Barry's Coilguns 

Mark 2

Wire Loop EquationsThis page describes equations to model the simplest possible solenoid  a single wire formed into a loop. It selects a few points of interest to help understand the magnetic field. A Java applet will be written to graphically illustrate this magnetic field  please check back each week so you can play with it when it's ready. When the Java applet is completed, you should see a graph (below) with slider controls to adjust the size of the loop. The graph shows you the magnetic field intensity along the main axis, and the force on a particle. To keep the controls very simple, we have normalized all curves so they peak at a value of 1.0. The only thing you can adjust is the loop's radius. Magnetic Field StrengthThe magnetic field has been normalized to the peak field. That is, the graph shows the ratio of B_{z} to B_{z=0}. To see how the equations are derived, please visit theory of wire loops. The normalized Bfield is given by the equation:
That was very nice, but what if you want to know the point z for a particular B_{rel}?
Solve the above equation for z in terms of B_{rel}, and you get: Force on a ParticleNow let's introduce a tiny particle, one small enough that the shape of the Bfield is not disturbed. The force on this small particle is the derivative of the Bfield squared. The normalized force equation plotted above is:
ObservationsThis equation maintains the same general shape for various sizes of a radius. It is merely scaled up or down according to its radius. You can see how quickly the field falls off as you move away from the center of the loop. The force on a tiny particle in this magnetic field is shown in the simulator above. It is proportional to the gradient of the Bfield squared. You can see the force is positive on the left, negative on the right, and zero at the center of the wire loop. The sign (+ or ) shows the tendency to pull a projectile from either side back into the middle. We kept this applet as simple as possible. It would be nice to find the exit speed of a particle in this field, but that would require factors such as starting position, current over time, mass and permeability. (Maybe later!) Some Selected PointsThe Bfield equation can be solved for a few selected points using algebra. It would be interesting to know the field strength at one radius away from the center, and where the field is onehalf peak value, and where the field is negligible. At exactly one radius from the center, the point z is equal to the radius b. Substituting this
value of z, and solving for B: If we want to find point z for a certain value of B_{rel}, we use algebra to reverse the equation: When the field is onehalf of the peak value, B_{rel} = 0.5. Substituting this value for B_{rel} and evaluating z:
This result is very interesting! It says the magnetic field is practically gone at a distance of about twice the radius from the midpoint of the loop! And we know that 'twice the radius' is another name for its diameter. An old ruleofthumb has said exactly this, and now we know the analytical reasoning behind it. One last point... let's find where the field disappears. The field strength can be assumed negligible when it is reduced to less than 10% of the peak value. So let's solve for z when B_{rel} = 0.1. Beyond this distance we can simply ignore the Bfield:




Last Update 20080615
©19982019 Barry Hansen 