Nikiya Anton Bettey 2021: 09-21-2021-1817 - Poynting's theorem continuity equation Volume charge density Energy flux

Wednesday, September 22, 2021

09-21-2021-1817 - Poynting's theorem continuity equation Volume charge density Energy flux

In electrodynamics, Poynting's theorem is a statement of conservation of energy for the electromagnetic field,^{[clarification needed]}, in the form of a partial differential equation developed by British physicist John Henry Poynting.^[1] Poynting's theorem is analogous to the work-energy theorem in classical mechanics, and mathematically similar to the continuity equation, because it relates the energy stored in the electromagnetic field to the work done on a charge distribution (i.e. an electrically charged object), through energy flux.

https://en.wikipedia.org/wiki/Poynting%27s_theorem

A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.

Continuity equations are a stronger, local form of conservation laws. For example, a weak version of the law of conservation of energy states that energy can neither be created nor destroyed—i.e., the total amount of energy in the universe is fixed. This statement does not rule out the possibility that a quantity of energy could disappear from one point while simultaneously appearing at another point. A stronger statement is that energy is locally conserved: energy can neither be created nor destroyed, nor can it "teleport" from one place to another—it can only move by a continuous flow. A continuity equation is the mathematical way to express this kind of statement. For example, the continuity equation for electric charge states that the amount of electric charge in any volume of space can only change by the amount of electric current flowing into or out of that volume through its boundaries.

Continuity equations more generally can include "source" and "sink" terms, which allow them to describe quantities that are often but not always conserved, such as the density of a molecular species which can be created or destroyed by chemical reactions. In an everyday example, there is a continuity equation for the number of people alive; it has a "source term" to account for people being born, and a "sink term" to account for people dying.

Any continuity equation can be expressed in an "integral form" (in terms of a flux integral), which applies to any finite region, or in a "differential form" (in terms of the divergence operator) which applies at a point.

Continuity equations underlie more specific transport equations such as the convection–diffusion equation, Boltzmann transport equation, and Navier–Stokes equations.

Flows governed by continuity equations can be visualized using a Sankey diagram.

https://en.wikipedia.org/wiki/Continuity_equation

In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m⁻³), at any point in a volume.^[1]^[2]^[3] Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m⁻²), at any point on a surface charge distribution on a two dimensional surface. Linear charge density (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m⁻¹), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative.

Like mass density, charge density can vary with position. In classical electromagnetic theory charge density is idealized as a continuous scalar function of position ${\boldsymbol {x}}$ , like a fluid, and $\rho ({\boldsymbol {x}})$ , $\sigma ({\boldsymbol {x}})$ , and $\lambda ({\boldsymbol {x}})$ are usually regarded as continuous charge distributions, even though all real charge distributions are made up of discrete charged particles. Due to the conservation of electric charge, the charge density in any volume can only change if an electric current of charge flows into or out of the volume. This is expressed by a continuity equation which links the rate of change of charge density $\rho ({\boldsymbol {x}})$ and the current density ${\boldsymbol {J}}({\boldsymbol {x}})$ .

Since all charge is carried by subatomic particles, which can be idealized as points, the concept of a continuous charge distribution is an approximation, which becomes inaccurate at small length scales. A charge distribution is ultimately composed of individual charged particles separated by regions containing no charge.^[4] For example, the charge in an electrically charged metal object is made up of conduction electrons moving randomly in the metal's crystal lattice. Static electricity is caused by surface charges consisting of ions on the surface of objects, and the space charge in a vacuum tube is composed of a cloud of free electrons moving randomly in space. The charge carrier density in a conductor is equal to the number of mobile charge carriers (electrons, ions, etc.) per unit volume. The charge density at any point is equal to the charge carrier density multiplied by the elementary charge on the particles. However, because the elementary charge on an electron is so small (1.6⋅10⁻¹⁹ C) and there are so many of them in a macroscopic volume (there are about 10²² conduction electrons in a cubic centimeter of copper) the continuous approximation is very accurate when applied to macroscopic volumes, and even microscopic volumes above the nanometer level.

At atomic scales, due to the uncertainty principle of quantum mechanics, a charged particle does not have a precise position but is represented by a probability distribution, so the charge of an individual particle is not concentrated at a point but is 'smeared out' in space and acts like a true continuous charge distribution.^[4] This is the meaning of 'charge distribution' and 'charge density' used in chemistry and chemical bonding. An electron is represented by a wavefunction $\psi ({\boldsymbol {x}})$ whose square is proportional to the probability of finding the electron at any point ${\boldsymbol {x}}$ in space, so $|\psi ({\boldsymbol {x}})|^{2}$ is proportional to the charge density of the electron at any point. In atomsand molecules the charge of the electrons is distributed in clouds called orbitals which surround the atom or molecule, and are responsible for chemical bonds.

https://en.wikipedia.org/wiki/Charge_density

Energy flux is the rate of transfer of energy through a surface. The quantity is defined in two different ways, depending on the context:

Total rate of energy transfer (not per unit area);^[1] SI units: W = J⋅s⁻¹.
Specific rate of energy transfer (total normalized per unit area); ^[2] SI units: W⋅m⁻² = J⋅m⁻²⋅s⁻¹:
- This is a vector quantity, its components being determined in terms of the normal (perpendicular) direction to the surface of measurement.
- This is sometimes called energy flux density, to distinguish it from the second definition.
- Radiative flux, heat flux, and sound energy flux are specific cases of this meaning.

Explanation[edit]

"Energy current" is a somewhat informal term that is used, on occasion, to describe the process of energy transfer in situations where the transfer can usefully be viewed in terms of a flow. It is particularly used when the transfer of energy is more significant to the discussion than the process by which the energy is transferred. For instance, the flow of fuel oil in a pipeline could be considered as an energy current, although this would not be a convenient way of visualising the fullness of the storage tanks.

The units of energy current are those of power (W). This is closely related to energy flux, which is the energy transferred per unit area per unit time (measured in W/m²).

Energy current in electromagnetism[edit]

A specific use of the concept of energy current was promulgated by Oliver Heaviside in the last quarter of the 19th century. Against heavy resistance from the engineering community,^[1] Heaviside worked out the physics of signal velocity/impedance/distortion on telegraph, telephone, and undersea cables. He invented the inductor-loaded "distortionless line" later patented by Michael Pupin in the USA.^[2] Building on the concept of the Poynting vector, which describes the flow of energy in a transverse electromagnetic wave as the vector product of its electric and magnetic fields (E × H), Heaviside sought to extend this by treating the transfer of energy due to the electric current in a conductor in a similar manner. In doing so he reversed the contemporary view of current, so that the electric and magnetic fields due to the current are the "prime movers", rather than being a result of the motion of the charge in the conductor.^[3]

Heaviside's approach had some adherents at the time—enough, certainly, to quarrel with the "traditionalists" in print. However, the "energy current" view presented a number of difficulties, most notably that in asserting that the energy flowed in the electric and magnetic fields around the conductor the theory is unable to explain why the charge appears to flow in the conductor. Another major flaw is that electrical science and engineering are built on solutions of Maxwell's Equations in which the electric current - expressed through the current-density vector J – is a fundamental quantity, while a so-called 'energy current' does not appear. Moreover, there are no equivalent equations describing the physical behaviour of the Poynting vector on which the concept of energy current is based.

After the discovery of the electron in 1897, the Drude model, which describes electrical conduction in metals, was developed very quickly. By associating the somewhat abstract concept of moving charge with the rather more concrete motion of the charged electrons, the Drude model effectively deals with the traditional "charge current" and the Heaviside "energy current" views simultaneously. With this achievement of "unification", the energy current approach has largely lost favour, because in omitting the concepts related to conduction it has no direct model for (for example) Ohm's Law. In consequence it is less convenient to use than the "traditional" charge current approach, which defines the concepts of current, voltage, resistance, etc., as commonly used for electrical work.

Poynting-flow diagrams are part of E&M engineering, transmission line theory, and antenna design, but rare in electronics texts.^[4]

References[edit]

^ "The Maxwellians" by Bruce J. Hunt 1991 Cornell University Press
^ "Invention" by Dr. Norbert Wiener 1993 ISBN 0-262-23167-0 MIT Press pp 69-76
^ "Digital Hardware Design" by Ivor Catt, David Walton, Malcolm Davidson 1979 ISBN 0-333-25981-5 p. 65 [1] [2]
^ "In a simple circuit, where does the energy flow?" by William Beaty

https://en.wikipedia.org/wiki/Energy_current

The Quintom scenario (derived from the words quintessence and phantom, as in phantom energy) is a hypothetical model of dark energy.

Equation of State[edit]

In this scenario, the equation of state $p=w\rho$ of the dark energy, relating its pressure and energy density, can cross the boundary $w=-1$ associated with the cosmological constant. The boundary separates the phantom-energy-like behavior with $w<-1$ from the quintessence-like behavior with $w>-1$ . A no-go theoremshows that this behavior requires at least two degrees of freedom for dark energy models involving ideal gases or scalar fields.^[1]

The Quintom scenario was applied in 2008 to produce a model of inflationary cosmology with a Big Bounce instead of a Big Bang singularity.^[2]

External links[edit]

https://en.wikipedia.org/wiki/Quintom_scenario

In physics, quintessence is a hypothetical form of dark energy, more precisely a scalar field, postulated as an explanation of the observation of an accelerating rate of expansion of the universe. The first example of this scenario was proposed by Ratra and Peebles (1988).^[1] The concept was expanded to more general types of time-varying dark energy and the term "quintessence" was first introduced in a 1998 paper by Robert R. Caldwell, Rahul Dave and Paul Steinhardt.^[2] It has been proposed by some physicists to be a fifth fundamental force.^[3]^[4]^[5]^[6] Quintessence differs from the cosmological constant explanation of dark energy in that it is dynamic; that is, it changes over time, unlike the cosmological constant which, by definition, does not change. Quintessence can be either attractive or repulsive depending on the ratio of its kinetic and potential energy. Those working with this postulate believe that quintessence became repulsive about ten billion years ago, about 3.5 billion years after the Big Bang.^[7]

https://en.wikipedia.org/wiki/Quintessence_(physics)

In cosmology, the cosmological constant problem or vacuum catastrophe is the disagreement between the observed values of vacuum energy density (the small value of the cosmological constant) and theoretical large value of zero-point energy suggested by quantum field theory.

Depending on the Planck energy cutoff and other factors, the discrepancy is as high as 120 orders of magnitude,^[1] a state of affairs described by physicists as "the largest discrepancy between theory and experiment in all of science"^[1]and "the worst theoretical prediction in the history of physics."^[2]

https://en.wikipedia.org/wiki/Cosmological_constant_problem

In physics, energy density is the amount of energy stored in a given system or region of space per unit volume. It may also be used for energy per unit mass, though a more accurate term for this is specific energy (or gravimetric energy density).

Often only the useful or extractable energy is measured, which is to say that inaccessible energy (such as rest massenergy) is ignored.^[1] In cosmological and other general relativistic contexts, however, the energy densities considered are those that correspond to the elements of the stress–energy tensor and therefore do include mass energy as well as energy densities associated with the pressures described in the next paragraph.

Energy per unit volume has the same physical units as pressure, and in many circumstances is a synonym: for example, the energy density of a magnetic field may be expressed as (and behaves as) a physical pressure, and the energy required to compress a compressed gas a little more may be determined by multiplying the difference between the gas pressure and the external pressure by the change in volume. A pressure gradient has the potential to perform work on the surroundings by converting internal energy to work until equilibrium is reached.

https://en.wikipedia.org/wiki/Energy_density

Scalar field[edit]

Quintessence (Q) is a scalar field with an equation of state where w_q, the ratio of pressure p_q and density $\rho$ _q, is given by the potential energy $V(Q)$ and a kinetic term:

w_{q}={\frac {p_{q}}{\rho _{q}}}={\frac {{\frac {1}{2}}{\dot {Q}}^{2}-V(Q)}{{\frac {1}{2}}{\dot {Q}}^{2}+V(Q)}}

Hence, quintessence is dynamic, and generally has a density and w_q parameter that varies with time. By contrast, a cosmological constant is static, with a fixed energy density and w_q = −1.

Tracker behavior[edit]

Many models of quintessence have a tracker behavior, which according to Ratra and Peebles (1988) and Paul Steinhardt et al. (1999) partly solves the cosmological constant problem.^[8] In these models, the quintessence field has a density which closely tracks (but is less than) the radiation density until matter-radiation equality, which triggers quintessence to start having characteristics similar to dark energy, eventually dominating the universe. This naturally sets the low scale of the dark energy.^[9] When comparing the predicted expansion rate of the universe as given by the tracker solutions with cosmological data, a main feature of tracker solutions is that one needs four parameters to properly describe the behavior of their equation of state,^[10]^[11] whereas it has been shown that at most a two-parameter model can optimally be constrained by mid-term future data (horizon 2015–2020).^[12]

Specific models[edit]

Some special cases of quintessence are phantom energy, in which w_q < −1,^[13] and k-essence (short for kinetic quintessence), which has a non-standard form of kinetic energy. If this type of energy were to exist, it would cause a big rip^[14] in the universe due to the growing energy density of dark energy which would cause the expansion of the universe to increase at a faster-than-exponential rate.

Holographic dark energy[edit]

Holographic dark energy models compared with cosmological constant models, imply a high degeneracy.^{[clarification needed]}^[15] It has been suggested that dark energy might originate from quantum fluctuations of spacetime, and are limited by the event horizon of the universe.^[16]

Studies with quintessence dark energy found that it dominates gravitational collapse in a spacetime simulation, based on the holographic thermalization. These results show that the smaller the state parameter of quintessence is, the harder it is for the plasma to thermalize.^[17]

Terminology[edit]

The name comes from quinta essentia (fifth element). So called in Latin starting from the Middle Ages, this was the element added by Aristotle to the other four ancient classical elements, because he thought it was the essence of the celestial world. Aristotle called this element aether, which he posited to be a pure, fine, and primigenial element. Similarly, modern quintessence would be the fifth known "dynamical, time-dependent, and spatially inhomogeneous" contribution to the overall mass–energy content of the universe.

Of course, the other four components are not the ancient Greek classical elements, but rather "baryons, neutrinos, dark matter, [and] radiation." Although neutrinos are sometimes considered radiation, the term "radiation" in this context is only used to refer to massless photons. Spatial curvature of the cosmos (which has not been detected) is excluded, because it is non-dynamical and homogeneous; the cosmological constant would not be considered a fifth component in this sense, because it is non-dynamical, homogeneous, and time-independent.^[2]