Railgun Physics

The underlying physics involved in the railgun are rather simple. Current flowing through an inductor creates a magnetic field. The current flowing through the field creates a Lorentz force on the inductor tending to push the coil apart. If one portion of the coil is free to move, this portion will slide away from the power source.Qualitatively, its a relatively straightforward process. Difficulty arrises in trying to quantitatively determine the time dynamics of the electric and magnetic fields present, and an analytical description of the motion of the slug. To do so, we must examine the relationship between the current in the loop, the induced magnetic field, the motion of the slug, and the geometry of the loop - all as functions of time.

We will break down the complex problem of determining the analytical equations of motion for the slug in the following way:

Determine the instantaneous induced magnetic field at any point as a function of loop geometry and current in the loop at a given time.Take Faraday's Law and derive the equation for the induced EMF in the loop. Calculate this based on the magnetic field, determined above.Solve Ohm's Law for an analytical formulation of current as a function of loop resistance, initial charge in the capacitor(s), capacitance of the power cap(s), and induced EMF. Take the derivative of this equation to get an equation for I'(t) with no integrals in it.Now we need to replace I(t) and I'(t) in the above equation with something we know how to calculate. To do this, we'll first manipulate the Lorentz Force law like this:Determine the instantaneous magnetic field at any point on the slug as a function of loop geometry and current in the loop at a given time.Integrate the magnetic field over the length of the slug and multiply by the instantaneous current to reveal the magnitude of the Lorentz Force at a given time.Set the Lorentz Force magnitude equal to mL" (F=ma). Solve this equation for I(t) and take its time derivative.Now we have have two equations for I'(t) which we can set equal to each other, leaving a differential equation for L(t). Solve this differential equation to reveal the time evolution of L(t) [position of slug on rails], L'(t) [velocity of slug], L''(t) [acceleration of slug], etc.1) Magnetic field in the current loop

Assuming we know the shape of our current loop and the magnitude of the current at a given instant, the instantaneous magnetic field caused by the current in the loop is given by the Biot-Savart law (Griffiths 5.28):

Where r is the vector from the source (dl) to the point at which we are evaluating the field. We can expect the displacement current term of Ampere's law,

to affect our magnetic field, because we expect to see an electric field that changes with time. However, its contribution is scaled by the permittivity of free space, a factor ~10-11. This means that in order for the displacement current to noticably affect the magnetic field, the change in electric field would have to be on the order of 1011 V/m s.

This would entail a change in voltage in the circuit of Px1011 V/s where P is the perimeter of the current loop. In the highly unrealistic case of a 10kV power source (much higher than we're likely to use), a loop perimeter of 0.1m (tiny) and a firing relay that switches in 1 milliseconds (unreasonably fast!!), the displacement current is then on the order of 1 Amp, but we are expecting power supply currents possibly as high as ~10000 Amps initially, making the displacement current contribution to B inconsequential.

Therefore, to get a value for B(t), we break the Biot-Savart integral up into four parts - the two rails, the slug, and the connection opposite the slug which we will approximate as a fixed armature across the rails. In reality, this connection will consist of the power source, the firing relay, and the connecting wires. Our assumption is that the field contribution of this portion of the circuit can, in fact, be manipulated to be very similar to a direct connection across the rails.

The direction of the magnetic field vector depends on which direction the current flows through the loop. For convenience, we'll assume that the current moves counter-clockwise. By the right-hand rule, the magnetic field is in the positive z direction, "up".

The magnetic field at some point, P, induced by current flowing through a straight wire can be derived analytically. A succinct example of this derivation is given in Griffiths, chapter 5, example 5. The following figure is taken from this example.

The end result is that

B = (mu0I/4z*pi) * [sin(theta2) - sin(theta1)]where z is the shortest distance from P to the wire, theta1 and theta2 are the initial and final angles between z and r. We will approximate the railgun as four straight wire field contributions summed together.

For the first rail:

z = ysin(theta1) = -(L - x) / sqrt((L - x)2 + y2)sin(theta2) = x / sqrt(x2 + y2)For the second rail:z = (W - y)sin(theta1) = x / sqrt(x2 + (W - y)2)sin(theta2) = -(L - x) / sqrt((L - x)2 + (W - y)2)For the slug:z = (L - x)sin(theta1) = (W - y) / sqrt((W - y)2 + (L - x)2)sin(theta2) = y / sqrt(y2 + (L - x)2)For the "back" connection:z = xsin(theta1) = -y / sqrt(x2 + y2)sin(theta2) = (W - y) / sqrt(x2 + (W - y)2)where L is the length of the rail to the slug, W is the rail separation, while x and y are the coordinates of the point inside the loop for which we are calculating the field. Plugging these four sets of values into the equation above and summing the results leads to the following analytical solution for the magnetic field at some point inside the loop:

Oops, screwed that equation graphic up bad. It's all wrong, but the general form is right. it's B(I,x,y,L,W)|slug = (mu_oI/4pi)[f(L,W,x,y)]

2) Inductive EMF in Current Loop

Faraday's law tells us that when we have a changing magnetic field, we have an induced electric field.

We take Faraday's law and act upon it with the curl theorem of vector calculus,

to achieve the following relation for the induced EMF, epsilon:

Again, this has been done analytically with Mathematica. [NOTE: Our initial railgun derivation used a slightly different form for epsilon, Griffiths 7.13. This is basically the same equation except the time derivative is pulled outside the integral because the magnetic field is assumed to be constant in time. This is obviously not the case for the railgun, and therefore the original derivation was incorrect.]

3) Ohm's Law

The induced EMF, calculated in section 3 above, opposes the power source voltage and therefore decreases the circuit current via Ohm's Law:

V(t) = I(t)R = V(t)power supply - epsilon(t)Where R is the circuit resistance. We'll assume that the resistance is constant with time, and we'll hope that it is very small. For this derevation, we will also assume that the power supply is a capacitor. This leads to the canonical form for I(t):

I(t) = (1/R)[Vcap - epsilon(t)]The capacitance of a capacitor is given by

C = q(t)/V(t)where q is the amount of charge on the capacitor and V is the voltage across the capacitor. The charge at any time is given by

q(t) = q(0) - integral[I(t)dt]Now we can write the current like this:

I(t) = (1/R)[(q0 - integral[I(t)dt])/C - epsilon(t)]Now we can take the derivative of this equation with respect to t to get a differential equation of I.

I'(t) = (1/R)(de/dI) - I(t)/RCSome thought needs to be given to bounday conditions here, since the factor of q0/RC has disappeared.

4) The Lorentz Force

The current in the loop flowing through magnetic field calculated in part 1 generates a Lorentz force outwards on the loop. The Lorentz force on the slug can be calculated analytically (Griffiths 5.17):

Fmag(I,L,W) = I(t) * integral[dy x Bslug(I,y,L,W)]The magnetic field in this equation is given the subscript 'slug' because we are only interested in the field at the slug. This is the only part of the field that contributes to the Lorentz force on the projectile itself.

To calculate this magnetic field, we'll use the same technique and assumptions used in part 1. However, we can simplify the analysis here a bit since we'll always be at x=L. Also, the contribution to the field from the slug itself can be ignored.

Intuitively, if the slug were just a piece of straight wire, no amount of current through that wire would cause it to move anywhere in the absence of external fields. In the actual system, where the slug is under the influence of large magnetic fields, it can still be shown that the slug current won't move the slug. It's a simple matter of momentum conservation. If the slug moves itself, then momentum has been created from nothing.

Since the magnetic field is uniformly vertical and the slug lies horizontally on the rails, it is clear that the direction of the force will be horizontal and perpendicular to the slug, i.e. in the x direction. The cross product coveniently drops out, and all vectors becomes scalars with the exception of an x-hat at the end.

Fmag(I,L,W) = (mu0L(t)I(t)2/pi) * (W2+L(t)2-L(t)*sqrt(W2+L(t)2)) / sqrt(W2+L(t)2)This is also equal to the mass of the slug times the acceleration of the slug (Newton's 2nd law of motion, F=ma). The acceleration of the slug can be written as the second time derivative of L, leading to this equation:

m * L''(t) = (mu0L(t)I(t)2/pi) * (W2+L(t)2-L(t)*sqrt(W2+L(t)2)) / sqrt(W2+L(t)2)This equation can be solved for the current:

I(t) = sqrt[pi * m * L''(t) * sqrt(W2+L(t)2) / (mu0 * L(t) * (W2+L(t)2-L(t) * sqrt(W2+L(t)2)))]5) Differential Equation of Motion

The results from parts 4 and 5 can be combined to replace all instances of I(t) and I'(t) and leave a differential equation in terms of L(t), L'(t), L''(t), etc. (slug position, velocity, acceleration, etc)

It is hoped that Mathematica will be able to solve this differential equation to give an analytical form of the equations of motion for the slug in terms of physical variables like the mass of the slug, rail separation, and capacitance.

This will represent the end of the theory effort for the railgun and we will then proceed to design considerations and begin construction. (assuming, of course, that the numbers don't indicate that we require 1000 Farads of capacitance or one million volts or something.)

Picture this: A massive destroyer receives the location coordinates of an enemy headquarters more than 200 miles away. Instead of launching a million-dollar Tomahawk cruise missile, it points a gun barrel in the direction of the target, diverts electric power from the ship's engine to the gun turret, and launches a 3-foot-long, 40-pound projectile up a set of superconducting rails. The projectile leaves the barrel at hypersonic velocity--Mach 7-plus--exits the Earth's atmosphere, re-enters under satellite guidance, and lands on the building less than six minutes later; its incredible velocity vaporizes the target with kinetic energy alone.The U.S. Navy is developing an electromagnetic railgun that will turn destroyers into super-long-range machine guns--able to fire up to a dozen relatively inexpensive projectiles every minute. The Navy is collaborating with the British Ministry of Defence, which has a similar effort under way. In 2003, its facility in Kirkcudbright, Scotland, hosted a 1/8-scale test of an electromagnetic railgun that produced stable flight in a projectile fired out of the barrel at Mach 6. But Capt. Roger McGinnis, program manager for directed energy weapons at Naval Sea Systems Command in Washington, D.C., estimates the U.S. version won't be "deliverable" until 2015 at the earliest.The technology behind the electromagnetic railgun has been around for more than 20 years, but early efforts wilted because of the huge power requirements: No ship could generate or store enough electricity to fire the gun. The concept was revived a few years ago when the Navy announced plans for its next-generation battleship, the all-electric DD(X). "In the past, destroyers had 90 percent of their power tied to propulsion," explains McGinnis. "But with DD(X), you can divert the power to whatever you need. We can stop the ship and fire the railgun as many times as we need, then divert the power back to the screws."The barrel of the electromagnetic railgun will contain two parallel conducting rails about 20 feet long, bridged by a sliding armature. In the current design, electric current travels up one rail, crosses the armature, and heads down the second rail. The loop induces a magnetic field that pushes the armature, and the projectile aboard it, up the rails.The challenges that remain include ensuring that the gun can target enemy sites with precision, and creating equipment that can withstand the gargantuan pressures the gun will create. "Right now, guns are only as accurate as the targeting of the bore, and now we're talking about 200-plus-mile ranges, so there has to be aerodynamic correction," says Fred Beach, the assistant program manager for the electromagnetic railgun at Naval Sea Systems Command. The projectile, he says, will receive course correction information from satellites and will steer itself with movable control surfaces. And because the projectile will be subjected to up to 45,000 Gs during firing, the onboard electronics must be strengthened to withstand the acceleration. Forces inside the gun itself--particularly getting the armature to move easily within the system--are also challenging the designers. "Getting two pieces of metal to slide past each other is pretty hard--we're getting a lot of damage to the rails," Beach says.The electromagnetic railgun's projectiles will cover 290 miles in six minutes--initially traveling 8,200 feet per second and hitting their target at 5,000 feet per second. Current Navy guns, which shoot powder-ignited explosive shells, have a maximum range of 12 miles and, because they are unguided, are difficult to aim. Though guided missiles, the current long-range alternative for destroyers, can achieve ranges comparable to that of the electromagnetic railgun, their cost and storage problems are what's driving the efforts to find an alternative. Ships can only carry up to 70 guided missiles and must return to port to restock because the missiles cannot be loaded at sea, whereas railgun projectiles can easily be loaded at sea, and by the hundreds. Also appealing is that the electromagnetic railgun's missiles do not contain volatile explosives; the weapon does its work with kinetic energy.

How Do Rail Guns Work?

The electromagnetic rail gun is used by the United States Navy and it is very unique in that it is a gun that does not use gunpowder. Instead, it is powered by electricity and a magnetic field. It has been a huge technological advancement because before it was invented, the maximum range a gun could shoot a projectile was 12 miles. A rail gun can reach a target 250 miles away in 6 minutes, traveling at a speed of around 16,000 meters per second.

The Basic Parts of a Rail Gun

A rail gun uses large amounts of power, so a power source has to be used that can generate millions of amps in order for a rail gun to function. The armature is the most important part of the rail gun because it is the metal piece (conductive sabot) that holds the projectile. On either side of the armature, there are two conductive rails which vary in length depending on the size and power capabilities of the rail gun. The rails are connected to the power source and the current from the source travels up one rail (the positive rail) and through the armature and back down to the power source through the other rail on the other side (the negative rail).

Simulation of How a Rail Gun Works

How it WorksIn the parts of the rail gun where an electric current is present, magnetic fields are present around rails. As shown in the diagram above, the magnetic field rotates in a counterclockwise direction around the positive rail. In the negative rail, the magnetic field rotates in a clockwise direction.The main reason that a rail gun works is because of the third right hand rule. As illustrated in the diagram below, the magnetic field is always pointing outwards towards the sky and the current travels from the positive side to the negative side of the power source, which is to the right. This means that the force of the rail gun projects straight out of the gun. However, in order for a rail gun to work effectively and travel long distances at large speeds, there must be a great amount of force inflicted on the projectile. This is why millions of amperes are required in the power source for the rail gun to work properly.

Due to the fact that a rail gun requires millions of amperes, the Navy does not have a solution for providing enough power to use a rail gun on a naval ship yet. However, they are currently working on building a type of battleship, the all-electric DD(X), which can temporarily take some power from the engine whenthe rail gun needs to be used. Not only can rail guns be used in the military, but they can also be used to launch satellites or space shuttles into outer space potentially because a rail gun can have multiple projectiles, not just a missile.

Rail Gun BasicsA rail gun is basically a large electric circuit, made up of three parts: a power source, a pair of parallel rails and a moving armature. Let's look at each of these parts in more detail.

For simulation (animation) http://science.howstuffworks.com/rail-gun1.htmThe power supplyis simply a source of electric current. Typically, the current used in medium- to large-caliber rail guns is in the millions of amps.The railsare lengths of conductive metal, such as copper. They can range from four to 30 feet (9 meters) long.The armaturebridges the gap between the rails. It can be a solid piece of conductive metal or a conductivesabot-- a carrier that houses a dart or other projectile. Some rail guns use aplasmaarmature. In this set-up a thin metal foil is placed on the back of a non-conducting projectile. When power flows through this foil it vaporizes and becomes a plasma, which carries the current.Here's how the pieces work together:An electric current runs from the positive terminal of the power supply, up the positive rail, across the armature, and down the negative rail back to the power supply.Current flowing in any wire creates a magnetic field around it -- a region where a magnetic force is felt. This force has both a magnitude and a direction. In a rail gun, the two rails act like wires, with a magnetic field circulating around each rail. The force lines of the magnetic field run in a counterclockwise circle around the positive rail and in a clockwise circle around the negative rail. The net magnetic field between the rails is directed vertically.Like a charged wire in an electric field, the projectile experiences a force known as theLorentz force(after the Dutch physicist Hendrik A. Lorentz). The Lorentz force is directed perpendicularly to the magnetic field and to the direction of the current flowing across the armature. You can see how this works in the diagram below.

Notice that the Lorentz force is parallel to the rails, acting away from the power supply. The magnitude of the force is determined by the equation F = (i)(L)(B), whereFis the net force,iis the current,Lis the length of the rails andBis the magnetic field. The force can be boosted by increasing either the length of the rails or the amount of current.Because long rails pose design challenges, most rail guns use strong currents -- on the order of a million amps -- to generate tremendous force. The projectile, under the influence of the Lorentz force, accelerates to the end of the rails opposite the power supply and exits through an aperture.The circuit is broken, which ends the flow of current.

USURPING POWERRail guns require tremendous currents to fire projectiles at speeds of Mach 5 or higher. This presents problems for a traditional battleship because power cannot be diverted from the ship's propulsion system. In the Navy's next-generation battleship, the all-electric DD(X), producing this kind of current will be possible. To launch a rail gun projectile, power would be diverted from the ship's engine to the gun turret. The gun would be fired, up to six rounds per minute, for as long as required. Then power would be shifted back to the engine.

INTRODUCTION Railguns are a means of accelerating an object based on an electromagnetic force. Such devices have been in use in laboratory settings for several decades, but the number of potential applications increases with advances in technology. Although railguns are quite complex, their basic underlying principles are simple enough for anyone with basic physics training to understand. This design proposal will outline the principles behind designing a low-velocity railgun for educational use in a high school classroom.1.1 Railgun Theory The components of the most basic railgun design include two parallel rails and a movable armature, all of which must be electrically conductive. Current is run down one rail, through the armature, and back up the other rail to complete the circuit. A magnetic field is induced between the rails by the current loop formed. A Dutch physicist named Hendrick Antoon Lorentz formulated an equation for magnetic force, called Lorentz force, by which the force on a current-carrying object by a magnetic field is given by:

whereFis the force on the object,Iis the electrical current,is the length vector of the current flow in the object, andis the magnetic field. A simplistic railgun setup can be seen in Figure 1.

Figure 1 Simple Railgun Schematic [1] Accurately describing railgun behavior is much more difficult than merely applying the Lorentz force equation. As the armature travels down the rails, the size of the current loop increases and thereby changes the magnitude of the induced magnetic field. The position-time behavior of the armature can therefore only be accurately described through the use of multiple-order differential equations. A large amount of current is required to produce a force large enough to accelerate even a lightweight projectile. The force is largely dependant on the electrical current because it is a product of the current running through it and the magnetic field it experiences, which is a function of that same current. The Biot-Savart Law describes the induced magnetic field magnitude of a semi-infinite length wire by

where0is the permeability of free space,Iis the current through the wire, andris the radius of the wire [2]. It can be seen from this law that, for a wire with a radius of ten millimeters, it would take a current of 50,000 Amps to produce a magnetic field of just 1 T at the wire surface! Even with parallel wires of this size each contributing to a magnetic field four millimeters away from the surface of both wires, 35,000 Amps would still be needed to induce a magnetic field strength of 1 T. The force on the armature is a function of the square of the current running through it. Capacitors are often employed to produce the high current needed because of their large charge storage and short discharge time. Even in materials with minimal resistivity, a fraction of electrical energy is lost in the form of heat due to impedance of the conductor. Heat buildup can become significant at the large electrical currents needed in a railgun. While a force on a current-carrying object is the inherent mechanism behind the railgun, it can also cause some problems. The same current that is passed through the armature also flows through both rails, causing a force on the rails that tries to push them apart. The rails are subject to the same Lorentz force equation as the armature, so a strong supporting structure is needed to contain the lateral pressure on the rails and prevent them from flying apart. The outward force on the rails, assuming the current in each rail is the same because they are in the same loop, can be calculated by

whereLis the length of the wire.1.2 Previously Constructed Railguns Engineers have built railguns for recreation, research, or real-world applications. Many military applications exist for the high kinetic energy that a railgun can produce, but many hobbyists have also created railguns out of sheer enjoyment. Railguns built by others can help predict and prematurely correct some of the problems that could be encountered in building such a device.1.2.1 Military Designs The military has been actively researching the applications of railguns in military situations for several years for a couple of reasons. First of all, railguns are capable of firing projectiles with a huge amount of kinetic energy. Recent tests at the University of Canberra in Australia [3] have produced a force of 250,000 times that of gravity on a mass of 16 grams. This translates to going from rest to 13,000 miles per hour in under one-fifth of a second! Projected naval railguns from Batelle will be able to fire up to six rounds per minute with an impact speed of Mach 5, producing penetration up to forty feet [4]. Secondly, railguns require no chemicals to use. Rather than using energy from controlled explosions like current weapons, railgun ammunition will consist of electrical energy firing inert slugs. This will almost eliminate the need for the military to store dangerous explosives on ships and bases, increasing safety and stealth. While railguns have not become standard issue as of yet, they do hold promise in many exciting areas. Naval ships and land bases will likely switch to railguns due to their large projected ranges (up to 200 nautical miles) and lack of explosives to store and protect [3]. The army is researching the possibility of using railguns on armored vehicles because of their huge destructive power and low armament storage needs. Launching supplies rather than projectiles from a railgun is also being explored, and NASA is even exploring this technique to launch supplies both from and towards Earth.1.2.2 Amateur Designs Numerous amateur railguns have been conceived and built for a variety of purposes. Some have built these as recreational devices, others for the challenge of building the railgun, and still others for demonstration purposes. Firing speeds and armature properties differ for each gun and each application. Some of the simplest railguns are for demonstration purposes and can be built in a span of minutes. These simple railguns, such as the one in Figure 2, simply use batteries or a DC laboratory power supply to cause a small section of wire to roll. These simple railguns demonstrate the principles of electromagnetic motion, just as the more powerful ones, but for much less time and money.

Figure 2 Simple Classroom Demonstration Railgun The majority of amateurs build railguns that are capable of muzzle velocities between 50 and 500 meters per second. Some have been for various high school and college classes, while others have been built in basements and garages as hobbies. One of the most well-known amateur railguns was made by Sam Barros, a student at Michigan Institute of Technology, and chronicled on www.powerlabs.org [1]. On his website, he outlines the theory behind railguns in general and his design in particular, as well as providing insight on some of the problems experienced in construction. Amazingly, many amateurs are able to fire railguns at velocities large enough to experience many of the problems of the larger military devices, such as metal vapor arcing (MVA) and rail damage by electrical arcing.1.2.3 Problems with Previous Railguns One of the most common problems with railguns built in the past has been rapid rail erosion. Erosion failure often occurs after only one or two uses in some cases [5]. Rail erosion occurs much more quickly than would be expected from purely frictional forces between the armature and rails. The mechanism of rail erosion due to electrical current is not fully understood and is under investigation. It is known that highly polished surfaces with low coefficients of friction are needed for maximum railgun efficiency. Another common problem is the armature fusing itself to the rails upon firing. When a stationary armature is fired by a railgun, it must overcome the coefficient of static frictionin order to begin motion. The coefficient of kinetic friction, which is much less than, is the only barrier to motion after the armature is already moving. If the armature is unable to overcome, the electric current energy put through the armature will not be converted to mechanical energy. This energy will instead transform into thermal energy, heating the armature significantly. Enough thermal energy can cause the armature to melt itself to the rails. Large contact surfaces and materials with high electrical and thermal conductivity can help to alleviate the melting problem. Some problems are a direct result of higher muzzle velocities that are beyond the scope of this project. The repulsion forces on the rails can cause the structural supports to fail at high currents, as dictated by the Biot-Savart Law. At very high velocities, a plasma can form between the rails by MVA. This plasma can greatly reduce the lifetime of the rails, but it can also act as an armature itself because it is highly conductive.1.3 Detailed Objectives For this project, a railgun will be designed and built for low velocities for use in high school classroom demonstrations and physics/mechanics based experiments. In order to remain safe for the client and all students involved, the railgun must shoot lightweight projectiles at slow speeds. Firing at low velocities with railguns poses its own unique problems not often experienced by other rail gun builders. Too much electrical power input will cause the projectile to accelerate too much, but too little will cause the gun not to work at all.1.3.1 Criteria for SuccessBecause the railgun is to be used in laboratory sessions, the gun will be designed to be educational and safe in every possible way. Clear plastic will be used as a casing material to allow visible inspection of the device at work. This device is also to be durable so that the client may use the railgun year after year in the laboratory. As such, much of the design will revolve around making a sturdy, robust device that will erode very little due to repeated use. Safety is the number one priority, however. All electrical components must be completely insulated from the outside world to prevent any electrical discharge into anything other than the railgun components, and all structural components must be sound enough to sustain even unexpected forces. The maximum budget for the entire project, including material and component costs, consultation fees, construction, testing, and implementation, is $700 as set by the client. The project can be considered a success if these objectives and goals are met in their entirety.1.3.2 Classroom Uses The entire purpose of the design and construction of this device is for demonstration of electromagnetic principles to high school physics students. Most of the electromagnetic theory behind railguns discussed in the theory section could easily be discussed and demonstrated with a railgun, including induced magnetic fields, electric current flow in materials, and more. Most railgun behavior, such as muzzle velocity, could be predicted through calculations and then verified by experiments. The actual demonstrations and laboratory sessions performed are to be decided by the client.