Physlib.Electromagnetism.PointParticle.OneDimension

Given the speed of light $c$, a charge $q \in \mathbb{R}$, and a position $r_0$ in 1D space, the distributional Lorentz current density $J$ for a stationary point particle is defined as the time-independent distribution: $$J = (cq \delta_{r_0}) \mathbf{e}_0$$ where $\delta_{r_0}$ is the Dirac delta distribution centered at $r_0$ and $\mathbf{e}_0$ is the basis vector corresponding to the temporal component of the Lorentz vector. This represents the current density of a point particle fixed at $r_0$ in one spatial dimension.

For a speed of light $c$, a point charge $q$ at position $r_0$ in one-dimensional space, the distributional Lorentz current density $J$ is equal to the spatial translation by $r_0$ of the stationary distribution $cq \delta_0 \mathbf{e}_0$. Here, $\delta_0$ is the Dirac delta distribution centered at the origin, $\mathbf{e}_0$ is the basis vector corresponding to the temporal component of the Lorentz vector, and the result is treated as a time-independent (stationary) distribution in space-time.

For any speed of light $c$, charge $q \in \mathbb{R}$, and position $r_0$ in 1D space, the spatial current density distribution $\vec{j}$ associated with the distributional Lorentz current density of a stationary point particle at $r_0$ is zero.

For a point particle with charge $q$ stationary at position $r_0$ in 1D space, the distributional charge density $\rho$ is the time-independent distribution given by $$\rho = q \delta_{r_0}$$ where $\delta_{r_0}$ is the Dirac delta distribution centered at $r_0$. This charge density is derived from the distributional Lorentz current density $J$ by taking its temporal component and scaling by the reciprocal of the speed of light $c$, i.e., $\rho = \frac{1}{c} J^0$.

In a free space $\mathcal{F}$ with magnetic permeability $\mu_0$ and speed of light $c$, the electromagnetic 4-potential $A$ of a stationary point particle with charge $q$ at position $r_0$ in 1D space is the time-independent distribution defined by the function: $$x \mapsto \left( -\frac{q \mu_0 c}{2} \|x - r_0\| \right) \mathbf{e}_0$$ where $\mathbf{e}_0$ denotes the basis vector for the temporal component of the Lorentz vector.

In a free space $\mathcal{F}$ with magnetic permeability $\mu_0$ and speed of light $c$, the electromagnetic 4-potential of a stationary point particle with charge $q$ at position $r_0$ in 1D space is equal to the time-independent distribution obtained by translating the distribution defined by the function $$x \mapsto \left( -\frac{q \mu_0 c}{2} \|x\| \right) \mathbf{e}_0$$ by the position vector $r_0$, where $\mathbf{e}_0$ denotes the temporal basis vector of the Lorentz vector space.

In a free space $\mathcal{F}$ with speed of light $c$, the vector potential $\mathbf{A}$ of the electromagnetic 4-potential associated with a stationary point particle of charge $q$ at position $r_0$ in 1D space is zero.

In a free space $\mathcal{F}$ with magnetic permeability $\mu_0$ and speed of light $c$, the scalar potential $\phi$ of a stationary point particle with charge $q$ located at position $r_0$ in 1D space is the time-independent distribution defined by the function: $$x \mapsto -\frac{q \mu_0 c^2}{2} \|x - r_0\|$$ where $x$ is the spatial coordinate and $\|x - r_0\|$ denotes the distance from the particle.

In a free space $\mathcal{F}$ with magnetic permeability $\mu_0$ and speed of light $c$, the electric field $\mathbf{E}$ of a stationary point particle with charge $q$ located at position $r_0$ in 1D space is the time-independent distribution defined by the function: $$x \mapsto \frac{q \mu_0 c^2}{2} \frac{x - r_0}{\|x - r_0\|}$$ where $x$ is the spatial coordinate and $\frac{x - r_0}{\|x - r_0\|}$ represents the unit vector (or sign) in the direction from the particle to the point $x$.

In a free space $\mathcal{F}$ with speed of light $c$, the electric field $\mathbf{E}$ of a stationary point particle with charge $q$ located at position $r_0$ in 1D space is time-independent, meaning its time derivative as a distribution is zero: $$\frac{\partial \mathbf{E}}{\partial t} = 0$$ where $\mathbf{E}$ is the electric field distribution derived from the electromagnetic 4-potential of the particle.

For a stationary point particle with charge $q$ at position $r_0$ in 1D space within a free space $\mathcal{F}$ with speed of light $c$, the magnetic field matrix associated with its electromagnetic 4-potential is zero.

In a free space $\mathcal{F}$ with magnetic permeability $\mu_0$ and speed of light $c$, let $\mathbf{E}$ be the electric field distribution of a stationary point particle with charge $q$ located at position $r_0$ in 1D space. The spatial divergence of $\mathbf{E}$ in the distributional sense is given by: $$\nabla \cdot \mathbf{E} = q \mu_0 c^2 \delta(x - r_0)$$ where $\delta(x - r_0)$ denotes the time-independent Dirac delta distribution centered at $r_0$.

In a one-dimensional free space $\mathcal{F}$ with magnetic permeability $\mu_0$ and speed of light $c$, let $A$ be the distributional electromagnetic 4-potential of a stationary point particle with charge $q$ at position $r_0$, and let $J$ be the corresponding distributional Lorentz current density defined by $J = (cq \delta_{r_0}) \mathbf{e}_0$. This theorem states that $A$ is an extremum of the electromagnetic action for the source $J$, which is equivalent to saying that the variational gradient of the electromagnetic Lagrangian density $\mathcal{L}$ vanishes: $$\frac{\delta \mathcal{L}}{\delta A} = 0$$ In physical terms, this confirms that the electromagnetic potential of the 1D stationary point particle satisfies the distributional Maxwell equations.