MATHEMATICS: Coulomb's Law
Purpose
Coulomb's law provides the mathematical relationship describing the electrostatic force between two electric charges.
It is therefore a natural starting point for the investigation of charge interaction.
The purpose of this article is not to assign a deeper physical mechanism to Coulomb's law. Its purpose is to establish the mathematical relationship itself so that subsequent investigation can build upon a clearly defined foundation.
Coulomb's Law
For two point charges separated by a distance \(r\), the magnitude of the electrostatic force is given by:
\[ F = k\frac{|q_1q_2|}{r^2} \]
where:
| Symbol | Meaning |
|---|---|
| \(F\) | Magnitude of the electrostatic force |
| \(q_1\) | First electric charge |
| \(q_2\) | Second electric charge |
| \(r\) | Separation between the charges |
| \(k\) | Coulomb's constant |
In vacuum,
\[ k=\frac{1}{4\pi\varepsilon_0}. \]
Therefore the law may also be written:
\[ F= \frac{1}{4\pi\varepsilon_0} \frac{|q_1q_2|}{r^2}. \]
Direction of the Force
The magnitude equation alone does not specify whether the force is attractive or repulsive.
The signs of the charges provide that distinction:
| Charge relationship | Interaction |
|---|---|
| \(q_1q_2>0\) | Repulsive |
| \(q_1q_2<0\) | Attractive |
Thus the sign of the charge product determines the character of the interaction, while the inverse-square term determines how the magnitude changes with separation.
The Inverse-Square Relationship
The most important structural feature of Coulomb's law for the present investigation is its dependence upon the square of the separation distance:
\[ F\propto\frac{1}{r^2}. \]
This means that increasing the separation between two charges reduces the interaction rapidly.
For example, if the separation is doubled,
\[ r\rightarrow2r, \]
then the force becomes:
\[ F\rightarrow \frac{1}{(2r)^2} = \frac{1}{4r^2}. \]
The force therefore becomes one-quarter of its original magnitude.
This inverse-square attenuation is not a secondary detail. It is a defining mathematical characteristic of the Coulomb interaction.
From Force to Field
Coulomb's law can also be expressed through the electric field produced by a point charge.
The electric field is defined as force per unit test charge:
\[ \mathbf{E}=\frac{\mathbf{F}}{q}. \]
For a point charge \(Q\),
\[ \mathbf{E} = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r^2}\hat{\mathbf r}. \]
The field therefore inherits the same inverse-square spatial dependence.
Force as an Interaction Between Charges
Coulomb's law can be viewed operationally as a relationship between two charge states.
A charge establishes an electric field. A second charge placed within that field experiences a force.
The relationship may therefore be represented as:
\[ \boxed{ \text{Charge} \rightarrow \text{Electric Field} \rightarrow \text{Force} } \]
This representation does not claim that the electric field is a material substance. It simply identifies the mathematical and operational sequence used to describe the interaction.
Energy Associated With the Electric Field
The electric field is also associated with an energy density.
In vacuum, the electric contribution to electromagnetic energy density is:
\[ u_E=\frac{1}{2}\varepsilon_0E^2. \]
This provides an important transition from the mathematics of charge interaction toward the mathematics of an electromagnetic energy field.
Coulomb's law describes the force associated with charge interaction. The energy-density relationship describes energy associated with the resulting electric field.
The distinction is important:
\[ \boxed{ \text{Force relationship} \neq \text{Energy-density relationship} } \]
They are related descriptions of the same electromagnetic system, but they should not be conflated.
Connection to the Lorentz Force
Coulomb's law describes electrostatic interaction between charges. The Lorentz force law extends the description to moving charges and magnetic effects.
The Lorentz force is:
\[ \mathbf F = q\mathbf E + q\mathbf v\times\mathbf B. \]
The first term,
\[ q\mathbf E, \]
represents the electric-force contribution.
Coulomb's law therefore provides the natural electrostatic foundation for the electric portion of the Lorentz force.
\[ \boxed{ \text{Coulomb} \rightarrow \text{Electric Interaction} \rightarrow \text{Lorentz} } \]
What Coulomb's Law Establishes
For the purposes of this investigation, Coulomb's law establishes a small number of important mathematical facts:
- Electric charges interact through a measurable force.
- The magnitude of the electrostatic interaction depends upon the product of the charges.
- The interaction attenuates according to an inverse-square relationship with separation.
- The sign relationship between charges determines attraction or repulsion.
- The interaction can be represented through an electric field.
- The electric field carries an associated energy density.
What Coulomb's Law Does Not Establish
Coulomb's law describes the observed relationship. It does not, by itself, establish the physical mechanism responsible for that relationship.
In particular, the equation does not establish:
- what charge physically is,
- what an electric field physically consists of,
- why opposite charges attract,
- why like charges repel,
- why the spatial dependence is inverse-square,
- or whether the field represents a deeper physical substrate.
Those are mechanism questions.
Audit Position
| Statement | Status |
|---|---|
| Electrostatic force follows an inverse-square relationship with separation. | Established relationship |
| Charge sign determines attraction or repulsion in the Coulomb description. | Established relationship |
| Electric fields are associated with energy density. | Established relationship |
| The inverse-square relationship represents attenuation of an interaction with distance. | Operational interpretation |
| The inverse-square relationship may reveal a deeper mechanism governing attraction. | Open investigation |
| A physical substrate produces the Coulomb interaction. | Open investigation |
Foundation for the Next Investigation
Coulomb's law establishes the electrostatic side of charge interaction.
The next step is to examine what happens when charge is not merely stationary.
Moving charge introduces the magnetic component of electromagnetic interaction.
That transition is expressed by the Lorentz force:
\[ \mathbf F = q\mathbf E + q\mathbf v\times\mathbf B. \]
The investigation can therefore proceed from the established electrostatic relationship toward the magnetic behavior observed in experiment.
\[ \boxed{ \text{Coulomb} \rightarrow \text{Charge Interaction} \rightarrow \text{Lorentz} \rightarrow \text{Magnetic Observation} } \]
Summary
Coulomb's law provides the mathematical foundation for describing electrostatic interaction between charges.
Its defining spatial characteristic is inverse-square attenuation:
\[ F\propto\frac{1}{r^2}. \]
The sign of the interacting charges determines whether the described interaction is attractive or repulsive.
From this foundation, the investigation can proceed toward moving charge, magnetic interaction, electromagnetic energy density, and ultimately the question of mechanism.
Coulomb's law is therefore not the conclusion. It is the first mathematical gear in the investigation of charge interaction.