Surface charges and external electric field in a toroid carrying a steady current

Abstract

We solve the problem of a resistive toroid carrying a steady azimuthal current. We use standard toroidal coordinates, in which case Laplace's equation is R-separable. We obtain the electric potential inside and outside the toroid, in two separate cases: 1) the toroid is solid; 2) the toroid is hollow (a toroidal shell). Considering these two cases, there is a difference in the potential inside the hollow and solid toroids. We also present the electric field and the surface charge distribution in the conductor due to this steady current. These surface charges generate not only the electric field that maintains the current flowing, but generate also the electric field outside the conductor. The problem of a toroid is interesting because it is a problem with finite geometry, with the whole system (including the battery) contained within a finite region of space. The problem is solved in an exact analytical form. We compare our theoretical results with an experimental figure demonstrating the existence of the electric field outside the conductor carrying steady current. Toroidal RingThe electric field outside conductors with steady currents has been studied in a number of cases. Here we consider the case of the conducting toroid with a steady current. Our goal is to find the electric potential inside and outside the toroid, and from the potential we can find the electric field and surface charges. More details about this problem and its analytical solution can be found in [12].Consider a toroidal conductor with uniform resistivity. It has greater radius R 0 and smaller radius r 0 and carries a steady current I in the azimuthal direction, flowing along the circular loop. The toroid has rotational symmetry around the z-axis and is centered in the plane z = 0. The battery that maintains the current is located at ϕ = π rad, see Fig. 1. Air or vacuum surrounds the conductor.The electric potential φ can be calculated using toroidal coordinates (η, ξ, ϕ) [13, p. 112], defined by:(1) Figure 1. A toroidal ohmic conductor with symmetry axis z, smaller radius r 0 (m) and greater radius R (m). A thin battery is located at ϕ = π rad maintaining constant potentials (represented as the "+" and "-" signs) in its extremities. A steady current flows azimuthally in this circuit loop in the clockwise direction, from ϕ = +π rad to ϕ = −π rad.Here, a is a constant such that when η → ∞ we have the circle x = a cos ϕ, y = a sin ϕ and z = 0. The toroidal coordinates can have the possible values: 0 ≤ η < ∞, −π rad ≤ ξ ≤ π rad and −π rad ≤ ϕ ≤ π rad. We take η 0 as a constant that described the toroid surface in toroidal coordinates. The internal (external) region of the toroid is characterized by η > η 0 (η < η 0 ).

Computational physics in action: the problem of a charged conducting toroid

“…Using equations (3)- (6) and relation E = −∇Φ we can find numerically the absolute value E of the electric field for two special cases discussed in the previous section. Since the series in equation ( 3) converges, we can hold a large, but a finite number of terms for numerical calculations.…”

“…where C 0 = 8 0 R. For the thin toroid approximation (r R) this equation can be simplified into the following form [6]…”

“…The electrostatics of such an object is considered in a number of original studies. Hernandes and Assis [5,6] have found the surface charges and external electric field in a toroid, carrying a steady current. Scharstein and Wilson [7] have considered the electrostatic excitation of a conducting toroid.…”

TE, TM Fields in Toroidal Electromagnetism

“…Toroidal functions have been known for a long time and an important bibliography is given in [1,2] with some applications to diffraction of acoustic and electromagntic plane waves by a torus. These functions have played a major role in the analysis of plasma confinement in a torus configuration [3], where further references can be found] and, more recently, important works have been devoted to their application in particular situations [4][5][6].…”