Magnetic Field Induced by Straight Currents on the Hyperboloid

Abstract

We consider the magnetic field induced by the steady or the quasi-steady electric currents that flow along the straight wires, which are equidistantly arranged on the hyperboloid. The spatial distribution of the magnetic field and the force acting on each wire are calculated. The continuum limit, where the wires are densely aligned, is also considered. We discuss the application of the hyperbolic current configuration to the generation of high magnetic fields.

Figures (8)

• Figure 1: The Kobe Port Tower, which is well known by the hyperbolic structure.
• Figure 2: Straight current flow on the hyperboloid, which is specified by Eq. (2). Only the finite area near the $xy$ plane is drawn from the infinitely extending lines.
• Figure 3: Left: Location of the $\ell$-th wire, which passes through the point P$\!_{\ell}^{~}$ on the $xy$-plane. Right: Orthonormal vectors ${\bm u}_{\ell}^{~}$, ${\bm v}_{\ell}^{~}$, and ${\bm w}_{\ell}^{~}$ defined in Eq. (4).
• Figure 4: The distance $d_{\ell}^{~}$ from the location ${\bm r}$ to the $\ell$-th wire.
• Figure 5: Field configuration on the $xy$-plane, when $N = 16$ and $\theta = \pi / 4$. Components parallel to the $xy$-plane are shown by the red arrows, and the $z$-components are shown by blue circles. Inside the gray area around the crossing points of the wires, which are shown by cross marks, the arrows and circles are not drawn.