What are the foundations of Maxwell’s equations and how do they relate to TENGs?

Gauss’s law for electricity, Gauss’s law for magnetism, Faraday’s law of induction, and Ampère’s circuital law are the foundations of Maxwell’s equations. But Maxwell saw what wasn’t there.

He added a term to one of the equations, providing a single mathematical model for electric, optical, and radio technologies and changing scientists’ understanding of the world. His impact extends from the prediction of electromagnetic waves to an explanation of triboelectric nanogenerators (TENGs) for energy harvesting and beyond.

Maxwell’s equations can be considered the culmination of an unplanned collaborative effort. An overview of the original four laws includes:

Gauss’s law for static electric fields
Gauss’s law for static magnetic fields
Faraday’s law describes how a magnetic field that changes over time produces an electric field.
Ampère’s circuital law applies to steady-state magnetic fields and electric currents.

Maxwell saw that Ampère’s circuital law could be improved. He added the concept of “displacement current” that describes how changing electric and magnetic fields interact with each other and with electric charges.

Like Ampère’s circuital law, the Ampère-Maxwell law starts with the induced magnetic field circulation on the left-hand side of the equation. On the right-hand side of the equation, Ampère’s law includes the product of the permeability of free space and the current. Maxwell’s modification kept the permeability of free space and added displacement current to the current term (Figure 1). Displacement current represents a current-like parameter arising from a changing electric field, even where no physical conduction current exists.

Figure 1. Maxwell added the concept of displacement current to the electric current (I_enclosed) on the right side of Ampère’s equation. (Image: Fosco)

What’s the big deal?

The Ampère-Maxwell law leads to several important conclusions. The addition of the displacement current enabled the prediction of electromagnetic waves. When combined with other parts of Maxwell’s equations, the extended law can derive the speed of light.

The concept of displacement current allows for a consistent explanation of magnetic field generation even when there is no physical movement of charge. It provides a solution to the capacitor charging/discharging operation that can’t be explained using Ampère’s circuital law. It’s also being used to improve the design of TENG energy harvesters and TENG-based near-field wireless communication.

New applications for displacement current

TENGs convert low-frequency and irregular mechanical energy into electricity. A basic TENG design consists of two materials that come into contact and separate, transferring charge and producing enough energy for low-power sensing and other applications.

Low-level energy harvesting is the most common application of the triboelectric effect, but it’s also being applied to displacement current-based near-field wireless communication. For example, high-frequency electromagnetic waves are mostly absorbed by water, but low-frequency electromagnetic waves produced using a TENG-based system can be effectively transmitted.

Fractal-structured nanogenerators (FSNGs) are also being developed to promise higher energy density than TENGs. Figure 2(a) shows the three-dimensional structure of the proposed FSNG, (b to e) illustrate the simulated charge and electric field distributions, and (f) details the charge transfer process during one cycle of operation.

Figure 2. Working principle and prototype of FSNG. (Image: Royal Society of Chemistry)

Summary

Gauss’s laws for electricity and magnetism, Faraday’s law of induction, and Ampère’s circuital law are the foundations of Maxwell’s equations. Maxwell added a factor to Ampère’s circuital law describing the concept of displacement current. That resulted in several unexpected developments, including the prediction of electromagnetic waves, calculation of the speed of light, understanding of how capacitors function, and the recent emergence of TENG-based energy harvesting and related near-field communication devices.