Welcome to the newly redesigned haven for the rail gun enthusiast.  This page covers some of the latest techniques in electromagnetic propulsion, but the construction of a rail gun is a perilous undertaking so use the information contained herein at your own risk.   NOTE: In the interest of simplicity vector directions are ignored, it is assumed that the magnitude is as calculated and in the desired direction.
Introduction Contents

A rail gun in it's simplest form is a pair of conducting rails separated by a distance L and with one rail connected to the positive and one the negative side of a power source supplying voltage V and current I. A conducting projectile bridges the gap L between the rails, completing the electrical circuit. As current I flows through the rails, a magnetic field B is generated with an orientation dictated by the right hand rule and with a magnitude governed by equation 1.

    (1)  B=NuI

When a current I moves through a conductor of length L in the presence of a magnetic field B, the conductor experiences a force F according to equation 2.

      (2) F=ILB

The direction of the force depends on the direction of the current through the projectile and the magnetic field since the force is truly a vector with direction dictated by the cross product of the vector quantities I and B. 
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How fast does a rail gun projectile go?

Short answer: currently about 4 km/s

The speed of a rail gun slug is determined by several factors; the applied force, the amount of time that force is applied, and friction.  Friction will be ignored in this discussion, as it's effects can only be determined through testing. If this concerns you, assume a friction force equal to 25% of driving force.   The projectile, experiencing a net force as described in the above section, will accelerate in the direction of that force as in equation 3.

     (3) a=F/m

Unfortunately, as the projectile moves, the magnetic flux through the circuit is increasing and thus induces a back EMF (Electro Magnetic Field) manifested as a decrease in voltage across the rails.  The theoretical terminal velocity of the projectile is thus the point where the induced EMF has the same magnitude as the power source voltage, completely canceling it out.  Equation 4 shows the equation for the magnetic flux.

     (4) H=BA

Equation 5 shows how the induced voltage V(i) is related to H and the velocity of the projectile.

     (5) V(i)=dH/dt=BdA/dt=BLdx/dt

Since the projectile will continue to accelerate until the induced voltage is equal to the applied, Equation 6 shows the terminal velocity v(max) of the projectile.

     (6) v(max)=V/(BL)

These calculations give an idea of the theoretical maximum velocity of a rail gun projectile, but the actual muzzle velocity is dictated by the length of the rails.  The length of the rails governs how long the projectile experiences the applied force and thus how long it gets to accelerate.  Assuming a constant force and thus a constant acceleration, the muzzle velocity (assuming the projectile is initially at rest) can be found using Equation 7.

     (7) v(muz)=(2DF/m)^.5=(2DILB/m)^.5=I(2DLu/m)^.5

These calculationsignore friction and air drag, which can be formidable at the speeds and forces applied to the rail gun slug.  Top rail gun designs currently can launch a 2kg projectile with a muzzle velocity of close to4km/s on roughly 6 meter rails. To reach this kind of velocity, the power source must provide roughly 6.5 million Amps.  Ouch.

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