Poynting Vector Flow in a Circular Circuit

TL;DR

This paper analyzes the Poynting vector in a circular circuit with a steady current, revealing energy flow both inside and outside the circuit, and provides analytical and numerical expressions for it.

Contribution

It derives a comprehensive expression for the Poynting vector in a realistic circular circuit, extending previous idealized models to include external energy flow.

Findings

Energy flows outside the circuit wire.

Total power matches energy flow into the wire.

Provides analytical and numerical evaluations of the Poynting vector.

Abstract

A circuit is considered in the shape of a ring, with a battery of negligible size and a wire of uniform resistance. A linear charge distribution along the wire maintains an electrostatic field and a steady current, which produces a constant magnetic field. Earlier studies of the Poynting vector and the rate of flow of energy considered only idealized geometries in which the Poynting vector was confined to the space within the circuit. But in more realistic cases the Poynting vector is nonzero outside as well as inside the circuit. An expression is obtained for the Poynting vector in terms of products of integrals, which are evaluated numerically to show the energy flow. Limiting expressions are obtained analytically. It is shown that the total power generated by the battery equals the energy flowing into the wire per unit time.