Quantum mechanics

Inhomogeneous electromagnetic wave equation

Differential equation

Optical field

Classical mechanics

Physics

Scalar (mathematics)

Mathematics

Equations of motion

Electromagnetic field

Geometry

Maxwell's equations

Materials Today

The conventional Maxwell’s equations are for media whose boundaries and volumes are fixed. But for cases that involve moving media and time-dependent configuration, the equations have to be expanded. Here, starting from the integral form of the Maxwell’s equations for general cases, we first derived the expanded Maxwell’s equations in differential form by assuming that the medium is moving as a rigid translation object. Secondly, the expanded Maxwell’s equations are further developed with including the polarization density term Ps in displacement vector owing to electrostatic charges on medium surfaces as produced by effect such as triboelectrification, based on which the first principle theory for the triboelectric nanogenerators (TENGs) is developed. The expanded equations are the most comprehensive governing equations including both electromagnetic interaction and power generation as well as their coupling. Thirdly, general approaches are presented for solving the expanded Maxwell’s equations using vector and scalar potentials as well as perturbation theory, so that the scheme for numerical calculations is set. Finally, we investigated the conservation of energy as governed by the expanded Maxwell’s equations, and derived the general approach for calculating the displacement current ∂∂tPs for the output power of TENGs. The current theory is general and it may impact the electromagnetic wave generation and interaction (reflection) with moving train/car, flight jets, missiles, comet, and even galaxy stars if observed from earth.

On the expanded Maxwell’s equations for moving charged media system – General theory, mathematical solutions and applications in TENG