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

What is a rail gun?

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

  • B=Magnetic field strength (Teslas)
  • N=Number of turns in solenoid (1 in our case)
  • u=1.26x10^-6 (The magnetic permeability of free space, Henries/Meter)
  • I=Current through rails and projectile (Amperes)

Figure 1:  Simple Rail Gun

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

  • F=Force on conductor (projectile, in Newtons)

  • I=Current through rails and projectile (Amperes)

  • L=Length of rail separation (Meters)

  • B=Magnetic field strength (Teslas)

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.  In Figure 1, the force is oriented down the rails, away from the power source.  See the section on the Right Hand Rule below for a detailed description of this.

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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

  • a=Acceleration (Meters/second^2)

  • F=Force on projectile (Newtons)

  • m=Mass of projectile (Kilograms)

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

  • H=Magnetic Flux (Teslas x Meter^2)

  • B=Magnetic field strength (Teslas) (Assuming uniform field)

  • A=Area (Meter^2)

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

  • V(i)=Induced voltage

  • dH/dt=Time rate of change in magnetic flux

  • B=Magnetic field strength (Teslas)

  • dA/dt=Time rate of change in area

  • L=Width of rails (Meters)

  • dx/dt=Time rate of change in position (velocity of projectile)

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)

  • v(max)=Terminal velocity of projectile (Meters/second)

  • V=Power source voltage (Volts)

  • B=Magnetic field strength (Teslas)

  • L=Width of rails (Meters)

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

  • v(muz)=Muzzle velocity (Meters/Second)

  • D=Length of rails (Meters)

  • F=Force applied (Newtons)

  • m=Mass of projectile (Kilograms)

  • I=Current through projectile (Amperes)

  • L=Width between rails (Meters)

  • B=Magnetic field strength (Teslas)

  • u=1.26x10^-6 (The magnetic permeability of free space, Henries/Meter)

These calculations ignore 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 to 4km/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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What is the right hand rule?

The right hand rule is a mnemonic for memorizing the orientation of fields, forces, or other vector quantities after the cross product of two vector quantities is taken. For example, the direction of the magnetic field around a conductor due to a current can be determined by pointing the right thumb in the direction of the current and curling the other four fingers as if grasping the conductor. The magnetic field similarly exists as a vector field circling the conductor in the direction indicated by your fingers. See Figure 2 for a visual representation.


Figure 2: Right hand rule for magnetic field from current through conductor

In addition, the right hand rule comes into play when performing cross products of vector quantities. For example, when figuring out which way the projectile in a rail gun will go, you look to Equation 2. Equation 2 is truly a cross product, but presented as a simple multiplication for the sake of simplicity. The force exerted on the projectile is the cross product of scalar length L, vector i the path of the current in the projectile, and vector field B the magnetic field. When determining the direction of this force we can use the right hand rule. Since all the angles involved are 90 degrees, the resultant force has a magnitude resulting from the simple multiplication of the magnitude of i and B and the value of L. (|F|=L|i||B|) To determine the direction, lay your right hand along the path of the current through the projectile, with your fingers pointing in the direction the current is travelling. Next, curl your fingers in the direction of the B field. Your thumb will now be pointing in the direction of the applied force. See Figure 3 for a visual representation.


Figure 3: Right hand rule for cross product



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Page last updated 4/24/02