Physlib.Electromagnetism.Kinematics.ElectricField

Let $A$ be an electromagnetic potential and $c$ be the speed of light. The electric field $\mathbf{E}$ is given by the gradient of the scalar potential $\phi$ and the time derivative of the vector potential $\mathbf{A}$ as: $$\mathbf{E}(t, \mathbf{x}) = -\nabla \phi(t, \mathbf{x}) - \frac{\partial \mathbf{A}}{\partial t}(t, \mathbf{x})$$ where $\phi(t, \mathbf{x})$ and $\mathbf{A}(t, \mathbf{x})$ are the scalar and vector potentials associated with $A$ at time $t$ and position $\mathbf{x}$.

For a differentiable electromagnetic potential $A$ in $d$ spatial dimensions and the speed of light $c$, the $i$-th spatial component of the electric field $\mathbf{E}$ at time $t$ and position $\mathbf{x}$ can be expressed in terms of the field strength matrix $F$ as: $$E_i(t, \mathbf{x}) = -c F_{0, i}(t, \mathbf{x})$$ where $F_{0, i}(t, \mathbf{x})$ is the component at the temporal index $0$ and spatial index $i$ of the field strength matrix evaluated at the spacetime point corresponding to $(t, \mathbf{x})$.

For a differentiable electromagnetic potential $A$ in $d$ spatial dimensions and a speed of light $c$, the component of the field strength matrix $F$ at the temporal index $0$ and spatial index $i$ at a spacetime point $x$ is related to the $i$-th component of the electric field $E_i$ by: $$F_{0, i}(x) = -\frac{1}{c} E_i(t, \mathbf{x})$$ where $t$ and $\mathbf{x}$ are the temporal and spatial coordinates of the spacetime point $x$ respectively.

For a differentiable electromagnetic potential $A$ in $d$ spatial dimensions and the speed of light $c$, the component of the field strength matrix $F$ at a spacetime point $x$ with spatial index $i$ and temporal index $0$ is given by: $$F_{i, 0}(x) = \frac{1}{c} E_i(t, \mathbf{x})$$ where $E_i(t, \mathbf{x})$ is the $i$-th spatial component of the electric field evaluated at the time $t$ and position $\mathbf{x}$ corresponding to the spacetime point $x$.

For an electromagnetic potential $A$ in $d$ spatial dimensions and a given speed of light $c$, if $A$ is $n+1$ times continuously differentiable (of class $C^{n+1}$), then the electric field $\mathbf{E}$ derived from $A$ is $n$ times continuously differentiable (of class $C^n$) as a function of time $t$ and spatial position $\mathbf{x}$.

Let $A$ be an electromagnetic potential in $d$ spatial dimensions and $c$ be the speed of light. If $A$ is $(n+1)$-times continuously differentiable (of class $C^{n+1}$), then for any spatial index $i \in \{1, \dots, d\}$, the $i$-th component of the electric field $E_i(t, \mathbf{x})$ is $n$-times continuously differentiable (of class $C^n$) as a joint function of time $t$ and spatial position $\mathbf{x}$.

Let $A$ be an electromagnetic potential in $d$ spatial dimensions, $c$ be the speed of light, and $t$ be a fixed time. If $A$ is $n+1$ times continuously differentiable (of class $C^{n+1}$), then the $i$-th component of the electric field $\mathbf{E}_i(t, \mathbf{x})$, considered as a function of the spatial position $\mathbf{x}$, is $n$ times continuously differentiable (of class $C^n$).

Let $A$ be an electromagnetic potential in $d$ spatial dimensions, and $c$ be the speed of light. If $A$ is $n+1$ times continuously differentiable ($C^{n+1}$), then for any fixed spatial position $\mathbf{x}$, the component $i$ of the electric field $\mathbf{E}_i(t, \mathbf{x})$, viewed as a function of time $t$, is $n$ times continuously differentiable ($C^n$).

For an electromagnetic potential $A$ in $d$ spatial dimensions and the speed of light $c$, if $A$ is twice continuously differentiable ($C^2$) with respect to its spacetime coordinates, then the electric field $\mathbf{E}$, viewed as a function of time $t$ and position $\mathbf{x}$, is differentiable.

Let $A$ be an electromagnetic potential in $d$ spatial dimensions and let $c$ be the speed of light. If $A$ is twice continuously differentiable ($C^2$), then for any fixed position $\mathbf{x} \in \text{Space}_d$, the electric field $\mathbf{E}(t, \mathbf{x})$, viewed as a function of time $t$, is differentiable with respect to $t$.

For an electromagnetic potential $A$ in $d$ spatial dimensions and the speed of light $c$, if $A$ is twice continuously differentiable ($C^2$) with respect to its spacetime coordinates, then for any fixed time $t$, the electric field $\mathbf{E}(t, \cdot)$ is differentiable as a function of the spatial coordinates.

For an electromagnetic potential $A$ in $d$ spatial dimensions and given the speed of light $c$, if $A$ is twice continuously differentiable ($C^2$) on spacetime, then for any spatial index $i \in \{0, \dots, d-1\}$, the $i$-th component of the electric field $E_i(t, \mathbf{x})$, viewed as a function of time $t$ and position $\mathbf{x}$, is differentiable with respect to $(t, \mathbf{x})$.

For an electromagnetic potential $A$ in $d$ spatial dimensions and the speed of light $c$, if $A$ is twice continuously differentiable ($C^2$) on spacetime, then for any fixed time $t$ and any spatial index $i \in \{0, \dots, d-1\}$, the $i$-th component of the electric field $E_i(t, \mathbf{x})$ is differentiable with respect to the spatial coordinates $\mathbf{x}$.

Let $A$ be an electromagnetic potential in $d$ spatial dimensions and let $c$ be the speed of light. If $A$ is twice continuously differentiable ($C^2$), then for any fixed position $\mathbf{x} \in \text{Space}_d$ and any spatial index $i \in \{0, \dots, d-1\}$, the $i$-th component of the electric field $E_i(t, \mathbf{x})$, viewed as a function of time $t$, is differentiable with respect to $t$.

For an electromagnetic potential $A$ in $d$-dimensional space, let $\vec{A}(t, x)$ denote the vector potential, $\phi(t, x)$ denote the scalar potential, and $\vec{E}(t, x)$ denote the electric field. For any time $t$ and position $x$, the partial derivative of the vector potential with respect to time satisfies the relation: $$\frac{\partial \vec{A}(t, x)}{\partial t} = -\vec{E}(t, x) - \nabla \phi(t, x)$$ where $\nabla$ is the spatial gradient operator.

For a differentiable electromagnetic potential $A$ in $d$-dimensional space, let $A_i(t, x)$ be the $i$-th component of the vector potential $\vec{A}$, $\phi(t, x)$ be the scalar potential, and $E_i(t, x)$ be the $i$-th component of the electric field $\vec{E}$. For any time $t$, spatial position $x$, and spatial index $i$, the time derivative of the $i$-th component of the vector potential satisfies: $$\frac{\partial A_i(t, x)}{\partial t} = -E_i(t, x) - \frac{\partial \phi(t, x)}{\partial x_i}$$ where $\frac{\partial \phi}{\partial x_i}$ is the $i$-th component of the spatial gradient $\nabla \phi$.

For an electromagnetic potential $A$ in $d$ spatial dimensions that is twice continuously differentiable ($C^2$), and given the speed of light $c$, the time derivative of the $i$-th component of the electric field $\mathbf{E}$ at time $t$ and spatial position $\mathbf{x}$ is given by: $$\frac{\partial E_i(t, \mathbf{x})}{\partial t} = -c^2 \partial_0 F_{0, i}(t, \mathbf{x})$$ where $\partial_0$ denotes the partial derivative with respect to the temporal coordinate and $F_{0, i}$ is the component of the field strength matrix evaluated at the spacetime point corresponding to $(t, \mathbf{x})$.

For an electromagnetic potential $A$ in $d$ spatial dimensions that is twice continuously differentiable ($C^2$), and given the speed of light $c$, the divergence of the electric field $\nabla \cdot \mathbf{E}$ at a time $t$ and spatial position $\mathbf{x}$ is given by: $$(\nabla \cdot \mathbf{E})(t, \mathbf{x}) = c \sum_{\mu=0}^{d} \partial_\mu F_{\mu, 0}(t, \mathbf{x})$$ where $\partial_\mu$ denotes the partial derivative with respect to the $\mu$-th spacetime coordinate and $F_{\mu, 0}$ are the components of the field strength matrix evaluated at the spacetime point corresponding to $(t, \mathbf{x})$.

Given a spatial dimension $d$ and the speed of light $c$, the electric field $\mathbf{E}$ associated with a distributional electromagnetic potential $A$ is defined as the $\mathbb{R}$-linear map from the space of distributional 4-potentials to the space of vector-valued distributions on spacetime $\text{Time} \times \text{Space}_d$. The electric field is given by the formula: $$\mathbf{E} = -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t}$$ where $\phi$ is the distributional scalar potential, $\mathbf{A}$ is the distributional vector potential, $\nabla$ is the distributional spatial gradient, and $\frac{\partial}{\partial t}$ is the distributional time derivative.

For a spatial dimension $d$ and speed of light $c$, let $A$ be a distributional electromagnetic potential. For any spatial index $i \in \{0, \dots, d-1\}$ and any Schwartz test function $\epsilon \in \mathcal{S}(\text{Time} \times \text{Space}_d, \mathbb{R})$, the $i$-th component of the distributional electric field $\mathbf{E}$ evaluated at $\epsilon$ is given by $$E_i(\epsilon) = -c F^{0i}(\epsilon)$$ where $F^{0i}(\epsilon)$ denotes the component of the distributional field strength tensor $F$ corresponding to the temporal index $0$ and the spatial index $i$, evaluated at the test function $\epsilon$.