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.
The current density of a point particle stationary at a point r₀
of 3d space.
Equations
- One or more equations did not get rendered due to their size.
Instances For
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.
The electromagnetic potential of a point particle stationary at r₀
of 3d space.
Equations
- One or more equations did not get rendered due to their size.
Instances For
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: $$\vec E(r) = \frac{q}{4 π \epsilon_0} \frac{\vec r - \vec r₀}{|\vec r - \vec r₀|^3}.$$
C.1. the time derivative of the electric field #
D. The magnetic field #
Given that the vector potential is zero, the magnetic field is also zero.
E. Maxwell's equations #
The divergence of the electric field of a point particle in 3d space is given by: $$∇ · \vec E(r) = \frac{1}{\epsilon_0} q \delta(r - r₀).$$
From this, it follows that the electromagnetic potential of a point particle in 3d space satisfies Maxwell's equations for a point particle at rest.