Nikiya Anton Bettey 2021: 09-18-2021-0944 - Maxwell stress tensor Poynting vector

Saturday, September 18, 2021

09-18-2021-0944 - Maxwell stress tensor Poynting vector

The Maxwell stress tensor (named after James Clerk Maxwell) is a symmetric second-order tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the charge from the Lorentz force law. When the situation becomes more complicated, this ordinary procedure can become impractically difficult, with equations spanning multiple lines. It is therefore convenient to collect many of these terms in the Maxwell stress tensor, and to use tensor arithmetic to find the answer to the problem at hand.

In the relativistic formulation of electromagnetism, the Maxwell's tensor appears as a part of the electromagnetic stress–energy tensor which is the electromagnetic component of the total stress–energy tensor. The latter describes the density and flux of energy and momentum in spacetime.

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

In physics, the Poynting vector represents the directional energy flux (the energy transfer per unit area per unit time) of an electromagnetic field. The SI unit of the Poynting vector is the watt per square metre (W/m²). It is named after its discoverer John Henry Poynting who first derived it in 1884.^[1]^: 132 Oliver Heaviside also discovered it independently in the more general form that recognises the freedom of adding the curl of an arbitrary vector field to the definition.^[2] The Poynting vector is used throughout electromagnetics in conjunction with Poynting's theorem, the continuity equation expressing conservation of electromagnetic energy, to calculate the power flow in electric and magnetic fields.

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

Electrostatics is a branch of physics that studies electric charges at rest (static electricity).

Since classical physics, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber, or electron, was thus the source of the word 'electricity'. Electrostatic phenomena arise from the forces that electric charges exert on each other. Such forces are described by Coulomb's law. Even though electrostatically induced forces seem to be rather weak, some electrostatic forces such as the one between an electron and a proton, that together make up a hydrogen atom, is about 36 orders of magnitude stronger than the gravitational force acting between them.

There are many examples of electrostatic phenomena, from those as simple as the attraction of plastic wrap to one's hand after it is removed from a package, to the apparently spontaneous explosion of grain silos, the damage of electronic components during manufacturing, and photocopier & laser printer operation. Electrostatics involves the buildup of charge on the surface of objects due to contact with other surfaces. Although charge exchange happens whenever any two surfaces contact and separate, the effects of charge exchange are usually noticed only when at least one of the surfaces has a high resistance to electrical flow, because the charges that transfer are trapped there for a long enough time for their effects to be observed. These charges then remain on the object until they either bleed off to ground, or are quickly neutralized by a discharge. The familiar phenomenon of a static "shock" is caused by the neutralization of charge built up in the body from contact with insulated surfaces.

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

A gyroscope (from Ancient Greek γῦρος gûros, "circle" and σκοπέω skopéō, "to look") is a device used for measuring or maintaining orientation and angular velocity.^[1]^[2] It is a spinning wheel or disc in which the axis of rotation (spin axis) is free to assume any orientation by itself. When rotating, the orientation of this axis is unaffected by tilting or rotation of the mounting, according to the conservation of angular momentum.

Gyroscopes based on other operating principles also exist, such as the microchip-packaged MEMS gyroscopes found in electronic devices (sometimes called gyrometers), solid-state ring lasers, fibre optic gyroscopes, and the extremely sensitive quantum gyroscope.^[3]

Applications of gyroscopes include inertial navigation systems, such as in the Hubble Telescope, or inside the steel hull of a submerged submarine. Due to their precision, gyroscopes are also used in gyrotheodolites to maintain direction in tunnel mining.^[4] Gyroscopes can be used to construct gyrocompasses, which complement or replace magnetic compasses (in ships, aircraft and spacecraft, vehicles in general), to assist in stability (bicycles, motorcycles, and ships) or be used as part of an inertial guidance system.

MEMS gyroscopes are popular in some consumer electronics, such as smartphones.

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

A control moment gyroscope (CMG) is an attitude control device generally used in spacecraft attitude control systems. A CMG consists of a spinning rotor and one or more motorized gimbals that tilt the rotor’s angular momentum. As the rotor tilts, the changing angular momentum causes a gyroscopic torque that rotates the spacecraft.^[1]^[2]

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

https://en.wikipedia.org/wiki/Accumulator_(energy)

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

https://en.wikipedia.org/wiki/Aircraft_catapult#Steam_catapult

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

https://en.wikipedia.org/wiki/Category:Magnetic_propulsion_devices

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

Aerodynamics, from Greek ἀήρ aero (air) + δυναμική (dynamics), is the study of motion of air, particularly when affected by a solid object, such as an airplane wing. It is a sub-field of fluid dynamics and gas dynamics, and many aspects of aerodynamics theory are common to these fields. The term aerodynamics is often used synonymously with gas dynamics, the difference being that "gas dynamics" applies to the study of the motion of all gases, and is not limited to air. The formal study of aerodynamics began in the modern sense in the eighteenth century, although observations of fundamental concepts such as aerodynamic drag were recorded much earlier. Most of the early efforts in aerodynamics were directed toward achieving heavier-than-air flight, which was first demonstrated by Otto Lilienthal in 1891.^[1] Since then, the use of aerodynamics through mathematical analysis, empirical approximations, wind tunnel experimentation, and computer simulations has formed a rational basis for the development of heavier-than-air flight and a number of other technologies. Recent work in aerodynamics has focused on issues related to compressible flow, turbulence, and boundary layers and has become increasingly computational in nature.

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

Compressible flow (or gas dynamics) is the branch of fluid mechanics that deals with flows having significant changes in fluid density. While all flows are compressible, flows are usually treated as being incompressible when the Mach number (the ratio of the speed of the flow to the speed of sound) is smaller than 0.3 (since the density change due to velocity is about 5% in that case).^[1]The study of compressible flow is relevant to high-speed aircraft, jet engines, rocket motors, high-speed entry into a planetary atmosphere, gas pipelines, commercial applications such as abrasive blasting, and many other fields.

Irregular reflection[edit]

An irregular reflection is much like the case described above, with the caveat that δ is larger than the maximum allowable turning angle. Thus a detached shock is formed and a more complicated reflection occurs.

Supersonic wind tunnels[edit]

Supersonic wind tunnels are used for testing and research in supersonic flows, approximately over the Mach number range of 1.2 to 5. The operating principle behind the wind tunnel is that a large pressure difference is maintained upstream to downstream, driving the flow.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

https://en.wikipedia.org/wiki/Category:Continuum_mechanics

https://en.wikipedia.org/wiki/Category:Temporal_rates

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

https://en.wikipedia.org/wiki/Category:Electric_and_magnetic_fields_in_matter

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

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

https://en.wikipedia.org/wiki/Domain_wall_(magnetism)

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

https://en.wikipedia.org/wiki/Spin–orbit_interaction

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

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

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

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

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

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

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

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

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

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

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

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

https://en.wikipedia.org/wiki/Void_(astronomy)

https://en.wikipedia.org/wiki/Equation_of_state_(cosmology)

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

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

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

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

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

Small molecule hyperfine structure[edit]

A typical simple example of the hyperfine structure due to the interactions discussed above is in the rotational transitions of hydrogen cyanide (¹H¹²C¹⁴N) in its ground vibrational state. Here, the electric quadrupole interaction is due to the ¹⁴N-nucleus, the hyperfine nuclear spin-spin splitting is from the magnetic coupling between nitrogen, ¹⁴N (I_N = 1), and hydrogen, ¹H (I_H = 1⁄2), and a hydrogen spin-rotation interaction due to the ¹H-nucleus. These contributing interactions to the hyperfine structure in the molecule are listed here in descending order of influence. Sub-doppler techniques have been used to discern the hyperfine structure in HCN rotational transitions.^[11]

The dipole selection rules for HCN hyperfine structure transitions are $\Delta J=1$ , $\Delta F=\{0,\pm 1\}$ , where $J$ is the rotational quantum number and $F$ is the total rotational quantum number inclusive of nuclear spin ( $F=J+I_{\text{N}}$ ), respectively. The lowest transition ( $J=1\rightarrow 0$ ) splits into a hyperfine triplet. Using the selection rules, the hyperfine pattern of $J=2\rightarrow 1$ transition and higher dipole transitions is in the form of a hyperfine sextet. However, one of these components ( $\Delta F=-1$ ) carries only 0.6% of the rotational transition intensity in the case of $J=2\rightarrow 1$ . This contribution drops for increasing J. So, from $J=2\rightarrow 1$ upwards the hyperfine pattern consists of three very closely spaced stronger hyperfine components ( $\Delta J=1$ , $\Delta F=1$ ) together with two widely spaced components; one on the low frequency side and one on the high frequency side relative to the central hyperfine triplet. Each of these outliers carry ~ ${\tfrac {1}{2}}J^{2}$ ( $J$ is the upper rotational quantum number of the allowed dipole transition) the intensity of the entire transition. For consecutively higher- $J$ transitions, there are small but significant changes in the relative intensities and positions of each individual hyperfine component.^[12]

Measurements[edit]

Hyperfine interactions can be measured, among other ways, in atomic and molecular spectra and in electron paramagnetic resonance spectra of free radicals and transition-metal ions.

Applications[edit]

Astrophysics[edit]

The hyperfine transition as depicted on the Pioneer plaque

As the hyperfine splitting is very small, the transition frequencies are usually not located in the optical, but are in the range of radio- or microwave (also called sub-millimeter) frequencies.

Hyperfine structure gives the 21 cm line observed in H I regions in interstellar medium.

Carl Sagan and Frank Drake considered the hyperfine transition of hydrogen to be a sufficiently universal phenomenon so as to be used as a base unit of time and length on the Pioneer plaque and later Voyager Golden Record.

In submillimeter astronomy, heterodyne receivers are widely used in detecting electromagnetic signals from celestial objects such as star-forming core or young stellar objects. The separations among neighboring components in a hyperfine spectrum of an observed rotational transition are usually small enough to fit within the receiver's IF band. Since the optical depth varies with frequency, strength ratios among the hyperfine components differ from that of their intrinsic (or optically thin) intensities (these are so-called hyperfine anomalies, often observed in the rotational transitions of HCN^[12]). Thus, a more accurate determination of the optical depth is possible. From this we can derive the object's physical parameters.^[13]

Nuclear spectroscopy[edit]

In nuclear spectroscopy methods, the nucleus is used to probe the local structure in materials. The methods mainly base on hyperfine interactions with the surrounding atoms and ions. Important methods are nuclear magnetic resonance, Mössbauer spectroscopy, and perturbed angular correlation.

Nuclear technology[edit]

The atomic vapor laser isotope separation (AVLIS) process uses the hyperfine splitting between optical transitions in uranium-235 and uranium-238 to selectively photo-ionize only the uranium-235 atoms and then separate the ionized particles from the non-ionized ones. Precisely tuned dye lasers are used as the sources of the necessary exact wavelength radiation.

Use in defining the SI second and meter[edit]

The hyperfine structure transition can be used to make a microwave notch filter with very high stability, repeatability and Q factor, which can thus be used as a basis for very precise atomic clocks. The term transition frequency denotes the frequency of radiation corresponding to the transition between the two hyperfine levels of the atom, and is equal to $f = Δ E / h$ , where $Δ E$ is difference in energy between the levels and $h$ is the Planck constant. Typically, the transition frequency of a particular isotope of caesium or rubidium atoms is used as a basis for these clocks.

Due to the accuracy of hyperfine structure transition-based atomic clocks, they are now used as the basis for the definition of the second. One second is now defined to be exactly 9192631770cycles of the hyperfine structure transition frequency of caesium-133 atoms.

On October 21, 1983, the 17th CGPM defined the metre as the length of the path travelled by light in a vacuum during a time interval of 1/299,792,458 of a second.^[14]^[15]

Precision tests of quantum electrodynamics[edit]

The hyperfine splitting in hydrogen and in muonium have been used to measure the value of the fine structure constant α. Comparison with measurements of α in other physical systems provides a stringent test of QED.

Qubit in ion-trap quantum computing[edit]

The hyperfine states of a trapped ion are commonly used for storing qubits in ion-trap quantum computing. They have the advantage of having very long lifetimes, experimentally exceeding ~10 minutes (compared to ~1 s for metastable electronic levels).

The frequency associated with the states' energy separation is in the microwave region, making it possible to drive hyperfine transitions using microwave radiation. However, at present no emitter is available that can be focused to address a particular ion from a sequence. Instead, a pair of laser pulses can be used to drive the transition, by having their frequency difference (detuning) equal to the required transition's frequency. This is essentially a stimulated Raman transition. In addition, near-field gradients have been exploited to individually address two ions separated by approximately 4.3 micrometers directly with microwave radiation.^[16]

In physics[edit]

In physics, a pseudoscalar denotes a physical quantity analogous to a scalar. Both are physical quantities which assume a single value which is invariant under proper rotations. However, under the parity transformation, pseudoscalars flip their signs while scalars do not. As reflections through a plane are the combination of a rotation with the parity transformation, pseudoscalars also change signs under reflections.

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

In geometry and physics, spinors /spɪnər/ are elements of a complex vector space that can be associated with Euclidean space.^[b] Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation.^[c] However, when a sequence of such small rotations is composed (integrated) to form an overall final rotation, the resulting spinor transformation depends on which sequence of small rotations was used. Unlike vectors and tensors, a spinor transforms to its negative when the space is continuously rotated through a complete turn from 0° to 360° (see picture). This property characterizes spinors: spinors can be viewed as the "square roots" of vectors (although this is inaccurate and may be misleading; they are better viewed as "square roots" of sections of vector bundles – in the case of the exterior algebra bundle of the cotangent bundle, they thus become "square roots" of differential forms).

It is also possible to associate a substantially similar notion of spinor to Minkowski space, in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geometry by Élie Cartan in 1913.^[1]^[d] In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other subatomic particles.^[e]

Spinors are characterized by the specific way in which they behave under rotations. They change in different ways depending not just on the overall final rotation, but the details of how that rotation was achieved (by a continuous path in the rotation group). There are two topologically distinguishable classes (homotopy classes) of paths through rotations that result in the same overall rotation, as illustrated by the belt trick puzzle. These two inequivalent classes yield spinor transformations of opposite sign. The spin group is the group of all rotations keeping track of the class.^[f] It doubly covers the rotation group, since each rotation can be obtained in two inequivalent ways as the endpoint of a path. The space of spinors by definition is equipped with a (complex) linear representation of the spin group, meaning that elements of the spin group act as linear transformations on the space of spinors, in a way that genuinely depends on the homotopy class.^[g] In mathematical terms, spinors are described by a double-valued projective representation of the rotation group SO(3).

Although spinors can be defined purely as elements of a representation space of the spin group (or its Lie algebra of infinitesimal rotations), they are typically defined as elements of a vector space that carries a linear representation of the Clifford algebra. The Clifford algebra is an associative algebra that can be constructed from Euclidean space and its inner product in a basis-independent way. Both the spin group and its Lie algebra are embedded inside the Clifford algebra in a natural way, and in applications the Clifford algebra is often the easiest to work with.^[h] A Clifford space operates on a spinor space, and the elements of a spinor space are spinors.^[3] After choosing an orthonormal basis of Euclidean space, a representation of the Clifford algebra is generated by gamma matrices, matrices that satisfy a set of canonical anti-commutation relations. The spinors are the column vectors on which these matrices act. In three Euclidean dimensions, for instance, the Pauli spin matrices are a set of gamma matrices,^[i] and the two-component complex column vectors on which these matrices act are spinors. However, the particular matrix representation of the Clifford algebra, hence what precisely constitutes a "column vector" (or spinor), involves the choice of basis and gamma matrices in an essential way. As a representation of the spin group, this realization of spinors as (complex^[j]) column vectors will either be irreducible if the dimension is odd, or it will decompose into a pair of so-called "half-spin" or Weyl representations if the dimension is even.^[k]

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

Nikiya Anton Bettey 2021

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