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All About Wave Equations
Published in Bahman Zohuri, Patrick J. McDaniel, Electrical Brain Stimulation for the Treatment of Neurological Disorders, 2019
Bahman Zohuri, Patrick J. McDaniel
Now tracing the derivation of Eq. (2.46) back to Maxwell’s Equations in sets of Eq. (2.31) through (2.34), we notice that the second term, or ∂2E→(r→,t)/∂t2 in Eq. (2.44), derives from the displacement current ∂D→/∂t in Eq. (2.34), whereas the third term, or ∂E→(r→,t)/∂t in Eq. (2.44), derives from the transport current density J→ in Eq. (2.34). Thus, the very existence of electromagnetic wave propagation depends on Maxwell’s introduction of the displacement current. Without it, only exponential decay of the fields could occur.
High-Voltage Measurements
Published in Mazen Abdel-Salam, Hussein Anis, Ahdab El-Morshedy, Roshdy Radwan, High-Voltage Engineering, 2018
Field measurements play an important role in the construction and problem solving of equipment using high voltages or subjected to high electric fields (IEEE Working Group, 1983). Most electric field measuring devices are based on the principle that the electric field is proportional to the electric flux density D. When this changes with time due to natural phenomena or by mechanical movement, a displacement current results. The displacement current density is given by () J¯d=∂D¯∂t
The static and quasistatic electromagnetic fields
Published in Edward J. Rothwell, Michael J. Cloud, Electromagnetics, 2018
Edward J. Rothwell, Michael J. Cloud
The magnetic field is secondary and dependent on the electric field through (3.166). As such, its time dependence is dictated both by the electric field and the current density. For example, in a conductor where J(r,t)=σE(r,t) $ \mathbf{J}(\mathbf{r},t)=\sigma \mathbf{E}(\mathbf{r},t) $ , the magnetic field will have terms dependent both on the electric field waveform (through the current) and on its derivative (through the displacement current).
Impulse-voltage Measurement of Distribution-class Surge Arresters by D-dot Probes
Published in Electric Power Components and Systems, 2011
The D-dot “electric field-coupled” probe can also serve as a capacitive divider, if the high-voltage and low-voltage capacitances (C1 and C2) and the connections have a negligible inductance [5]. According to the nature of the low-voltage capacitance, two kinds of electric-field probes are distinguished. A low-voltage capacitance (C2) of some tens of nanofarads can be realized. The measurement of fast-front voltages can be easily performed with electric field probes [5, 17, 22]. The probe can be terminated by the characteristic impedance of the cable (Rt = Zo) or by a high impedance (Rt ≫, i.e., in the order of MΩ). For the first case (where Rt = Zo), the capacitive current is much lower than the resistive one. Therefore, D-dot probes measure the time derivative of the electric-flux density (dD/dt = [Ddot]) at the surface of a conductor [5, 17, 22, 23]. This is equivalent to the measurement of the time rate of change of the surface charge density. The basic theory follows from the fact that the displacement current density (Jd) is equal to the time rate of change of the electric flux density, i.e., Jd = dD/dt = [Ddot]. To obtain the electric field, the measuring signal has to be integrated by, e.g., a passive resistive-capacitive (RC) integrator. The upper frequency limit is proportional to the reciprocal value of the time constant RtC2 [5, 17, 22, 23].
Inhomogeneous wave equation, Liénard-Wiechert potentials, and Hertzian dipoles in Weber electrodynamics
Published in Electromagnetics, 2022
When the displacement current is not neglected, the complete set of Maxwell’s equations is remarkably powerful because it describes how electromagnetic fields propagate in space and time. As one can assert, the addition of the displacement current by J. C. Maxwell ushered in the age of technological modernity. However, thought experiments and actual experiments indicate that the Lorentz force might be invalid in the presence of a displacement current (Kühn 2021). In other words, by adding the displacement current, degrees of freedom are lost in the force law, and it is not self-evident that the Graßmann formula, which is perfectly appropriate for electro- and magnetostatics, retains its validity.