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