Physlib.Electromagnetism.Vacuum.Constant

Given a spatial dimension $d$, a speed of light $c$, a constant electric field vector $E_0 \in \mathbb{R}^d$, and an antisymmetric constant magnetic field matrix $B_0 \in \mathbb{R}^{d \times d}$, the electromagnetic potential $A$ is defined as a function that assigns a Lorentz vector to each spacetime point $x$. For a spacetime point $x$ with spatial components $\mathbf{x} = (x_1, \dots, x_d)$, the components of the potential $A(x)$ are given by: - The temporal component $A^0(x) = -\frac{1}{c} \langle E_0, \mathbf{x} \rangle$, where $\langle \cdot, \cdot \rangle$ denotes the standard Euclidean inner product. - The spatial components $A^i(x) = \frac{1}{2} \sum_{j=1}^d (B_0)_{ij} x_j$ for each $i \in \{1, \dots, d\}$.

Given a spatial dimension $d$, a speed of light $c$, a constant electric field vector $E_0 \in \mathbb{R}^d$, and an antisymmetric constant magnetic field matrix $B_0 \in \mathbb{R}^{d \times d}$, the electromagnetic potential $A$ associated with these constant fields is infinitely continuously differentiable ($C^\infty$). The potential $A(x)$ is defined at each spacetime point $x$ by its temporal component $A^0(x) = -\frac{1}{c} \langle E_0, \mathbf{x} \rangle$ and its spatial components $A^i(x) = \frac{1}{2} \sum_{j=1}^d (B_0)_{ij} x_j$, where $\mathbf{x}$ is the spatial part of $x$.

For a given spatial dimension $d$, speed of light $c$, constant electric field vector $E_0 \in \mathbb{R}^d$, and antisymmetric constant magnetic field matrix $B_0 \in \mathbb{R}^{d \times d}$, let $A$ be the corresponding constant electromagnetic potential. Then the scalar potential $\phi$ associated with $A$ is given by the function $\phi(t, \mathbf{x}) = -\langle E_0, \mathbf{x} \rangle$, where $\mathbf{x}$ is the spatial position vector and $\langle \cdot, \cdot \rangle$ denotes the standard Euclidean inner product.

Given a spatial dimension $d$, a speed of light $c$, a constant electric field vector $E_0 \in \mathbb{R}^d$, and an antisymmetric constant magnetic field matrix $B_0 \in \mathbb{R}^{d \times d}$ (where $(B_0)_{ij} = -(B_0)_{ji}$), the vector potential component $\mathbf{A}$ of the electromagnetic potential defined for these constant fields at any time $t$ and spatial position $\mathbf{x} = (x_1, \dots, x_d)$ is given by: $$\mathbf{A}_i(t, \mathbf{x}) = \frac{1}{2} \sum_{j=1}^d (B_0)_{ij} x_j$$ for each $i \in \{1, \dots, d\}$.

For a given speed of light $c$, a constant electric field vector $E_0 \in \mathbb{R}^d$, and an antisymmetric constant magnetic field matrix $B_0 \in \mathbb{R}^{d \times d}$, let $\mathbf{A}$ be the vector potential component of the electromagnetic potential defined for these constant fields. For any time $t$ and spatial position $\mathbf{x}$, the partial derivative of the vector potential with respect to time is zero: $$\frac{\partial \mathbf{A}}{\partial t}(t, \mathbf{x}) = 0$$

Consider a spatial dimension $d$, a speed of light $c$, a constant electric field vector $E_0 \in \mathbb{R}^d$, and an antisymmetric constant magnetic field matrix $B_0 \in \mathbb{R}^{d \times d}$ such that $(B_0)_{ij} = -(B_0)_{ji}$ for all $i, j \in \{1, \dots, d\}$. Let $\mathbf{A}$ be the vector potential associated with the electromagnetic potential defined for these constant fields. For any time $t$, spatial position $\mathbf{x} = (x_1, \dots, x_d)$, and coordinate indices $i, j$, the partial derivative of the $j$-th component of the vector potential $A_j$ with respect to the $i$-th spatial coordinate $x_i$ is given by: $$\frac{\partial A_j}{\partial x_i} = \frac{1}{2} (B_0)_{ji}$$

For a given spatial dimension $d$, speed of light $c$, constant electric field vector $E_0 \in \mathbb{R}^d$, and antisymmetric constant magnetic field matrix $B_0 \in \mathbb{R}^{d \times d}$, let $A$ be the corresponding constant electromagnetic potential. The electric field $\mathbf{E}$ derived from this potential $A$ is constant in both time and space, and is equal to $E_0$. That is, for any time $t$ and spatial position $\mathbf{x}$, $$\mathbf{E}(t, \mathbf{x}) = E_0$$

Given a spatial dimension $d$, a speed of light $c$, a constant electric field vector $E_0 \in \mathbb{R}^d$, and an antisymmetric constant magnetic field matrix $B_0 \in \mathbb{R}^{d \times d}$ (where $(B_0)_{ij} = -(B_0)_{ji}$ for all $i, j \in \{1, \dots, d\}$), let $A$ be the electromagnetic potential defined for these constant fields. Then, the magnetic field matrix derived from this potential $A$ is constant and equal to $B_0$ for all time $t$ and spatial position $\mathbf{x}$.

In a $d$-dimensional free space $\mathcal{F}$ with speed of light $c$, consider a constant electric field vector $E_0 \in \mathbb{R}^d$ and an antisymmetric constant magnetic field matrix $B_0 \in \mathbb{R}^{d \times d}$. Let $A$ be the electromagnetic potential defined by these fields, where its components are given by the temporal part $A^0(x) = -\frac{1}{c} \langle E_0, \mathbf{x} \rangle$ and the spatial parts $A^i(x) = \frac{1}{2} \sum_{j=1}^d (B_0)_{ij} x_j$. Then $A$ is an extremum of the free-space electromagnetic action for a zero Lorentz current density ($J = 0$).