The Coulomb Force – By Kobi Hall

Inside the reactor, the vacuum chamber will be filled with a gas of positively charged ions. Whether it’s Helium atoms to produce plasma or Hydrogen isotopes for fusion, it’s helpful for us to study the kinetics and dynamics of individual particles. There are four fundamental forces in nature: gravity, electromagnetism, and the strong and weak nuclear forces. In our reactor, gravity is negligible, and the weak nuclear force isn’t present at all. In another blog post Lukas talks about the strong nuclear force. The electromagnetic force is the force responsible for the large scale dynamics of the gas. Within the electromagnetic force, properties can usually be split up according to electrostatics and electrodynamics; whether or not there is motion among charged particles. Sophia is writing about dynamics in her post about charged particle scattering. The static case is often described by coulomb’s law, which says that like charges repel and opposite charges attract.

The electromagnetic force, specifically the Coulomb force, is most responsible for the forces we experience on a day-to-day basis. On a microscopic scale, it describes the attraction and repulsion of charged particles. On a macroscopic scale, this is what allows objects to collide and transfer momentum. In general, the force on a charged particle is described by the Lorentz force law

The force acts proportional to the electric field and perpendicular to the particle’s velocity and the magnetic field. The electric and magnetic fields are found by solving Maxwell’s equations, a system of coupled partial differential equations. For those unfamiliar with multivariable calculus, don’t worry, these equations aren’t important to understanding what follows.

The electric field is caused by charged particles moving through time, and the magnetic field is caused by charged particles moving through space. These equations are linear, so they can be split up according to different charges. A charged particle feels these forces due to the fields of other charged particles. In general computing the kinetics of multiple particles and fields is a very difficult process. Luckily Maxwell’s equations are compatible with special relativity. This means that electromagnetic effects change with relative motion. The mathematics of special relativity can allow us to reframe and simplify a number of scenarios. We can often find a reference frame where a given particle is approximately at rest.

Coulomb’s law describes the case of a static charged particle and no present currents. The magnetic field vanishes, and the electric field is proportional to the inverse square of the distance from the charge.

k is a proportionality constant that depends on the units of measurement called Coulomb’s constant. In the SI unit system Coulomb’s constant is 8.988*10^9 N m^2/C^2. For two charges, the force between them is

For like charges the product of the two charge values will be positive, resulting in a repulsive force. The closer the two charges are, the stronger the force between them is. The electromagnetic force exists at all scales. No matter how far away, charged particles can still exert nonzero forces on each other. This is contrasted with the two nuclear forces which only act on finite and small length scales. As the particles get infinitesimally close, the force approaches infinity. In a reactor, the coulomb force prevents atomic nuclei from getting close enough to fuse on the nuclear scale. Accelerating particles to high speeds is needed to overcome the high coulomb forces

Figure 1. Schematic of all of the variables in Coulomb's law (Wolfson, 2006).

The graph below shows the Coulomb force between two positively charged particles. Drag the positions of the particles, to see the forces change. Open the graph in desmos to play with the other values. It should be noted that none of the values in this graph have units to them, desmos doesn’t play well with very small or large numbers, this is meant to just visualize.

The formulation of forces may be intuitive to some, but in practice, we rarely calculate explicitly the forces between particles when working on the reactor. This is especially the case when considering more than two charged particles; the vector algebra quickly becomes tedious. It is often okay to neglect the forces of particles far enough away if there are two particles orders of magnitudes closer together than to others. Inside the reactor, this is a safe approximation to make when considering ions that are approaching the distances for fusion to occur. In this sense we talk about the coulomb force as a barrier to fusion. Deuterium ions will all be positively charged, so the coulomb force will be repulsive, preventing them from getting close enough to fuse. Daniel’s post about quantum tunneling outlines one of the mechanisms that occurs in order to overcome the Coulomb barrier and cause fusion. 

If we want to look at the large scale electric properties of the reactor, then we look at Coulomb’s law in a different form. A more scalable version of Coulomb’s law is expressed in terms of potential energy. In general, the (difference in) potential energy of a particle moving from a point A to a point B is

This integral does not depend on the specific path taken between A and B if F is what is called a conservative force. A singular particle moving through an electric field will experience a change of potential energy

Only the electric field shows up in this formula because the infinitesimal nudge along the particle’s trajectory dr will always be perpendicular to the magnetic force. (dr is proportional to the velocity, the cross product of v and B will be perpendicular to both v and B. The dot product between two perpendicular vectors will be zero). This value depends on the charge of the particle in motion, and often we want to work with something more general. The electric potential, V, can be thought of as the potential energy per unit charge (divide the formula for ΔU by q). The electric potential is just a number (a function of space), and it can be used to work backwards to find the electric field using derivatives. Coulomb’s law for the electric potential is

Here, r is the distance to some specified point in space. We can represent an arbitrary number of particles by summing together multiple terms, each particle having a disting charge and distance from the point. The electric potential is arguably the most useful quantity discussed in this post, and you may be more familiar with its other name, voltage. The voltage in an electrical system is the difference in electric potential between two points, ΔV, and has units of volts(V). The graph below shows two positively charged particles (red) and one negatively charged particle (purple). The purple lines are called equipotentials, lines of constant electric potential. Again, you are encouraged to move the positions of the particles and observe how the electric potential changes.

References

Wolfson, Richard. Essential University Physics, Pearson,  2006.

Griffiths, David. Introduction to Electrodynamics, Prentice Hall, 1999.