Induced voltage in an open wire

TL;DR

This paper resolves the puzzle of induced voltage in open wires within a time-varying magnetic field, revealing the contributions of electric field components and linking the voltage to geometric symmetry points or lines.

Contribution

It introduces a method to determine induced voltage in open wires considering electric field components and symmetry, extending Faraday's law to open circuits.

Findings

Longitudinal electric field contributes 1/3 to the voltage.

Transverse electric field contributes 2/3 to the voltage.

Induced voltage relates to the area with respect to symmetry points or lines.

Abstract

A puzzle arising from Faraday's law is considered and solved concerning the question which voltage is induced in an open wire with a time-varying homogeneous magnetic field. In contrast to closed wires where the voltage is determined by the time variance of magnetic field and enclosed area, in an open wire we have to integrate the electric field along the wire. It is found that the longitudinal electric field contributes with 1/3 and the transverse field with 2/3 to the induced voltage. In order to find the electric fields the sources of the magnetic fields are necessary to know. The representation of a homogeneous and time-varying magnetic field implies unavoidably a certain symmetry point or symmetry line which depend on the geometry of the source. As a consequence the induced voltage of an open wire is found to be the area covered with respect to this symmetry line or point perpendicular to the magnetic field. This in turn allows to find the symmetry points of a magnetic field source by measuring the voltage of an open wire placed with different angles in the magnetic field. We present exactly solvable models for a symmetry point and for a symmetry line, respectively. The results are applicable to open circuit problems like corrosion and for astrophysical applications.