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I am wondering if someone could provide me with a formula that would tell me at what velocity a projectile can be launched from something using an electromagnetic field. The idea is much like a rail gun or Gauss rifle, but not exactly.

Just looking to find a formula to determine the speed of the projectile.

Thank you, Michael Vanderpool

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find a formula — Hm, conservation of energy? – kennytm Nov 2 '10 at 20:14
Thanks for the comment, but I dont see how it was very helpful to answering the question. – Rekar Nov 2 '10 at 20:18
@Rekar: Because the specific detail isn't given, so I don't know if it's possible to give a more specific answer. (I know it isn't too helpful, so I commented instead of answered.) – kennytm Nov 2 '10 at 20:21
Ah Ok, Well... Lets say we have a ball that has a mass of 50g. If I have a EMF of, idk, 10T, how fast would that 50g ball travel when using the EMF to launch that 50g ball? All Im really looking for here is the formula, as stated above. I will be using the formula for object much much larger than 50g, but you get the Idea I hope. – Rekar Nov 2 '10 at 20:26
If the ball is made of wood, then the EMF will have no effect on it =) – mtrencseni Nov 2 '10 at 21:28

As KennyTM stated, use the conservation of energy. Say for example you have a constant electric field E to accelerate your particle of charge q and mass m, this will mean that the electrical energy E d q accumulated over the distance d will convert entirely into kinetic energy $\frac12 m v^2$ (assuming you stay non-relativistic, otherwise it's $\sqrt{m^2c^4+p^2c^2}$). Just solve the equation for v (or the momentum p in the relativistic case). Things get a bit more complicated for a non-static EM field.

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First, from your comment: a magnetic field will never accelerate anything (I am more specifically thinking about a charge "particle" as this is trivial for a golf ball) (Edit: see the comments bellow about that statement).

The magnetic field is always perpendicular to the trajectory, thus giving no contribution in a change of the kinetic energy.

To accelerate a particle, you need an electric field.

Then the way to proceed is the following: if you know the electric field and the path, you can integrate E dot dx and multiply by the charge. This will give you an energy difference, and this energy is added as kinetic energy, thus from the kinetic energy, if you are non relativistic you can use $E_k = \frac{1}{2} m v^2$ and make a difference.

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Actually a magnetic field can accelerate particles (as in a particle accelerator), but only perpendicular to the direction of motion, so that the particle's speed doesn't change. – David Z Nov 4 '10 at 23:01
OK, I took the shortcut "acceleration = change in the magnitude of the speed", but you're right. – Cedric H. Nov 4 '10 at 23:14

This source discusses Ampere's Force Law. The force between two parallel conductors is equal to two times a constant, times the current in wire A, times the current in wire B, divided by the spacing between wires. The constant is known as the magnetic force constant and is 1 x 10E-7 newtons in the SI system.

For parallel conductors one meter in length, one meter apart and carrying one ampere, the force between them is exactly 2 x 10E-7 newtons.

The force in a rail gun will be directly proportional to its current and inversely proportional to the spacing between its rails.

But what length should you assume when calculating a rail gun? The length of the conductor should be considered to be the width of the projectile. This is equal to the distance between rails. As the distance between rails decreases, the unit forces increase, but the projectile gets smaller. Those two considerations fall out and it makes no difference what the rail spacing is. In other words, make the rail spacing as small as possible in order to accommodate your projectile.

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The Lorentz Force Law could also help you out,

F= q(E+ v X B)

where E is the electric field, B is the magenetic field strength, v is the velocity, and q is the charge of the object.

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