An Exploration of the Physics Behind Rail Guns

An Exploration of the Physics Behind Rail Guns

Daniel Lenord

University of Alaska Fairbanks

Physics 212 Web Project, Spring 2003

SOURCE: http://ffden-2.phys.uaf.edu/212_fall2003.web.dir/Daniel_Lenord/index.html

INDEX

Electromagnetic Gun

The concept of a rail gun is simple: two parallel bars connected to a power source

produce an electric field. This electric field can be used to propel a projectile along the

bars. If enough amperage is provided, the projectile can achieve velocities of up to 4

km/s. The U.S. Army has been interested in the potential of electromagnetic guns for

quite some time. In 1988 the University of Texas Center for Electromagnetics began

work on a 9 MJ range gun, which would be designed to launch 2-4 kg projectiles at

velocities of up to 4 km/s. Design features of this range gun included up to 230 MJ

stored energy, 6 kV and 3 MA peak output ratings, and a 9 shot repetitive fire

capability. In the early 1990's, the U.S. Army, along with the Marine Corps, showed

much interest in the Cannon Caliber Electromagnetic Gun System (CCEMG). This

project seeks to demonstrate an electromagnetic gun system designed from a

system/mission prospective. The CCEMG showed a 3 times increase in energy and

power density over the range gun system.

The U.S. Army plans to produce electric tanks by the year 2015 with electromagnetic

guns mounted on them. The all-electric tank would include electric vehicle drive and

suspension and electric armaments. In this concept, the flywheel energy can be used

as a flywheel battery to provide power for vehicle acceleration and regenerative

braking, and will also produce electrical power for the electromagnetic armaments.

This is the perfect example of how physics is still impacting the world in which we

live.

INTRODUCTION

A rail gun in its 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 the following equation.

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)

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 the following.

F=ILB

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

• I=Current through rails and projectile (Amperes)

• L=Length of rail separation (Meters)


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.

VELOCITY

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 1.

(1) 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 2 shows the equation for the magnetic

flux.

(2) H=BA

• H=Magnetic Flux (Teslas x Meter^2)

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

• A=Area (Meter^2)

Equation 3 shows how the induced voltage V(i) is related to H and the velocity of the

projectile.

(3) V(i)=dH/dt=BdA/dt=BLdx/dt

• V(i)=Induced voltage

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


• 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 4 shows the terminal velocity v(max) of the projectile.

(4) v(max)=V/(BL)

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

• V=Power source voltage (Volts)


• 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 5.

(5) 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)


• 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.

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.

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.

Right hand rule for cross product

BIBLIOGRAPHY

1. Jengel and Fatro's Rail Gun Page. http://home.insightbb.com/~jmengel4/rail/rail-intro.html

2. Pappas, John. University of Texas Center for Electromechanics http://www.utexas.edu/research/cem/

3. MIT students. Amateur Rail Gun Production Journal http://www.railgun.org

4. United States Patent and Trademark Office http://www.uspto.gov/

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An Exploration of the Physics Behind Rail Guns