PhyslibAlpha.Electromagnetism.Distributional.WireJunction

Let $\mathbf{f}$ be a vector-valued distribution defined on the spacetime domain $\text{Time} \times \text{Space } d$. The temporal derivative $\partial_t$ and the spatial divergence operator $\nabla \cdot$ commute, such that: $$\frac{\partial}{\partial t} (\nabla \cdot \mathbf{f}) = \nabla \cdot \left( \frac{\partial \mathbf{f}}{\partial t} \right)$$ where $\nabla \cdot \mathbf{f}$ is the distributional spatial divergence and $\frac{\partial}{\partial t}$ is the distributional temporal derivative.

For any dimension $d$, let $\chi \in \mathcal{S}(\text{Space } d, \mathbb{R})$ be a Schwartz function and $w \in \text{Space } d$ be a non-zero vector. Then there exists a Schwartz function $g \in \mathcal{S}(\mathbb{R}, \mathbb{R})$ such that for all $s \in \mathbb{R}$, $g(s) = \chi(s \cdot w)$. Here, $s \cdot w$ represents the parametrization of a line through the origin in the direction of $w$.

Let $\psi$ be a Schwartz function on $\mathbb{R}^d$ (denoted by $\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{R})$) and let $w \in \mathbb{R}^d$ be a non-zero vector. The integral along the ray $(0, \infty)$ in the direction of $w$ of the directional derivative of $\psi$ in the direction $w$ is equal to the negative value of $\psi$ at the origin: $$\int_{0}^{\infty} D\psi(sw)[w] \, ds = -\psi(0)$$ where $D\psi(x)[w]$ denotes the Fréchet derivative of $\psi$ at the point $x$ evaluated in the direction $w$. This result is a version of the fundamental theorem of calculus applied to a function restricted to a ray.

Let $\psi: \mathbb{R}^d \to \mathbb{R}$ be a Schwartz function and $u \in \mathbb{R}^d$ be a non-zero vector with components $u_i$ relative to the standard orthonormal basis $\{e_i\}$. Then the sum over the components of $u$ multiplied by the integral along the ray $s \mapsto su$ of the partial derivatives of $\psi$ equals $-\psi(0)$: $$\sum_i \left( \int_0^\infty \frac{\partial \psi}{\partial x_i}(su) \, ds \right) u_i = -\psi(0)$$ where $\frac{\partial \psi}{\partial x_i}$ denotes the directional derivative of $\psi$ in the direction of the basis vector $e_i$.

Let $\eta \in \mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$ be a Schwartz function on spacetime. For any index $i \in \{0, \dots, d-1\}$ of the standard orthonormal basis $\{e_i\}$ of $\text{Space } d$ and any point $x \in \text{Space } d$, the integral over time of the partial spatial derivative of $\eta$ is equal to the partial spatial derivative of the time integral of $\eta$: $$\int_{\text{Time}} \frac{\partial \eta}{\partial x_i}(t, x) \, dt = \frac{\partial}{\partial x_i} \left( \int_{\text{Time}} \eta(t, x) \, dt \right)$$ where $\frac{\partial \eta}{\partial x_i}$ denotes the Fréchet derivative of $\eta$ at $(t, x)$ evaluated in the direction of the spatial basis vector $(0, e_i)$.

There exists a Schwartz function $\eta$ defined on spacetime $\text{Time} \times \mathbb{R}^d$ such that its integral over time at the spatial origin $0 \in \mathbb{R}^d$ is non-zero, i.e., $$\int_{\text{Time}} \eta(t, 0) \, dt \neq 0.$$

Let $d$ be the spatial dimension and $\mathcal{F}$ be a free space environment with speed of light $c$. Suppose $A$ is a distributional electromagnetic potential and $J$ is a distributional Lorentz current density such that $A$ is an extremum of the electromagnetic Lagrangian density (i.e., Maxwell's equations are satisfied). Then the distributional charge density $\rho$ and spatial current density $\mathbf{j}$ satisfy the continuity equation. Specifically, for any Schwartz test function $\varepsilon$ on spacetime $\text{Time} \times \text{Space}^d$: $$\frac{\partial \rho}{\partial t}(\varepsilon) + (\nabla \cdot \mathbf{j})(\varepsilon) = 0$$ where $\rho = \frac{1}{c} J^0$ is the distributional charge density and $\mathbf{j}$ is the distributional spatial current density.

A distributional Lorentz current density $J$ is a **junction of thin wires** with directions $\vec{u}_k \in \mathbb{R}^d$ and steady currents $I_k \in \mathbb{R}$ (where $k$ ranges over a finite index set $\iota$) if, given the speed of light $c$, its spatial current density distribution $\vec{j}$ satisfies: $$\langle \vec{j}, \eta \rangle = \sum_{k \in \iota} \left( I_k \int_0^\infty \bar{\eta}(s \vec{u}_k) \, ds \right) \vec{u}_k$$ for all Schwartz test functions $\eta \in \mathcal{S}(\text{Time} \times \text{Space}_d, \mathbb{R})$, where $\bar{\eta}(x) = \int_{\text{Time}} \eta(t, x) \, dt$ is the time-integrated test function. Physically, this defines $J$ such that each wire $k$ is a semi-infinite ray $s \mapsto s \vec{u}_k$ for $s > 0$ carrying a constant current $I_k$ flowing away from the origin.

Let $d$ be the spatial dimension and $c$ be the speed of light. Suppose $J$ is a distributional Lorentz current density representing a junction of thin wires with non-zero direction vectors $\vec{u}_k \in \mathbb{R}^d$ and steady currents $I_k \in \mathbb{R}$ for $k$ in a finite index set $\iota$. For any Schwartz test function $\eta$ defined on spacetime $\text{Time} \times \text{Space}_d$, the spatial divergence $\nabla \cdot \mathbf{j}$ of the spatial current density $\mathbf{j}$ satisfies: $$\langle \nabla \cdot \mathbf{j}, \eta \rangle = \left( \sum_{k \in \iota} I_k \right) \int_{\text{Time}} \eta(t, \mathbf{0}) \, dt$$ where $\mathbf{0}$ is the origin in $\text{Space}_d$. This identifies the divergence as a point source at the origin with a strength equal to the total current leaving the junction.

Consider a junction of thin wires in a $d$-dimensional space, where each wire $k$ in a finite set $\iota$ has a non-zero direction vector $\vec{u}_k \in \mathbb{R}^d$ and carries a steady current $I_k \in \mathbb{R}$. Let $J$ be the distributional Lorentz current density associated with this junction. Suppose there exists a distributional electromagnetic potential $A$ such that $A$ is an extremum of the electromagnetic Lagrangian density in free space $\mathcal{F}$ (implying Maxwell's equations hold). If the distributional charge density $\rho = \frac{1}{c} J^0$ satisfies the condition of no charge accumulation, expressed as the vanishing of its distributional time derivative $\partial_t \rho = 0$, then the sum of the currents leaving the junction is zero: $$\sum_{k \in \iota} I_k = 0$$