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Static Electric Fields
Published in G. Jagadeeswar Reddy, T. Jayachandra Prasad, Basics of Electromagnetics and Transmission Lines, 2020
G. Jagadeeswar Reddy, T. Jayachandra Prasad
D¯=∊0E¯. Then the flux of D¯, i.e., ψ=∮SD¯.ds¯, where ψ is the electric flux. Which can be defined according to SI units as one line of flux originates from +1 Coloumb and terminates at −1 Coloumb. So the unit of Electric flux is also Coloumb and D¯ is the electric flux density whose unit is columb/m2.
Fundamentals of Electric Field
Published in Sivaji Chakravorti, Electric Field Analysis, 2017
Similarly, the amount of electric fieldlines that pass through an area is the electric flux through that area. Consider the case of a source point charge of positive polarity, as shown in Figure 1.1a. If the source charge magnitude is Q C, then the total amount of electric fieldlines coming out of the source charge will be also Q C. Now, if a fictitious sphere of radius r is considered such that the source charge is located at the centre of the sphere, then the electric flux through the surface of the sphere will be Q C, as the surface of the sphere completely encloses the source charge, and all the electric fieldlines coming out radially from the source point charge passes through the spherical surface. Electric flux is typically denoted by ψ.
Introduction to Electrochemistry
Published in Caroline Desgranges, Jerome Delhommelle, A Mole of Chemistry, 2020
Caroline Desgranges, Jerome Delhommelle
However, there are no “magnetic charges” (also called magnetic monopoles), analogous to electric charges. Indeed, magnetic fields are generated by a dipole. It is during his explorations in South America that von Humboldt (1769–1859) notices strange things happening with his compass. Indeed, he observes irregular disturbances that he names “magnetic storms”. To better understand these phenomena, Gauss (1777–1855) starts to study the Earth’s magnetic field. To do so, he uses a method similar to that used in celestial mechanics to study the effect of gravity. It consists of defining the Earth as a sphere, and the magnetic field generated by the core of Earth as a sum of “north–south” multipoles: “two-pole” (like a magnet) whose strength decreases as 1/r3 + “four-pole” whose strength decreases 1/r4 + “eight-pole” decreasing as 1/r5 + … Von Humboldt’s and Gauss’ results show that the dipole part is the most important! It also provides a new map of the magnetic field on Earth and an answer to von Humboldt’s observations of magnetic storms! Gauss also demonstrates that magnetic monopoles do not exist. Indeed, one proof can be found in the now so-called Gauss’ law of magnetism. It states that the magnetic flux through any closed surface is equal to zero. In other words, the quantity of magnetic field passing through a closed surface is zero! If we look at its correspondence in electricity, the Gauss’ law of electricity states that the electric flux through any closed surface is proportional to the net electric charge enclosed by that surface. Comparing these two laws, it can be seen that there is no “magnetic charge” (since it is equal to zero), and thus no existence of magnetic monopoles! It also means that the basic entity for magnetism is the magnetic dipole!
Effective medium properties of a ferromagnetic microwire grid
Published in Waves in Random and Complex Media, 2022
Tarun Kumar, Natarajan Kalyanasundaram
For anisotropic materials, the permittivity is a matrix which is known as the permittivity tensor. In such a case, the electric flux density and electric field intensity are not in the same direction. They can be expressed in the form of permittivity tensor as given in [19] or in terms of components The permittivity tensor is a matrix defined as [19] where each element of the matrix may not be necessarily a real and nonzero number. If the coordinate system has same orientation as that of the crystal, the medium possesses only the diagonal elements (i.e., and ) in the permittivity and permeability matrix [19].
Simulating the induction heating of cross-ply C/PEKK laminates – sensitivity and effect of material variability
Published in Advanced Composite Materials, 2021
Wouter J. B. Grouve, Francisco Sacchetti, Evan J. Vruggink, Remko Akkerman
in which is the electric charge density, is the electric field, is the magnetic induction, is the electric flux density, denotes the electric current density and represents the magnetic field, while time is denoted with . The Maxwell equations are complemented with the following constitutive equations:
Experimental and numerical implementation of auxetic substrate for enhancing voltage of piezoelectric sandwich beam harvester
Published in Mechanics of Advanced Materials and Structures, 2022
Mohamad Hossein Fatahi, Mohsen Hamedi, Majid Safarabadi
Another way to increase performance of piezoelectric material is to excite it in a manner that two in-plane strains have same signs. According to the linear piezoelectricity equation (Eqs. 1, 2), generated electric charge is depend on piezoelectric constants and strains [23]. Normally, compressing material in a direction will cause expansion in other one (positive Poisson’s ratio); it means strains have different signs and therefore it results a decrease in electric charge. When the main loading is in first direction, nearing strain in second direction to first will increase performance of piezoelectric material. By doing this work, stress in direction 2 is no more zero and have same sign with stress in direction 1. This does not matter much in equivalent stress and safety factor according to Tresca or von Mises failure criteria [24] where is stress vector, is electric flux density vector, is strain vector, is electric field intensity vector, is elasticity matrix, is dielectric permittivity matrix, and is piezoelectric matrix. Directions 1 and 2 are in-plane and 3 is in polarization direction. These directions are orthogonal and relying on x, z, and y respectively (shown in Figure 3).