorentz force.

Magnetic rules of thumb are probably deep magic that would take an engineer to explain the rules of. As to the form of the rule, probably not as you state it. It would probably come from the displacement and orientation of magnetic dipoles.

[Mr Bean]

Question time
The following is from an STGOD

At 3% efficiency 650J input results in 19.5J output to shoot a 4.5G projectile at 75m/s.

At 5% efficiency 650J input results in 32.5J output to shoot a 4.5G projectile at 125 m/sec.

To shoot a 453.6G (1 pound) projectile at 10 km/s you would need the following:
A = 4.5 * 100.8 (4.5G projectile vs 453.6G projectile)
B = 80 (10000 m/s / 125 m/s)
C = 32.5 (Joules output for 5% efficiency)
D = 20 (Difference between input and output at 5% efficiency)

A * B * C = Output
Output * D = Input

(4.5 * 100.8 ) * (80) * (32.5) = 1,179,360W Output
1,179,360 * 20 = 23,587,200W Input or ~ 23.6Mw

The above math is from a mythical coil-gun that has been constructed, and I'm wondering about what if any effect the "Ampère's force law" would have on said gun.

What I'm trying to figure out, given the amount of power stated, at the efficiency provided(5%) would this coilgun be possible, side note what would it's length be.

Before I checked this thread I would have taken these numbers at face value, but after reading it I have to ask if the numbers are correct, is this amount of current enough to shoot a 1 pound projectile at 10km/sec. And a follow-up how long would said coil-gun be?

Assume material science is to the point that making such a gun is possible. Ignore that factor and concentrate simply physics of the problem.

Thanks in advance.

[Kuroneko]

The full dynamic situation is quite complicated, but if you're just interested in some sort of upper bound on acceleration given a magnetic field, that's a lot simpler. First, the hand-waving. The work to establish a magnetic field is W = (1/8π)Int[ H·B d³x ]; back in SI units, this gives a density of B²/2μ₀. Thus we should expect the force to be bounded by B²A/2μ₀, where A is the cross-section of the projectile.

For a situation with general magnetization, taking a vacuum with magnetic induction B₀ and field intensity H₀ and introducing a material of permeability μ, the difference will be (1/8π)Int[ (1-μ)H·H₀ d³x ] = (1/2)Int[ M·B₀ d³x ], where M is the magnetization. Note the implicit assumption of B = μH, which is not true for ferromagnetic materials, but may be an acceptable approximation in some cases. If the source currents are fixed, the force for displacement x is just F_x = ∂W/∂x.

[Ariphaos]

The full dynamic situation is quite complicated, but if you're just interested in some sort of upper bound on acceleration given a magnetic field, that's a lot simpler. First, the hand-waving. The work to establish a magnetic field is W = (1/8π)Int[ H·B d³x ]; back in SI units, this gives a density of B²/2μ₀. Thus we should expect the force to be bounded by B²A/2μ₀, where A is the cross-section of the projectile.

So, for something that reached saturation at 2.5 Teslas, and a cross-sectional area of one square meter, we get about five meganewtons per square meter?

That's a respectable amount of force, to be sure...

And yes, this is pretty much all I need, thank you very much.

[Kuroneko]

So, for something that reached saturation at 2.5 Teslas, and a cross-sectional area of one square meter, we get about five meganewtons per square meter?

Well, it could be--that depends on the magnetizing field. If it matches the induction field by the same permeability factor as the surrounding medium, the force should vanish completely. The above expression is an upper limit in the idealization of constant source currents with projectile far from saturation and high permeability. Let's refine the bound a bit. In the same situation as before,
W = (1/8π)Int[ (H₀·B - H·B₀) d³x ] = (1/8π)Int[ (1/μ₀-1/μ)B·B₀ d³x ], μ₀ = 1 free space (cgs).
The corresponding force, F_x = ∂W/∂x, should now be obvious; SI units gain a 4π, etc. As you can see, when saturated (internal B maximal but constant), the force on the projectile will be linear in original field B₀ (and thus also linear in current), rather than quadratic as before.

[starslayer]

Wyrm, I've never seen the force between two current-carrying wires called Ampere's Force Law. Interesting. Must be an engineer's thing. For my conjecture at the magentic attraction force, I basically assumed that we can treat the ends of the magnets as monopoles (i.e., we have infinite bar magnets). The actual relation would involve dipole orientation as you say.

The Ampere's Law I know of is one of Maxwell's equatioms: cint(B*dl) = mu_0 * I_enclosed + mu_0*e_0*d(phi_E)/dt.

[Wyrm]

The work to establish a magnetic field is W = (1/8π)Int[ H·B d³x ]; back in SI units, this gives a density of B²/2μ₀. Thus we should expect the force to be bounded by B²A/2μ₀, where A is the cross-section of the projectile.

For a situation with general magnetization, taking a vacuum with magnetic induction B₀ and field intensity H₀ and introducing a material of permeability μ, the difference will be (1/8π)Int[ (1-μ)H·H₀ d³x ] = (1/2)Int[ M·B₀ d³x ], where M is the magnetization.

When you say difference, do you mean taking the work equation for setting up the magnetic field around the coil without the projectile (found in your W = ... equation above) and subtracting the work equation with the projectile?

If the source currents are fixed, the force for displacement x is just F_x = ∂W/∂x.

Makes sense.

If I recall, in general, B = μ₀(H + M) where M is magnetization. Given that ferromagnets have hysterisis, I suspect that the derivitive wrt time for magnetization is (d/dt)M = f(M, H, (d/dt)H) for some f. Correct?

W = (1/8π)Int[ (H₀·B - H·B₀) d³x ] = (1/8π)Int[ (1/μ₀-1/μ)B·B₀ d³x ], μ₀ = 1 free space (cgs).
The corresponding force, F_x = ∂W/∂x, should now be obvious; SI units gain a 4π, etc. As you can see, when saturated (internal B maximal but constant), the force on the projectile will be linear in original field B₀ (and thus also linear in current), rather than quadratic as before.

Is B₀ the magnetic field absent the slug, and B the magnetic field with the slug, or what?

[starslayer]

Mr. Bean, the force between two current carrying wires shouldn't have any effect on that coilgun. However, General Deathdealer's math is wrong. This math should be correct to within an order of magnitude:

First, using the output power given (it's wrong, but I'll get to that later):

The object's final kinetic energy is K = 1/2*m*v^2 = .5*(.5 kg)*(10 km/s)^2 ~ 2.5E7 J. With our given power output, this implies an acceleration time of Pt = W = K; t = K/P = 2.5E7 J/1.2E6 W ~ 20 s. Now, to find the distance over which the energy will be transmitted.

Pdt = dW = Fdx; W = Pt = Fx = max. Now, here's a step which is blatantly false for a constant power input, but will still keep us within an order of magnitude: that our acceleration is constant. In reality, it will decrease over time for constant power. Continuing, Pt = m(x/t^2)x; x = sqrt(Pt^3/m) = sqrt(1.2E6 W * (20 s)^3/.5 kg) ~ 1.5E5 m. In other words, we need our projectile to accelerate over about 150 km, and thus our coil must be about that long on top of that to generate enough of a field gradient to accelerate the projectile. So, the entire system is over 300 km long, or on the same length scale as the Death Star.

However, this estimate is incorrect, as General Deathdealer messed up a bit on his math. Here is the corrected version:

We know that our projectile is 100 times more massive than a small one, goes 80 times as fast as a small one, and has the same efficiency. We also know the amount of energy output by the small one. We want the power output of the larger system. Our scaling:

Since the large projectile has 100 times the mass and 80 times the velocity of the small one, it has 100 * 80^2 = 160,000 times the small object's kinetic energy. Since I am assuming we want these guns to fire in the same amount of time, the power should scale with the kinetic energy, so it should be 160,000 times the power input of the small one. Now, in his output/input calculations, GD didn't check his units, and ended up with grams * Joules, which are not Watts. To correct this, we must wither specify a time in which the gun is fired or a power output; I will say that the gun fires in .01 s. Now, we repeat our calculations, with a twist or two:

Pt = K; P = K/t = 2.5E7 J/.01 s = 2.5E9 W
Pt = Fx = max = mx^2/t^2
x = sqrt(Pt^3/m) = sqrt(2.5E9 W * (.01 s)^3/.5 kg) ~ 2.5E2 m.

This length requirement is much more reasonable, as it only requires a system that is about 500 meters long, but it requires a power input 2.5E9 W * 20 ~ 50 GW. This is about 3 orders of magnitude higher than the original supposed input. To make a system with the same kinetic energy output on the 50-100 meter length scale, the power requirements are even higher, but not impossible with room-temperature superconductor and sufficiently advanced capacitors. However, these may break the limits of the STGOD.

[Wyrm]

Wyrm, I've never seen the force between two current-carrying wires called Ampere's Force Law.

Funny, considering the parallel-wires experiment to be Ampère's most famous experiment in classical electrodynamics.

At least there isn't as many Ampère's laws as there are Kirchhoff's.

For my conjecture at the magentic attraction force, I basically assumed that we can treat the ends of the magnets as monopoles (i.e., we have infinite bar magnets). The actual relation would involve dipole orientation as you say.

The energy of a dipole μ in a magnetic field B is U = -μ·B, if we assume the dipole moment to be neglible. The force would just be the gradient of the same: F = ∇U(x). (Expressing this in terms of μ and B gets funky.)

The Ampere's Law I know of is one of Maxwell's equatioms: cint(B*dl) = mu_0 * I_enclosed + mu_0*e_0*d(phi_E)/dt.

Ampère's circuital law is just the enclosed current: cInt[ B·dl ] = µ₀ I_enc. Maxwell's contribution is the addition of the displacement current term: … + µ₀ε₀ dΦ_E/dt

[Mr Bean]

So if I'm reading you right, in order to fire the gun with a 1 lb projection it must be 500 meters long and 50 GW's worth of energy?

I'm think I'm reading you wrong on what sized projectile the secondary calculations account fore.