Physlib.Electromagnetism.Kinematics.MagneticField

For an electromagnetic potential $A$ and a given speed of light $c$, the magnetic field $\mathbf{B}$ is equal to the curl of the vector potential $\mathbf{A}$. That is, for any time $t$ and spatial position $\mathbf{x}$, the field is given by: $$\mathbf{B}(t, \mathbf{x}) = \nabla \times \mathbf{A}(t, \mathbf{x})$$

For an electromagnetic potential $A$ and a speed of light $c$, let $\mathbf{B}(t, \mathbf{x})$ be the magnetic field at time $t$ and spatial position $\mathbf{x}$, and let $F$ be the field strength matrix (Faraday tensor). If $A$ is differentiable, then the first component (the $x$-component) of the magnetic field is equal to the negative of the $(y, z)$ spatial component of the field strength matrix: $$B_x(t, \mathbf{x}) = -F_{yz}(t, \mathbf{x})$$ Here, the index $0$ of the magnetic field corresponds to the $x$-direction, and the indices $1$ and $2$ of the spatial part of the field strength matrix correspond to the $y$ and $z$ directions, respectively.

For an electromagnetic potential $A$ and a given speed of light $c$, let $\mathbf{B}$ be the associated magnetic field and $F$ be the field strength matrix (Faraday tensor). If $A$ is differentiable, then the second component of the magnetic field $\mathbf{B}$ at time $t$ and spatial position $\mathbf{x}$ is equal to the entry of the field strength matrix corresponding to the first and third spatial indices. Mathematically, this is expressed as: $$B_1(t, \mathbf{x}) = F_{\text{inr } 0, \text{inr } 2}(P)$$ where $P = (t, \mathbf{x})$ is the coordinate in spacetime, and indices $0, 1, 2$ represent the spatial dimensions.

Let $c$ be the speed of light and $A$ be a differentiable electromagnetic potential. For any time $t$ and spatial position $\mathbf{x}$, the third component of the magnetic field $\mathbf{B}$ is equal to the negative of the component of the field strength matrix $F$ corresponding to the first and second spatial indices. That is, $$B_2(t, \mathbf{x}) = -F_{01}(t, \mathbf{x})$$ where the indices $0$ and $1$ on the right-hand side refer to the first two spatial dimensions in the spacetime index set.

Let $A$ be an electromagnetic potential that is twice continuously differentiable ($\mathcal{C}^2$) with respect to the real numbers. For any given time $t$, the spatial divergence of the magnetic field $\mathbf{B}$ at that time is zero. That is: $$\nabla \cdot \mathbf{B}(t, \mathbf{x}) = 0$$ where $\mathbf{B}$ is the magnetic field defined by the curl of the vector potential $\mathbf{A}$ associated with $A$.

Let $c$ be the speed of light and $A$ be a differentiable electromagnetic potential. For any time $t$ and spatial position $\mathbf{x} \in \mathbb{R}^3$, the field strength matrix (Faraday tensor) $F_{\mu\nu}$ at the spacetime point $(t, \mathbf{x})$ is expressed in terms of the components of the electric field $\mathbf{E} = (E_0, E_1, E_2)$ and the magnetic field $\mathbf{B} = (B_0, B_1, B_2)$ as follows: 1. The time-time component is zero: $F_{00} = 0$. 2. The time-space and space-time components are given by the electric field: $F_{0, i+1} = -\frac{E_i}{c}$ and $F_{i+1, 0} = \frac{E_i}{c}$ for $i \in \{0, 1, 2\}$. 3. The spatial components $F_{i+1, j+1}$ for $i, j \in \{0, 1, 2\}$ are given by: $$F_{spatial} = \begin{pmatrix} 0 & -B_2 & B_1 \\ B_2 & 0 & -B_0 \\ -B_1 & B_0 & 0 \end{pmatrix}$$ Specifically: - $F_{11} = F_{22} = F_{33} = 0$ - $F_{12} = -B_2, \quad F_{21} = B_2$ - $F_{13} = B_1, \quad F_{31} = -B_1$ - $F_{23} = -B_0, \quad F_{32} = B_0$

Let $c$ be the speed of light and $A$ be a differentiable electromagnetic potential. For any point $x$ in spacetime, let $(t, \mathbf{x})$ be the corresponding time and spatial coordinates. The components of the field strength matrix (Faraday tensor) $F_{\mu\nu}$ at the point $x$ are determined by the electric field $\mathbf{E}(t, \mathbf{x}) = (E_0, E_1, E_2)$ and the magnetic field $\mathbf{B}(t, \mathbf{x}) = (B_0, B_1, B_2)$ according to the following relations: 1. The temporal component is zero: $F_{00} = 0$. 2. The mixed temporal-spatial components are given by the electric field: $F_{0, i+1} = -\frac{E_i}{c}$ and $F_{i+1, 0} = \frac{E_i}{c}$ for $i \in \{0, 1, 2\}$. 3. The spatial components $F_{i+1, j+1}$ for $i, j \in \{0, 1, 2\}$ form the following antisymmetric matrix: $$\begin{pmatrix} F_{11} & F_{12} & F_{13} \\ F_{21} & F_{22} & F_{23} \\ F_{31} & F_{32} & F_{33} \end{pmatrix} = \begin{pmatrix} 0 & -B_2 & B_1 \\ B_2 & 0 & -B_0 \\ -B_1 & B_0 & 0 \end{pmatrix}$$ Specifically, $F_{12} = -B_2$, $F_{13} = B_1$, $F_{23} = -B_0$, and the diagonal elements are zero.

For a speed of light $c$ and an electromagnetic potential $A$ in $d$ spatial dimensions, the magnetic field matrix $B$ at time $t$ and spatial position $\mathbf{x}$ is defined by the spatial components of the field strength matrix $F$. Specifically, for any spatial indices $i, j \in \{1, \dots, d\}$, the entry $B_{ij}(t, \mathbf{x})$ is equal to the component of the field strength matrix $F$ evaluated at the spacetime point corresponding to $(t, \mathbf{x})$ with the indices $\mu, \nu$ corresponding to the $i$-th and $j$-th spatial components: $$B_{ij}(t, \mathbf{x}) = F_{ij}(t, \mathbf{x})$$

For a given speed of light $c$ and an electromagnetic potential $A$ in $d$ spatial dimensions, the spatial-spatial components of the field strength matrix $F$ at a spacetime point $x$ are equal to the components of the magnetic field matrix $B$ at the corresponding time $t$ and spatial position $\mathbf{x}$. Specifically, for any spatial indices $i, j \in \{1, \dots, d\}$, it holds that: $$F_{ij}(x) = B_{ij}(t, \mathbf{x})$$ where $t = \text{time}(x, c)$ and $\mathbf{x} = \text{space}(x)$.

For an electromagnetic potential $A$ in $d$ spatial dimensions and a speed of light $c$, the magnetic field matrix $B$ evaluated at time $t$ and spatial position $\mathbf{x}$ is antisymmetric. Specifically, for any spatial indices $i, j \in \{1, \dots, d\}$, the entries of the matrix satisfy: $$B_{ij}(t, \mathbf{x}) = -B_{ji}(t, \mathbf{x})$$

For an electromagnetic potential $A$ in $d$ spatial dimensions and a speed of light $c$, the diagonal entries of the magnetic field matrix $B(t, \mathbf{x})$ are zero for any time $t$ and spatial position $\mathbf{x}$. Specifically, for any spatial index $i \in \{0, \dots, d-1\}$, the entry $B_{ii}(t, \mathbf{x})$ satisfies: $$B_{ii}(t, \mathbf{x}) = 0$$

For an electromagnetic potential $A$ in 3 spatial dimensions and a given speed of light $c$, if $A$ is differentiable, then the magnetic field vector $\mathbf{B}$ at time $t$ and spatial position $\mathbf{x}$ is determined by the entries of the magnetic field matrix $B_{ij}$ as follows: $$B_0(t, \mathbf{x}) = -B_{1, 2}(t, \mathbf{x})$$ $$B_1(t, \mathbf{x}) = B_{0, 2}(t, \mathbf{x})$$ $$B_2(t, \mathbf{x}) = -B_{0, 1}(t, \mathbf{x})$$ where the indices $0, 1, 2$ correspond to the three spatial dimensions.

For an electromagnetic potential $A$ in 3-dimensional space that is twice continuously differentiable ($C^2$), and given a speed of light $c$, the $i$-th component of the curl of the magnetic field $\mathbf{B}$ at time $t$ and spatial position $\mathbf{x}$ is equal to the sum of the partial derivatives of the entries of the magnetic field matrix $B_{ji}$ with respect to the $j$-th spatial coordinate: $$(\nabla \times \mathbf{B})_i(t, \mathbf{x}) = \sum_{j=0}^2 \frac{\partial}{\partial x_j} B_{ji}(t, \mathbf{x})$$ where $\mathbf{B}$ is the magnetic field vector and $B_{ji}$ denotes the components of the magnetic field matrix.

For an electromagnetic potential $A$ in $d$ spatial dimensions that is differentiable, and given a speed of light $c$, the components of the magnetic field matrix $B$ at time $t$ and spatial position $\mathbf{x}$ are given by the difference of the partial derivatives of the vector potential components: $$B_{ij}(t, \mathbf{x}) = \partial_j A_i(t, \mathbf{x}) - \partial_i A_j(t, \mathbf{x})$$ where $A_k(t, \mathbf{x})$ is the $k$-th component of the vector potential $\vec{A}$ at time $t$ and position $\mathbf{x}$, and $\partial_k$ denotes the partial derivative with respect to the $k$-th spatial coordinate for indices $i, j \in \{1, \dots, d\}$.

Let $d$ be the spatial dimension and $c$ be the speed of light. For an electromagnetic potential $A$, if $A$ is $C^{n+1}$ smooth, then for any indices $i$ and $j$, the $(i, j)$-th component of the magnetic field matrix, $B_{ij}(t, x)$, is $C^n$ smooth as a function of time $t$ and spatial position $x$.

Let $d$ be the spatial dimension and $c$ be the speed of light. For an electromagnetic potential $A$, if $A$ is $C^{n+1}$ smooth, then for any fixed time $t$ and any indices $i$ and $j$, the $(i, j)$-th component of the magnetic field matrix, $B_{ij}(t, x)$, is $C^n$ smooth as a function of the spatial position $x$.

Let $d$ be the spatial dimension and $c$ be the speed of light. For an electromagnetic potential $A$, if $A$ is $C^{n+1}$ smooth, then for any fixed spatial position $x$ and any indices $i$ and $j$, the $(i, j)$-th component of the magnetic field matrix $B_{ij}(t, x)$ is $C^n$ smooth as a function of time $t$.

For a given spatial dimension $d$ and speed of light $c$, let $A$ be an electromagnetic potential that is twice continuously differentiable ($C^2$) over the space-time manifold. Then, for any indices $i$ and $j$, the $(i, j)$-th component of the magnetic field matrix $B_{ij}(t, x)$ is differentiable with respect to the pair $(t, x)$, where $t$ denotes time and $x$ denotes the spatial position.

For a given spatial dimension $d$ and speed of light $c$, let $A$ be an electromagnetic potential that is twice continuously differentiable ($C^2$). For any fixed time $t$ and indices $i$ and $j$, the $(i, j)$-th component of the magnetic field matrix $B_{ij}(t, x)$ is differentiable with respect to the spatial position $x \in \text{Space } d$.

For a given spatial dimension $d$ and speed of light $c$, let $A$ be an electromagnetic potential that is twice continuously differentiable ($C^2$). Then, for any fixed spatial position $x$ and any indices $i$ and $j$, the $(i, j)$-th component of the magnetic field matrix $B_{ij}(t, x)$ is differentiable with respect to time $t$.

For an electromagnetic potential $A$ in $d$ spatial dimensions that is twice continuously differentiable ($C^2$), let $B_{ij}(t, \mathbf{x})$ denote the components of the magnetic field matrix at time $t$ and spatial position $\mathbf{x}$. For any spatial indices $i, j, k \in \{1, \dots, d\}$, the following identity holds: $$\partial_k B_{ij}(t, \mathbf{x}) = \partial_i B_{kj}(t, \mathbf{x}) - \partial_j B_{ki}(t, \mathbf{x})$$ where $\partial_m$ denotes the partial derivative with respect to the $m$-th spatial coordinate.

For a given spatial dimension $d$ and speed of light $c$, let $A$ be an electromagnetic potential that is twice continuously differentiable ($C^2$). For any time $t$, spatial position $\mathbf{x}$, and spatial indices $i, j \in \{1, \dots, d\}$, the time derivative of the $(i, j)$-th component of the magnetic field matrix $B_{ij}$ is given by the difference of the spatial derivatives of the electric field components: $$\frac{\partial B_{ij}(t, \mathbf{x})}{\partial t} = \partial_i E_j(t, \mathbf{x}) - \partial_j E_i(t, \mathbf{x})$$ where $E_k(t, \mathbf{x})$ is the $k$-th component of the electric field at time $t$ and position $\mathbf{x}$, and $\partial_m$ denotes the partial derivative with respect to the $m$-th spatial coordinate.

For an electromagnetic potential $A$ in $d$ spatial dimensions that is thrice continuously differentiable ($C^3$), let $B_{ij}(t, \mathbf{x})$ be the components of the magnetic field matrix and $E_k(t, \mathbf{x})$ be the components of the electric field at time $t$ and spatial position $\mathbf{x}$. For any spatial indices $i, j \in \{1, \dots, d\}$, the second partial derivative of the magnetic field matrix components with respect to time is given by: $$\frac{\partial^2 B_{ij}(t, \mathbf{x})}{\partial t^2} = \partial_i \left( \frac{\partial E_j(t, \mathbf{x})}{\partial t} \right) - \partial_j \left( \frac{\partial E_i(t, \mathbf{x})}{\partial t} \right)$$ where $\partial_k$ denotes the partial derivative with respect to the $k$-th spatial coordinate and $\partial_t$ denotes the partial derivative with respect to time.

Given a spatial dimension $d$ and speed of light $c$, let $A$ be an electromagnetic potential that is twice continuously differentiable ($C^2$). For any time $t$, spatial position $x$, and spatial index $i \in \{1, \dots, d\}$, the divergence-like sum of the magnetic field matrix $B$ satisfies the following identity: $$\sum_{j=1}^d \partial_j B_{ji}(t, x) = \frac{1}{c^2} \frac{\partial E_i(t, x)}{\partial t} + \sum_{\mu=0}^d \partial_\mu F_{\mu i}(t, x)$$ where $B_{ji}$ are the components of the magnetic field matrix, $E_i$ is the $i$-th component of the electric field, and $F_{\mu i}$ are the components of the field strength matrix. The index $j$ ranges over the spatial dimensions, while $\mu$ ranges over both time (index 0) and space.

For a given spatial dimension $d$ and speed of light $c$, the magnetic field matrix is a linear map that associates a distributional electromagnetic potential $A$ with a distribution mapping test functions on spacetime (Time $\times$ Space $d$) to the tensor product space $\mathbb{R}^d \otimes \mathbb{R}^d$. This map is defined by extracting the spatial-spatial components of the field strength distribution $F_{\mu\nu}$. For a test function $\varepsilon$, the $(i, j)$-th component of the resulting matrix corresponds to the distributional derivative $\partial_j A_i - \partial_i A_j$, where $A_i$ and $A_j$ are the spatial components of the vector potential.

For a given spatial dimension $d$ and speed of light $c$, let $A$ be a distributional electromagnetic potential and $\varepsilon$ be a test function in the Schwartz space $\mathcal{S}(\text{Time} \times \text{Space}^d, \mathbb{R})$. The magnetic field matrix distribution evaluated at $\varepsilon$ is given by the sum over spatial indices $i, j \in \{1, \dots, d\}$: $$B(\varepsilon) = \sum_{i=1}^d \sum_{j=1}^d \left( \partial_j A_i(\varepsilon) - \partial_i A_j(\varepsilon) \right) \mathbf{e}_i \otimes \mathbf{e}_j$$ where $A_i$ is the $i$-th component of the distributional vector potential, $\partial_j$ denotes the distributional partial derivative with respect to the $j$-th spatial coordinate, and $\mathbf{e}_i \otimes \mathbf{e}_j$ are the basis elements of the tensor product space $\mathbb{R}^d \otimes \mathbb{R}^d$.

For a given spatial dimension $d$ and speed of light $c$, let $A$ be a distributional electromagnetic potential and $\varepsilon \in \mathcal{S}(\text{Time} \times \mathbb{R}^d, \mathbb{R})$ be a test function in the Schwartz space. Let $B(\varepsilon)$ denote the magnetic field matrix distribution evaluated at $\varepsilon$. Then, for any spatial indices $i, j \in \{1, \dots, d\}$, the $(i, j)$-th component of the magnetic field matrix in the standard tensor basis is given by: $$(B(\varepsilon))_{ij} = \partial_j A_i(\varepsilon) - \partial_i A_j(\varepsilon)$$ where $A_k$ is the $k$-th component of the distributional vector potential and $\partial_k$ denotes the distributional partial derivative with respect to the $k$-th spatial coordinate.

For a given spatial dimension $d$ and speed of light $c$, let $A$ be a distributional electromagnetic potential and $\varepsilon \in \mathcal{S}(\text{Time} \times \mathbb{R}^d, \mathbb{R})$ be a test function in the Schwartz space. Let $B$ denote the magnetic field matrix distribution associated with $A$. For any spatial indices $i, j, k \in \{1, \dots, d\}$, the $(i, j)$-th component of the $k$-th distributional spatial partial derivative of the magnetic field matrix evaluated at $\varepsilon$ is given by: $$(\partial_k B(\varepsilon))_{ij} = \partial_k \partial_j A_i(\varepsilon) - \partial_k \partial_i A_j(\varepsilon)$$ where $A_m$ is the $m$-th component of the distributional vector potential and $\partial_n$ denotes the distributional partial derivative with respect to the $n$-th spatial coordinate.

For a given spatial dimension $d$ and speed of light $c$, let $A$ be a distributional electromagnetic potential and $\varepsilon \in \mathcal{S}(\text{Time} \times \text{Space}^d, \mathbb{R})$ be a test function in the Schwartz space. For any spatial indices $i, j \in \{0, \dots, d-1\}$, the $(i, j)$-th component of the magnetic field matrix distribution $B$ evaluated at $\varepsilon$ is equal to the spatial-spatial $(\text{inr } i, \text{inr } j)$-th component of the field strength tensor distribution $F$ evaluated at $\varepsilon$. This is expressed as: $$(B(\varepsilon))_{ij} = (F(\varepsilon))_{\text{inr } i, \text{inr } j}$$ where $\text{inr } i$ and $\text{inr } j$ denote the indices in the spacetime manifold corresponding to the spatial dimensions.

For a distributional electromagnetic potential $A$ in one spatial dimension ($d = 1$) and a given speed of light $c$, the magnetic field matrix $B$ is zero ($B = 0$).