On the first principle theory of nanogenerators from Maxwell's equations

Nanogenerators (NGs) are a field that uses Maxwell's displacement current as the driving force for effectively converting mechanical energy into electric power/signal, which have broad applications in energy science, environmental protection, wearable electronics, self-powered sensors, medical science, robotics and artificial intelligence. NGs are usually based on three effects: piezoelectricity, triboelectricity (contact electrification), and pyroelectricity. In this paper, a formal theory for NGs is presented starting from Maxwell's equations. Besides the general expression for displacement vector D = εE used for deriving classical electromagnetic dynamics, we added an additional term Ps in D, which represents the polarization created by the electrostatic surface charges owing to piezoelectricity and/or triboelectricity as a result of mechanical triggering in NG. In contrast to P that is resulted from the electric field induced medium polarization and vanishes if E = 0, Ps remains even when there is no external electric field. We reformulated the Maxwell equations that include both the medium polarizations due to electric field (P) and non-electric field (such as strain) (Ps) induced polarization terms, from which, the output power, electromagnetic behavior and current transport equation for a NG are systematically derived. A general solution is presented for the modified Maxwell equations, and analytical solutions about the output potential are provided for a few cases. The displacement current arising from ε∂E/∂t is responsible for the electromagnetic waves, while the newly added term ∂Ps/∂t is the application of Maxwell's equations in energy and sensors. This work sets the first principle theory for quantifying the performance and electromagnetic behavior of a nanogenerator in general.