Electrostatics of a stationary point particle in 3d

i. Overview

In this module we give the electromagnetic properties of a point particle sitting at the origin in 3d space.

ii. Key results

threeDimPointParticle : The electromagnetic potential of a point particle stationary at a point in 3d space.

threeDimPointParticle_isExterma : The electric field of a point particle stationary at a point of 3d space satisfies Maxwell's equations

iii. Table of contents

A. The current density

A.1. The charge density

A.2. The 3-current density

B. The Potentials

B.1. The electromagnetic potential

B.2. The scalar potential

B.3. The vector potential is zero

C. The electric field

C.1. the time derivative of the electric field

D. The magnetic field

E. Maxwell's equations

iv. References

A. The current density

The current density of a point particle in 3d space is given by:

$$J(r) = (c q \delta(r - r₀), 0, 0, 0) $$

where $c$ is the speed light, $q$ is the charge of the particle and $r₀$ is the position of the particle in 3d space.

A.1. The charge density

The charge density of a point particle in 3d space is given by: $$ρ(r) = q \delta(r - r₀) $$

where $q$ is the charge of the particle and $r₀$ is the position of the particle in 3d space.

A.2. The 3-current density

The 3-current density of a point particle in 3d space is given by: $$\vec J(r) = 0.$$

In other words, there is no current flow for a point particle at rest.

B. The Potentials

B.1. The electromagnetic potential

The 4-potential of a point particle in 3d space is given by:

$$A(r) = \frac{q μ₀ c}{4 π |r - r₀|} (1, 0, 0, 0) $$

where $μ₀$ is the permeability of free space, $c$ is the speed of light, $q$ is the charge of the particle and $r₀$ is the position of the particle in 3d space.

B.2. The scalar potential

The first component of the 4-potential is the scalar potential, once one has taken account of factors of the speed of light. It is given by:

$$V(r) = \frac{q}{4 π \epsilon_0 |r - r_0|}.$$

B.3. The vector potential is zero

The spatial components of the 4-potential give the vector potential, which is zero for a stationary point particle.

$$\vec A(r) = 0.$$

C. The electric field

The electric field of a point particle in 3d space is given by: $$\vec E(r) = \frac{q}{4 π \epsilon_0} \frac{\vec r - \vec r₀}{|\vec r - \vec r₀|^3}.$$