Electric energy density

The value of the dot product is a measure of the coalignment of two vectors and is independent of the coordinate system. The dot product therefore is a true scalar, the simplest invariant which can be formed from the two vectors. It provides a useful form for expressing many physical properties the work done in moving a body equals the dot product of the force and the displacement the electrical energy density in space is proportional to the dot product of electrical intensity and electrical displacement quantum mechanical observations are dot products of an operator and a state vector the invariants of special relativity are the dot products of four-vectors. The invariants involving a set of quantities may be used to establish if these quantities are the components of a vector. For instance, if AiBi forms an invariant and Bi are the components of a vector, then Az must be the components of another vector. 

In piezoelectric media, the strain energy is multiplied by the factor (1 + due to piezoelectric stiffening of the elastic constants. Thus, the strain energy includes electrical stored energy in this case. To calculate the magnitude of the electrical energy density Ue independent of the strain energy, the relation... 

This agrees with the velocity found by solving the wave equation. It is instructive to calculate the electrical energy density for this problem. As found in Ex-... 

Dividing the electrical energy density by the strain energy density yields... 

If a C02-free energy conversion is required, only hydrogen and electrical energy can fulfill this demand. In spite of progress in battery technology, at present electrical energy densities are still quite low. Of course, both hydrogen and electricity would have to be produced in a C02-free way. 

Using relation (4.47), for the electric energy density w, we obtain ... 

Furthermore, the electrical energy density We inside the capacitor is given by... 

We begin with the electric displacement vector Dj = ZyEi where i, j = x , y, / are Cartesian coordinates and the summation over repeated indices is inferred. The tensor of dielectric permittivity is symmetric Sy = Ej,and generally (even for biaxial medium) has six independent components. If an insulator is placed in the electric field, the stored electric energy density is given by... 

B. Chu, X. Zhou, K. Ren, B. Neese, M. Lin, Q. Wang, E. Bauer, Q.M. Zhang, A dielectric polymer with high electric energy density and fast discharge speed. Science 313 (5785) (2006)334-336. 

When the liquid crystal undergoes a configurational change, 6 changes, and thus E changes. The electric energy density is... 

Electrical energy density High-voltage composite bipolar cathode... 

Here, the real part is directly related to the electrical energy density that can be stored reversibly in a capacitor according to... 

It must be stressed that, unlike our expansion of Eq. [30], which is valid only for vanishingly small values of the order parameter, Eqs. [38] and [39] are valid for arbitrary values of the order parameter. In fact, the magnetic and electric energy density expressions of the elastic continuum theory (Eqs. [3] and [4] of Chapter 8) are special cases of Eqs. [38] and [39]. For example, consider the magnetic energy density of the low temperature, anisotropic phase of a system in which only spatial variation of ri(r) is important. S f) in Eq. [38] can then be replaced by its equilibrium value , where < > denotes thermal averaging. The term proportional to H can be neglected because of its spatial invariance, and one obtains Eq. [3] of Chapter 8 directly. 

This is the Poynting theorem S is the Poynting vector. The first two terms in Eq. (1.2.4) represent rate of change of the magnetic and electric energy densities... 

The total electric energy density Weiec arising when a fixed voltage is maintained and applied on external conductors is given by (see the formulae 10.05.4 and 10.08.1 in Guggenheim [120] and de Gennes and Frost [110, p.l34])... 

The magnetic induction B, which plays a similar role to that of the electric displacement D in the construction of the electric energy density discussed above, is defined in SI units by... 

Such reformulated boundary conditions will be used in Section 3.6 below in the case of an elementary weak Freedericksz transition. Notice also that wf appearing in the above formulations may be replaced by any applicable total bulk energy density which may also incorporate any relevant magnetic and/or electric energy densities. 

If, as above, the contribution independent of the orientation of a and c is neglected then the electric energy density follows from analogous reasoning using equation (2.87) to give... 

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