The Analogy between Electromagnetics and Hydrodynamics

TL;DR

This paper explores a detailed analogy between electromagnetics and hydrodynamics, establishing correspondences for energy, fields, and forces, and deriving the Navier-Stokes equation for magnetic field evolution.

Contribution

It introduces a comprehensive analogy linking electromagnetic phenomena to fluid dynamics, including derivation of Navier-Stokes equations for magnetic fields.

Findings

Magnetic induction is analogous to fluid velocity

Lorenz force corresponds to the Lamb vector

Navier-Stokes equation is derived for magnetic field evolution

Abstract

The similarity between electromagnetics and hydrodynamics has been noticed for a long time. Maxwell developed an analogy, where the magnetic field and the vector potential in electromagnetics are compared to the vorticity and velocity in hydrodynamics, respectively. However, this theory cannot make a correspondence of energy between two subjects. In the present work, the electromagnetic fields in a conducting medium are compared to the flow fields of an incompressible Newtonian fluid. The result shows that the magnetic induction intensity, current density, Lorenz force, superconductor boundary, Ohm's law and Ampere force in electromagnetics are analogous to the velocity, vorticity, Lamb vector, solid boundary, Newton's law of viscosity and Kutta-Joukowski theorem of lift force in hydrodynamics, respectively. The Navier-Stokes equation is derived for the evolution of magnetic field in the medium by using the Maxwell equations, Lorenz force and Ohm's law. The work is useful for a deep understanding of electromagnetics and hydrodynamics.