## Background

The electromagnetic force is the interaction between charged particles, through electric fields and magnetic fields. This section discusses the electric force, which is responsible for the electric field. The magnetic force is discussed separately.

Electricity is a good example of the electric force. Electrons are pushed through wires in electricity as they repel each other, continually forcing electrons down the wire. However, since we can’t measure individual electrons in the wire, electricity is a measurement of a collection of electrons. The force that moves electrons is referred to as a voltage (V), the flow of electrons as a current (I), and its opposition is called resistance (R). Georg Ohm established the relationship between these as V=IR, known as Ohm’s law. Since electricity is the study of a collection of particles, it is described separately on this site, including a relationship between Ohm’s law and Newton’s 2nd law. Refer to the **What is Electricity?** page.

On this page, we’ll focus on the law established by Charles-Augustin de Coulomb of the electric force at the particle-level. Force (F) is the measurement of how charged particles repel or attract each other and is based on the total charge of the particles (q), a constant (k) and the distance between particles (r), shown below. This is in Coulomb’s law.

*Credit: hyperphysics.phy-astr.gsu.edu*

The electric force has a feature, unlike gravity, that allows particles to be attracted or repelled. At a subatomic particle level, two electrons are known to repel each other. However, an electron and positron will attract each other, as was shown in the annihilation page.

**Explanation**

The energy wave explanation for the electric force, like all of the forces, is based on a difference in wave amplitude as particles move to minimize amplitude. When a particle reflects longitudinal waves (out-waves), it is a standing wave structure as it combines with in-waves. This forms a particle. Beyond the particle’s perimeter, the standing wave structure breaks down and the out-wave becomes a traveling wave. These waves are traveling, spherical, **longitudinal waves** that have an amplitude that decreases with the square of the distance from the particle core – the inverse square law seen in the electric force. This will have an effect on other particles based on constructive wave interference.

### Constructive Wave Interference – Repel

Gabriel LaFreniere has modeled the wave mechanics of two electrons in proximity. Due to the phase shift at the electron core, the waves constructively add between the particles but destructively add beyond them. This is seen in the simulation as amplitude is much higher between the particles. As particles move to minimize amplitude, this forces them to repel.

### Destructive Wave Interference – Attract

In a similar animation, LaFreniere has also modeled the electron and positron interaction. However, these results are different because the positron is model as a 180 degree phase difference from the electron, causing the destructive wave interference. Because there are only two stable nodes on the wave where amplitude is minimal, particles and their anti-particles will be on one of these nodes. The anti-particle will always be on the anti-phase of the wave – 180 degree phase difference and cancel waves with its counterpart.

In this case, waves are destructive between the particles and constructive beyond them. The minimal amplitude between the particles causes an attraction effect as each particle moves to minimize its amplitude.

This is the electric force. Traveling, longitudinal waves that affect particles as they move to minimize their amplitude.

### Particle Groups

Measurements are rarely a single particle affecting another particle. More often, it is a group of particles separated at distance to determine the electric force. In the upcoming equation, a dimensionless particle count (Q) is used to count the number of particles that separates two groups being measured, at a distance r.

*Note: Particle count (Q) is similar to the variable used for Coulomb’s law (q), as it represents the number of particles. However, it is dimensionless, expressed in a numerical value as opposed to a Coulomb charge. Therefore it was given a capital letter (Q) as opposed to lower case (q).*

**Equation**

In simple terms using two groups (Q) of particles separated at distance (r), and the properties of the electron’s energy, mass and radius (E_{e}, m_{e }and r_{e}), the electric force of an electron is shown below. It is the “fundamental” force.

When expressed in wave constant terms, it is the electric force:

**Electric Force**

**Proof**

Proof of the energy wave explanation for the electric force is the derivations of:

- Derived the electron’s mass and energy –
*from the energy form of the force equation* - Derived Coulomb’s Law and Coulomb’s constant.
- Calculations of electric forces –
*see example calculation below*

### Single Proton and Electron at Bohr Radius (*Hydrogen*) – Calculation

**Equation: **Electric Force equation**Variables:**

- Q
_{1}**=**1 (*proton*) - Q
_{2}**= –**1 (*electron*) - r = a
_{0}= 5.2918E-11 m (*Bohr radius*)

**Result: –**8.239E-8 newtons (kg m/s

^{2})

**Comments:**No difference (0.000%) from Coulomb force. Also, the value is equal to the orbital force at this location, for the electron’s position in hydrogen.

A summary of various electric force calculations is found on this site; more detailed calculations with instructions to reproduce these calculations is found in the *Forces* paper.