Physlib.Electromagnetism.Current.InfiniteWire

For a given speed of light $c$, this linear map associates a current $I \in \mathbb{R}$ with the 4-current density distribution $J$ of an infinite wire positioned along the $x$-axis. The wire is located at $y=0$ and $z=0$ in three-dimensional space. The resulting distribution $J$ is steady (independent of time $t$) and has a vanishing charge density component. The spatial current density is concentrated on the $x$-axis, represented by a Dirac delta distribution $\delta(y, z)$ in the transverse plane. In components, the 4-current density is given by $J = (0, I \delta(y, z), 0, 0)$.

For an infinite wire carrying a steady current $I$ along the $x$-axis, the associated distributional charge density $\rho$ (defined as the temporal component of the 4-current density $J$) is zero.

For a given speed of light $c$ and current $I$, let $J$ be the distributional 4-current density of an infinite wire aligned with the $x$-axis ($y=0, z=0$). Let $\vec{j} = (j_x, j_y, j_z)$ be the spatial current density distribution extracted from $J$. For any Schwartz test function $\eta \in \mathcal{S}(\mathbb{R} \times \mathbb{R}^3, \mathbb{R})$, the $x$-component $j_x$ evaluated at $\eta$ is given by: $$(j_x, \eta) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} I \eta(t, x, 0, 0) \, dt \, dx$$ This represents a steady current density $j_x(t, x, y, z) = I \delta(y, z)$ that is constant in time and along the direction of the wire.

For a given speed of light $c$, a current $I \in \mathbb{R}$, and a test function $\varepsilon$ in the Schwartz space $\mathcal{S}(\text{Time} \times \text{Space}_3, \mathbb{R})$, the $y$-component (corresponding to index 1) of the distributional spatial current density $\vec{j}$ of an infinite wire carrying current $I$ along the $x$-axis is zero. That is, $\vec{j}_y(\varepsilon) = 0$.

For a given speed of light $c$, a current $I \in \mathbb{R}$, and a Schwartz test function $\varepsilon$ defined on spacetime $\text{Time} \times \text{Space}_3$, the $z$-component (the third spatial component, indexed by 2) of the spatial current density distribution $\vec{j}$ associated with an infinite wire carrying current $I$ along the $x$-axis is zero.

Given a free space $\mathcal{F}$ with magnetic permeability $\mu_0$ and a current $I \in \mathbb{R}$, the electromagnetic 4-potential $A$ of an infinite wire carrying a steady current $I$ along the $x$-axis is defined as the distribution corresponding to the function: $$A(t, x, y, z) = \left(0, -\frac{\mu_0 I}{2\pi} \log(\sqrt{y^2 + z^2}), 0, 0\right)$$ Here, the first component represents the scalar potential $V$, and the remaining three components represent the vector potential $\vec{A}$ in the $x, y, z$ directions respectively. The potential is independent of time $t$ and the $x$-coordinate, reflecting the translational symmetry of the infinite wire.

For any free space $\mathcal{F}$ and any steady current $I \in \mathbb{R}$, the scalar potential $V$ of the electromagnetic 4-potential associated with an infinite wire carrying the current $I$ along the $x$-axis is zero.

In a free space $\mathcal{F}$ with magnetic permeability $\mu_0$, let an infinite wire carry a steady current $I$ along the $x$-axis. The vector potential $\vec{A}$ of the corresponding electromagnetic 4-potential distribution is constant in time and independent of the $x$-coordinate. It is given by the distribution corresponding to the function: $$\vec{A}(t, x, y, z) = \left( -\frac{\mu_0 I}{2\pi} \log(\sqrt{y^2 + z^2}), 0, 0 \right)$$ where the vector points in the direction of the current (the $x$-axis) and its magnitude depends on the distance from the wire in the $y$-$z$ plane.

Consider a free space $\mathcal{F}$ with magnetic permeability $\mu_0$ and an infinite wire carrying a steady current $I$ along the $x$-axis. Let $\vec{A}$ be the vector potential of the electromagnetic 4-potential distribution associated with this wire. The $x$-component (the component with index $0$) of this vector potential distribution, acting on a test function $\eta \in \mathcal{S}(\text{Time} \times \text{Space}^3, \mathbb{R})$, is given by: $$\vec{A}_x(\eta) = \left( \text{constantTime} \left( \text{constantSliceDist}_0 \left( -\frac{\mu_0 I}{2\pi} \log(\sqrt{y^2 + z^2}) \right) \right) \right) (\eta)$$ This indicates that the $x$-component of the vector potential is constant in time and along the $x$-axis, and its value at any point $(y, z)$ in the transverse plane is proportional to $\log(\sqrt{y^2 + z^2})$.

In a free space $\mathcal{F}$, for an infinite wire carrying a steady current $I$ along the $x$-axis, the $y$-component (index 1) of the vector potential distribution $\vec{A}$ is zero.

For an infinite wire in a free space $\mathcal{F}$ carrying a steady current $I$ along the $x$-axis, let $\vec{A}$ be the magnetic vector potential distribution. The $z$-component (the third component) of this vector potential is zero, i.e., $A_z = 0$.

In a free space $\mathcal{F}$, for an infinite wire carrying a steady current $I$ along the $x$-axis, let $\vec{A}$ be the associated vector potential distribution. The distributional time derivative of this vector potential is zero: $$\frac{\partial \vec{A}}{\partial t} = 0$$ This reflects the fact that the electromagnetic field of a wire carrying a steady current is static.

In free space $\mathcal{F}$, consider an infinite wire carrying a steady current $I$ along the $x$-axis. The spatial derivative of the resulting vector potential $\vec{A}$ with respect to the $x$-coordinate is zero: $$\frac{\partial \vec{A}}{\partial x} = 0$$ This result reflects the translational symmetry of the system along the direction of the wire.

For any free space $\mathcal{F}$ and any steady current $I \in \mathbb{R}$, the electric field $\vec{E}$ produced by an infinite wire carrying the current $I$ along the $x$-axis is zero: $$\vec{E} = 0$$ This result is consistent with the fact that for a steady current, the scalar potential is zero and the vector potential is time-independent, leading to a vanishing electric field.

In a free space environment $\mathcal{F}$, for an infinite wire carrying a steady current $I$ along the $x$-axis, the associated distributional electromagnetic 4-potential $A$ is an extremum of the electromagnetic Lagrangian density $\mathcal{L}$ with respect to the distributional 4-current density $J$. This condition, $\frac{\delta \mathcal{L}}{\delta A} = 0$, signifies that the potential $A$ satisfies Maxwell's equations for the source $J$.