[Paper Review] Maxwell's equations for a mechano-driven, shape-deformable, charged-media system, slowly moving at an arbitrary velocity field v(r,t)

Zhong Lin Wang|arXiv (Cornell University)|Feb 21, 2022

Mechanical and Optical Resonators26 references38 citations

TL;DR

This paper derives modified Maxwell’s equations for slow-moving, shape-deformable, charged media under arbitrary velocity fields v(r,t), using integral forms of fundamental laws instead of differential forms. The approach enables accurate modeling of mechano-electromagnetic coupling in non-inertial frames, offering a practical alternative to Lorentz-covariant electrodynamics for systems with acceleration, such as triboelectric nanogenerators.

ABSTRACT

The differential form of the Maxwell's equations was first derived based on an assumption that the media are stationary, which is the foundation for describing the electro-magnetic coupling behavior of a system. For a general case in which the medium has a time-dependent volume, shape and boundary and may move at an arbitrary velocity field v(r,t) and along a general trajectory, we derived the Maxwell's equations for a mechano-driven slow-moving media system directly starting from the integral forms of four physics laws, which should be accurate enough for describing the coupling among mechano-electro-magnetic interactions of a general system in practice although it may not be Lorentz covarance. Our key point is directly from the four physics laws by describing all of the fields, the space and the time in the frame where the observation is done. The equations should be applicable to not only moving charged solid and soft media that has acceleration, but also charged fluid/liquid media, e.g., fluid electrodynamics. This is a step toward the electrodynamics in non-inertia frame of references. General strategies for solving the Maxwell's equations for mechano-driven slowing moving medium are presented using the perturbation theory both in time and frequency spaces. Finally, approaches for the electrodynamics of moving media are compared, and related discussions are given about a few interesting questions.

Motivation & Objective

• To extend Maxwell’s equations beyond stationary media to systems with time- and space-dependent velocity fields v(r,t), enabling modeling of moving, deformable, charged media.
• To address the lack of practical electrodynamics frameworks for non-inertial reference frames where media experience acceleration, contrary to standard inertial-frame assumptions.
• To provide a non-Lorentz-covariant but accurate approach for engineering applications involving slow-moving, mechano-driven systems like triboelectric nanogenerators.
• To establish a foundation for Galilean electromagnetism in complex, non-uniform motion scenarios, bridging theory and practical electrodynamics.
• To challenge the dominance of special relativity in moving media electrodynamics by proposing a more feasible, non-relativistic framework for real-world systems with acceleration.

Proposed method

• Derives modified Maxwell’s equations directly from the integral forms of four fundamental laws (Gauss, Faraday, Ampere, Gauss for B) in the lab frame, without assuming stationary media.
• Uses a general mathematical framework based on time differentiation of surface integrals over moving, deforming surfaces, incorporating velocity field v(r,t) via a derived general flux theorem.
• Applies perturbation theory in both time and frequency domains to solve the resulting equations for slow-moving systems.
• Introduces a traveling-wave-like formulation equivalent to extended Hertz equations, valid in the low-velocity limit.
• Validates the approach through consistency checks with prior works (e.g., [14], [16], [17]) and demonstrates applicability to dielectrics, conductors, and fluids.
• Uses Stokes’ theorem and volume integral transformations to handle surface motion and deformation, deriving a key equation (A6) for flux time derivative in moving media.

Experimental results

Research questions

• RQ1 How can Maxwell’s equations be consistently extended to charged media undergoing arbitrary, time- and space-dependent motion with deformation?
• RQ2 What are the limitations of Lorentz-covariant electrodynamics for systems with non-uniform acceleration and non-inertial motion?
• RQ3 Can a non-relativistic, Galilean-based formulation accurately describe mechano-electromagnetic coupling in practical engineering systems?
• RQ4 How does the inclusion of a general velocity field v(r,t) affect the form of Faraday’s law and Ampere’s law in moving media?
• RQ5 What are the conditions under which the low-velocity approximation remains valid, especially when wave numbers and field gradients are involved?

Key findings

• The paper derives a generalized form of Maxwell’s equations for slow-moving, shape-deformable, charged media using integral laws, valid for arbitrary velocity fields v(r,t) and non-inertial frames.
• The derived equations are not Lorentz-covariant but are shown to be accurate and practical for systems with acceleration, such as triboelectric nanogenerators.
• A key mathematical result (equation A6) is established for the time derivative of flux through a moving, deforming surface, incorporating velocity field effects via surface and volume integrals.
• The approach is validated through consistency with prior works, including Li et al. [16] and Sheng et al. [17], who confirmed the validity of the low-velocity limit under specific conditions.
• The theory supports the use of Galilean electromagnetism as a practical alternative to special relativity for engineering applications involving slow, complex motion.
• The framework enables the modeling of fluid, soft, and solid charged media under mechanical driving, expanding the scope of electrodynamics beyond inertial frames.