<![CDATA[Math Is Fun Forum / Important Laws/Principles in Physics]]> 2022-09-06T10:28:39Z FluxBB http://www.mathisfunforum.com/viewtopic.php?id=26872 <![CDATA[Re: Important Laws/Principles in Physics]]> 31) Lambert's cosine law

In optics, Lambert's cosine law says that the radiant intensity or luminous intensity observed from an ideal diffusely reflecting surface or ideal diffuse radiator is directly proportional to the cosine of the angle

between the direction of the incident light and the surface normal;
. The law is also known as the cosine emission law or Lambert's emission law. It is named after Johann Heinrich Lambert, from his Photometria, published in 1760.

A surface which obeys Lambert's law is said to be Lambertian, and exhibits Lambertian reflectance. Such a surface has the same radiance when viewed from any angle. This means, for example, that to the human eye it has the same apparent brightness (or luminance). It has the same radiance because, although the emitted power from a given area element is reduced by the cosine of the emission angle, the solid angle, subtended by surface visible to the viewer, is reduced by the very same amount. Because the ratio between power and solid angle is constant, radiance (power per unit solid angle per unit projected source area) stays the same.

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<![CDATA[Re: Important Laws/Principles in Physics]]> 30) Poiseuille's Law

Hagen–Poiseuille equation

In nonideal fluid dynamics, the Hagen–Poiseuille equation, also known as the Hagen–Poiseuille law, Poiseuille law or Poiseuille equation, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section. It can be successfully applied to air flow in lung alveoli, or the flow through a drinking straw or through a hypodermic needle. It was experimentally derived independently by Jean Léonard Marie Poiseuille in 1838 and Gotthilf Heinrich Ludwig Hagen, and published by Poiseuille in 1840–41 and 1846. The theoretical justification of the Poiseuille law was given by George Stokes in 1845.

The assumptions of the equation are that the fluid is incompressible and Newtonian; the flow is laminar through a pipe of constant circular cross-section that is substantially longer than its diameter; and there is no acceleration of fluid in the pipe. For velocities and pipe diameters above a threshold, actual fluid flow is not laminar but turbulent, leading to larger pressure drops than calculated by the Hagen–Poiseuille equation.

Poiseuille's equation describes the pressure drop due to the viscosity of the fluid; other types of pressure drops may still occur in a fluid (see a demonstration here). For example, the pressure needed to drive a viscous fluid up against gravity would contain both that as needed in Poiseuille's law plus that as needed in Bernoulli's equation, such that any point in the flow would have a pressure greater than zero (otherwise no flow would happen).

Another example is when blood flows into a narrower constriction, its speed will be greater than in a larger diameter (due to continuity of volumetric flow rate), and its pressure will be lower than in a larger diameter (due to Bernoulli's equation). However, the viscosity of blood will cause additional pressure drop along the direction of flow, which is proportional to length traveled (as per Poiseuille's law). Both effects contribute to the actual pressure drop.

Equation

In standard fluid-kinetics notation:

where:

is the pressure difference between the two ends,
L is the length of pipe,
is the dynamic viscosity,
Q is the volumetric flow rate,
R is the pipe radius,
A is the cross sectional area of pipe.

The equation does not hold close to the pipe entrance.:

The equation fails in the limit of low viscosity, wide and/or short pipe. Low viscosity or a wide pipe may result in turbulent flow, making it necessary to use more complex models, such as the Darcy–Weisbach equation. The ratio of length to radius of a pipe should be greater than one forty-eighth of the Reynolds number for the Hagen–Poiseuille law to be valid. If the pipe is too short, the Hagen–Poiseuille equation may result in unphysically high flow rates; the flow is bounded by Bernoulli's principle, under less restrictive conditions, by

because it is impossible to have negative (absolute) pressure (not to be confused with gauge pressure) in an incompressible flow.

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<![CDATA[Re: Important Laws/Principles in Physics]]> 29) Dalton's Law

Dalton's law (also called Dalton's law of partial pressures) states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of the individual gases. This empirical law was observed by John Dalton in 1801 and published in 1802. Dalton's law is related to the ideal gas laws.

Formula

Mathematically, the pressure of a mixture of non-reactive gases can be defined as the summation:

where

represent the partial pressures of each component.

where

is the mole fraction of the ith component in the total mixture of n components .

Volume-based concentration

The relationship below provides a way to determine the volume-based concentration of any individual gaseous component

where
is the concentration of component i.

Dalton's law is not strictly followed by real gases, with the deviation increasing with pressure. Under such conditions the volume occupied by the molecules becomes significant compared to the free space between them. In particular, the short average distances between molecules increases intermolecular forces between gas molecules enough to substantially change the pressure exerted by them, an effect not included in the ideal gas model.

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<![CDATA[Re: Important Laws/Principles in Physics]]> 28) Wien's displacement law

Wien's displacement law states that the black-body radiation curve for different temperatures will peak at different wavelengths that are inversely proportional to the temperature. The shift of that peak is a direct consequence of the Planck radiation law, which describes the spectral brightness of black-body radiation as a function of wavelength at any given temperature. However, it had been discovered by Wilhelm Wien several years before Max Planck developed that more general equation, and describes the entire shift of the spectrum of black-body radiation toward shorter wavelengths as temperature increases.

Formally, Wien's displacement law states that the spectral radiance of black-body radiation per unit wavelength, peaks at the wavelength

given by:

where T is the absolute temperature. b is a constant of proportionality called Wien's displacement constant, equal to

or
. This is an inverse relationship between wavelength and temperature. So the higher the temperature, the shorter or smaller the wavelength of the thermal radiation. The lower the temperature, the longer or larger the wavelength of the thermal radiation. For visible radiation, hot objects emit bluer light than cool objects. If one is considering the peak of black body emission per unit frequency or per proportional bandwidth, one must use a different proportionality constant. However, the form of the law remains the same: the peak wavelength is inversely proportional to temperature, and the peak frequency is directly proportional to temperature.

Wien's displacement law may be referred to as "Wien's law", a term which is also used for the Wien approximation.

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<![CDATA[Re: Important Laws/Principles in Physics]]> 27) Stokes' law

In 1851, George Gabriel Stokes derived an expression, now known as Stokes law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.

Statement of the law

The force of viscosity on a small sphere moving through a viscous fluid is given by:

where:

is the frictional force – known as Stokes' drag – acting on the interface between the fluid and the particle
is the dynamic viscosity (some authors use the symbol
)
R is the radius of the spherical object
v is the flow velocity relative to the object.

In SI units,

is given in newtons
in Pa·s
, R in meters, and v in m/s.

Stokes' law makes the following assumptions for the behavior of a particle in a fluid:

* Laminar flow
* Spherical particles
* Homogeneous (uniform in composition) material
* Smooth surfaces
* Particles do not interfere with each other.

Particles do not interfere with each other.

For molecules Stokes' law is used to define their Stokes radius and diameter.

The CGS unit of kinematic viscosity was named "stokes" after his work.

Applications

Stokes' law is the basis of the falling-sphere viscometer, in which the fluid is stationary in a vertical glass tube. A sphere of known size and density is allowed to descend through the liquid. If correctly selected, it reaches terminal velocity, which can be measured by the time it takes to pass two marks on the tube. Electronic sensing can be used for opaque fluids. Knowing the terminal velocity, the size and density of the sphere, and the density of the liquid, Stokes' law can be used to calculate the viscosity of the fluid. A series of steel ball bearings of different diameters are normally used in the classic experiment to improve the accuracy of the calculation. The school experiment uses glycerine or golden syrup as the fluid, and the technique is used industrially to check the viscosity of fluids used in processes. Several school experiments often involve varying the temperature and/or concentration of the substances used in order to demonstrate the effects this has on the viscosity. Industrial methods include many different oils, and polymer liquids such as solutions.

The importance of Stokes' law is illustrated by the fact that it played a critical role in the research leading to at least three Nobel Prizes.

Stokes' law is important for understanding the swimming of microorganisms and sperm; also, the sedimentation of small particles and organisms in water, under the force of gravity.

In air, the same theory can be used to explain why small water droplets (or ice crystals) can remain suspended in air (as clouds) until they grow to a critical size and start falling as rain (or snow and hail). Similar use of the equation can be made in the settling of fine particles in water or other fluids.

For molecules Stokes' law is used to define their Stokes radius and diameter.

The CGS unit of kinematic viscosity was named "stokes" after his work.

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<![CDATA[Re: Important Laws/Principles in Physics]]> 26) Hubble's Law

Hubble's law, also known as the Hubble–Lemaître law or Lemaître's law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving away from Earth. The velocity of the galaxies has been determined by their redshift, a shift of the light they emit toward the red end of the visible spectrum.

Hubble's law is considered the first observational basis for the expansion of the universe, and today it serves as one of the pieces of evidence most often cited in support of the Big Bang model. The motion of astronomical objects due solely to this expansion is known as the Hubble flow. It is described by the equation

with
the constant of proportionality—the Hubble constant—between the "proper distance" D to a galaxy, which can change over time, unlike the comoving distance, and its speed of separation v, i.e. the derivative of proper distance with respect to cosmological time coordinate.

The Hubble constant is most frequently quoted in (km/s)/Mpc, thus giving the speed in km/s of a galaxy 1 megaparsec

away, and its value is about 70 (km/s)/Mpc. However, the SI unit of H0 is simply
, and the SI unit for the reciprocal of
is simply the second. The reciprocal of
is known as the Hubble time. The Hubble constant can also be interpreted as the relative rate of expansion. In this form H0 = 7%/Gyr, meaning that at the current rate of expansion it takes a billion years for an unbound structure to grow by 7%.

Although widely attributed to Edwin Hubble, the notion of the universe expanding at a calculable rate was first derived from general relativity equations in 1922 by Alexander Friedmann. Friedmann published a set of equations, now known as the Friedmann equations, showing that the universe might be expanding, and presenting the expansion speed if that were the case. Then Georges Lemaître, in a 1927 article, independently derived that the universe might be expanding, observed the proportionality between recessional velocity of, and distance to, distant bodies, and suggested an estimated value for the proportionality constant; this constant, when Edwin Hubble confirmed the existence of cosmic expansion and determined a more accurate value for it two years later, came to be known by his name as the Hubble constant. Hubble inferred the recession velocity of the objects from their redshifts, many of which were earlier measured and related to velocity by Vesto Slipher in 1917. Though the Hubble constant H0 is roughly constant in the velocity-distance space at any given moment in time, the Hubble parameter H, which the Hubble constant is the current value of, varies with time, so the term constant is sometimes thought of as somewhat of a misnomer.

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<![CDATA[Re: Important Laws/Principles in Physics]]> 25) Bernoulli's principle

In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy.  The principle is named after Daniel Bernoulli who published it in his book Hydrodynamica in 1738. Although Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonhard Euler in 1752 who derived Bernoulli's equation in its usual form. The principle is only applicable for isentropic flows: when the effects of irreversible processes (like turbulence) and non-adiabatic processes (e.g. heat radiation) are small and can be neglected.

Bernoulli's principle can be applied to various types of fluid flow, resulting in various forms of Bernoulli's equation. The simple form of Bernoulli's equation is valid for incompressible flows (e.g. most liquid flows and gases moving at low Mach number). More advanced forms may be applied to compressible flows at higher Mach numbers (see the derivations of the Bernoulli equation).

Bernoulli's principle can be derived from the principle of conservation of energy. This states that, in a steady flow, the sum of all forms of energy in a fluid along a streamline is the same at all points on that streamline. This requires that the sum of kinetic energy, potential energy and internal energy remains constant.  Thus an increase in the speed of the fluid – implying an increase in its kinetic energy (dynamic pressure) – occurs with a simultaneous decrease in (the sum of) its potential energy (including the static pressure) and internal energy. If the fluid is flowing out of a reservoir, the sum of all forms of energy is the same on all streamlines because in a reservoir the energy per unit volume (the sum of pressure and gravitational potential

is the same everywhere.

Bernoulli's principle can also be derived directly from Isaac Newton's Second Law of Motion. If a small volume of fluid is flowing horizontally from a region of high pressure to a region of low pressure, then there is more pressure behind than in front. This gives a net force on the volume, accelerating it along the streamline.

Fluid particles are subject only to pressure and their own weight. If a fluid is flowing horizontally and along a section of a streamline, where the speed increases it can only be because the fluid on that section has moved from a region of higher pressure to a region of lower pressure; and if its speed decreases, it can only be because it has moved from a region of lower pressure to a region of higher pressure. Consequently, within a fluid flowing horizontally, the highest speed occurs where the pressure is lowest, and the lowest speed occurs where the pressure is highest.

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<![CDATA[Re: Important Laws/Principles in Physics]]> 24) Sabine's Formula

Wallace Clement Sabine (June 13, 1868 – January 10, 1919) was an American physicist who founded the field of architectural acoustics. Sabine was the architectural acoustician of Boston's Symphony Hall, widely considered one of the two or three best concert halls in the world for its acoustics.

Career

After graduating, Sabine became an assistant professor of physics at Harvard in 1889. He became an instructor in 1890 and a member of the faculty in 1892. In 1895, he became an assistant professor and in 1905, he was promoted to professor of physics. In October 1906, he became dean of the Lawrence Scientific School, succeeding Nathaniel Shaler.

Sabine's career is the story of the birth of the field of modern architectural acoustics. In 1895, acoustically improving the Fogg Lecture Hall, part of the recently constructed Fogg Art Museum, was considered an impossible task by the senior staff of the physics department at Harvard. (The original Fogg Museum was designed by Richard Morris Hunt and constructed in 1893. After the completion of the present Fogg Museum the building was repurposed for academic use and renamed Hunt Hall in 1935.)  The assignment was passed down until it landed on the shoulders of a young physics professor, Sabine. Although considered a popular lecturer by the students, Sabine had never received his Ph.D. and did not have any particular background dealing with sound.

Sabine tackled the problem by trying to determine what made the Fogg Lecture Hall different from other, acoustically acceptable facilities. In particular, the Sanders Theater was considered acoustically excellent. For the next several years, Sabine and his assistants spent each night moving materials between the two lecture halls and testing the acoustics. On some nights they would borrow hundreds of seat cushions from the Sanders Theater. Using an organ pipe and a stopwatch, Sabine performed thousands of careful measurements (though inaccurate by present standards) of the time required for different frequencies of sounds to decay to inaudibility in the presence of the different materials. He tested reverberation time with several different types of Oriental rugs inside Fogg Lecture Hall, and with various numbers of people occupying its seats, and found that the body of an average person decreased reverberation time by about as much as six seat cushions. Once the measurements were taken and before morning arrived, everything was quickly replaced in both lecture halls, in order to be ready for classes the next day.

Sabine was able to determine, through the experiments, that a definitive relationship exists between the quality of the acoustics, the size of the chamber, and the amount of absorption surface present. He formally defined the reverberation time, which is still the most important characteristic currently in use for gauging the acoustical quality of a room, as number of seconds required for the intensity of the sound to drop from the starting level, by an amount of 60 dB (decibels).

His formula is

where

T = the reverberation time
V = the room volume
A = the effective absorption area

By studying various rooms judged acoustically optimal for their intended uses, Sabine determined that acoustically appropriate concert halls had reverberation times of 2-2.25 seconds (with shorter reverberation times, a music hall seems too "dry" to the listener), while optimal lecture hall acoustics featured reverberation times of slightly under 1 second. Regarding the Fogg Museum lecture room, Sabine noted that a spoken word remained audible for about 5.5 seconds, or about an additional 12-15 words if the speaker continued talking. Listeners thus contended with a very high degree of resonance and echo.

Using what he discovered, Sabine deployed sound absorbing materials throughout the Fogg Lecture Hall to cut its reverberation time and reduce the "echo effect." This accomplishment cemented Wallace Sabine's career, and led to his hiring as the acoustical consultant for Boston's Symphony Hall, the first concert hall to be designed using quantitative acoustics. His acoustic design was successful and Symphony Hall is generally considered one of the best symphony halls in the world.

The unit of sound absorption, the Sabin, was named in his honor.

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<![CDATA[Re: Important Laws/Principles in Physics]]> 23) Faraday's law of induction

Faraday's law of induction (briefly, Faraday's law) is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (emf)—a phenomenon known as electromagnetic induction. It is the fundamental operating principle of transformers, inductors, and many types of electrical motors, generators and solenoids.

The Maxwell–Faraday equation (listed as one of Maxwell's equations) describes the fact that a spatially varying (and also possibly time-varying, depending on how a magnetic field varies in time) electric field always accompanies a time-varying magnetic field, while Faraday's law states that there is emf (electromotive force, defined as electromagnetic work done on a unit charge when it has traveled one round of a conductive loop) on the conductive loop when the magnetic flux through the surface enclosed by the loop varies in time.

Faraday's law had been discovered and one aspect of it (transformer emf) was formulated as the Maxwell–Faraday equation later. The equation of Faraday's law can be derived by the Maxwell–Faraday equation (describing transformer emf) and the Lorentz force (describing motional emf). The integral form of the Maxwell–Faraday equation describes only the transformer emf, while the equation of Faraday's law describes both the transformer emf and the motional emf.

History

Electromagnetic induction was discovered independently by Michael Faraday in 1831 and Joseph Henry in 1832. Faraday was the first to publish the results of his experiments. In Faraday's first experimental demonstration of electromagnetic induction (August 29, 1831), he wrapped two wires around opposite sides of an iron ring (torus) (an arrangement similar to a modern toroidal transformer). Based on his assessment of recently discovered properties of electromagnets, he expected that when current started to flow in one wire, a sort of wave would travel through the ring and cause some electrical effect on the opposite side. He plugged one wire into a galvanometer, and watched it as he connected the other wire to a battery. Indeed, he saw a transient current (which he called a "wave of electricity") when he connected the wire to the battery, and another when he disconnected it.  This induction was due to the change in magnetic flux that occurred when the battery was connected and disconnected. Within two months, Faraday had found several other manifestations of electromagnetic induction. For example, he saw transient currents when he quickly slid a bar magnet in and out of a coil of wires, and he generated a steady (DC) current by rotating a copper disk near the bar magnet with a sliding electrical lead ("Faraday's disk"). 191–195

Michael Faraday explained electromagnetic induction using a concept he called lines of force. However, scientists at the time widely rejected his theoretical ideas, mainly because they were not formulated mathematically.  An exception was James Clerk Maxwell, who in 1861–62 used Faraday's ideas as the basis of his quantitative electromagnetic theory. In Maxwell's papers, the time-varying aspect of electromagnetic induction is expressed as a differential equation which Oliver Heaviside referred to as Faraday's law even though it is different from the original version of Faraday's law, and does not describe motional emf. Heaviside's version (see Maxwell–Faraday equation below) is the form recognized today in the group of equations known as Maxwell's equations.

Lenz's law, formulated by Emil Lenz in 1834, describes "flux through the circuit", and gives the direction of the induced emf and current resulting from electromagnetic induction.

The most widespread version of Faraday's law states:

The electromotive force around a closed path is equal to the negative of the time rate of change of the magnetic flux enclosed by the path.

Mathematical statement

The definition of surface integral relies on splitting the surface Σ into small surface elements. Each element is associated with a vector dA of magnitude equal to the area of the element and with direction normal to the element and pointing "outward" (with respect to the orientation of the surface).

For a loop of wire in a magnetic field, the magnetic flux ΦB is defined for any surface Σ whose boundary is the given loop. Since the wire loop may be moving, we write Σ(t) for the surface. The magnetic flux is the surface integral:

where dA is an element of surface area of the moving surface Σ(t), B is the magnetic field, and B · dA is a vector dot product representing the element of flux through dA. In more visual terms, the magnetic flux through the wire loop is proportional to the number of magnetic field lines that pass through the loop.

When the flux changes—because B changes, or because the wire loop is moved or deformed, or both—Faraday's law of induction says that the wire loop acquires an emf, defined as the energy available from a unit charge that has traveled once around the wire loop.[(Some sources state the definition differently. This expression was chosen for compatibility with the equations of Special Relativity.) Equivalently, it is the voltage that would be measured by cutting the wire to create an open circuit, and attaching a voltmeter to the leads.

Faraday's law states that the emf is also given by the rate of change of the magnetic flux:

where
is the electromotive force (emf) and ΦB is the magnetic flux.

The direction of the electromotive force is given by Lenz's law.

The laws of induction of electric currents in mathematical form was established by Franz Ernst Neumann in 1845.

Faraday's law contains the information about the relationships between both the magnitudes and the directions of its variables. However, the relationships between the directions are not explicit; they are hidden in the mathematical formula.

A Left Hand Rule for Faraday's Law. The sign of ΔΦB, the change in flux, is found based on the relationship between the magnetic field B, the area of the loop A, and the normal n to that area, as represented by the fingers of the left hand. If ΔΦB is positive, the direction of the emf is the same as that of the curved fingers (yellow arrowheads). If ΔΦB is negative, the direction of the emf is against the arrowheads.

It is possible to find out the direction of the electromotive force (emf) directly from Faraday’s law, without invoking Lenz's law. A left hand rule helps doing that, as follows:

* Align the curved fingers of the left hand with the loop (yellow line).
* Stretch your thumb. The stretched thumb indicates the direction of n (brown), the normal to the area enclosed by the loop.
* Find the sign of ΔΦB, the change in flux. Determine the initial and final fluxes (whose difference is ΔΦB) with respect to the normal n, as indicated by the stretched thumb.
* If the change in flux, ΔΦB, is positive, the curved fingers show the direction of the electromotive force (yellow arrowheads).
* If ΔΦB is negative, the direction of the electromotive force is opposite to the direction of the curved fingers (opposite to the yellow arrowheads).

For a tightly wound coil of wire, composed of N identical turns, each with the same ΦB, Faraday's law of induction states that

where N is the number of turns of wire and ΦB is the magnetic flux through a single loop.

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<![CDATA[Re: Important Laws/Principles in Physics]]> 22) Laws of Reflection

If the reflecting surface is very smooth, the reflection of light that occurs is called specular or regular reflection. The laws of reflection are as follows:

* The incident ray, the reflected ray and the normal to the reflection surface at the point of the incidence lie in the same plane.
* The angle which the incident ray makes with the normal is equal to the angle which the reflected ray makes to the same normal.
* The reflected ray and the incident ray are on the opposite sides of the normal.

These three laws can all be derived from the Fresnel equations.

Mechanism

In classical electrodynamics, light is considered as an electromagnetic wave, which is described by Maxwell's equations. Light waves incident on a material induce small oscillations of polarisation in the individual atoms (or oscillation of electrons, in metals), causing each particle to radiate a small secondary wave in all directions, like a dipole antenna. All these waves add up to give specular reflection and refraction, according to the Huygens–Fresnel principle.

In the case of dielectrics such as glass, the electric field of the light acts on the electrons in the material, and the moving electrons generate fields and become new radiators. The refracted light in the glass is the combination of the forward radiation of the electrons and the incident light. The reflected light is the combination of the backward radiation of all of the electrons.

In metals, electrons with no binding energy are called free electrons. When these electrons oscillate with the incident light, the phase difference between their radiation field and the incident field is

(180°), so the forward radiation cancels the incident light, and backward radiation is just the reflected light.

Light–matter interaction in terms of photons is a topic of quantum electrodynamics, and is described in detail by Richard Feynman in his popular book QED: The Strange Theory of Light and Matter.

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<![CDATA[Re: Important Laws/Principles in Physics]]> 21) Mass–energy equivalence

In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame, where the two values differ only by a constant and the units of measurement. The principle is described by the physicist Albert Einstein's famous formula:

The formula defines the energy E of a particle in its rest frame as the product of mass (m) with the speed of light squared :

Because the speed of light is a large number in everyday units (approximately 300000 km/s or 186000 mi/s), the formula implies that a small amount of rest mass corresponds to an enormous amount of energy, which is independent of the composition of the matter. Rest mass, also called invariant mass, is the mass that is measured when the system is at rest. It is a fundamental physical property that is independent of momentum, even at extreme speeds approaching the speed of light (i.e., its value is the same in all inertial frames of reference). Massless particles such as photons have zero invariant mass, but massless free particles have both momentum and energy. The equivalence principle implies that when energy is lost in chemical reactions, nuclear reactions, and other energy transformations, the system will also lose a corresponding amount of mass. The energy, and mass, can be released to the environment as radiant energy, such as light, or as thermal energy. The principle is fundamental to many fields of physics, including nuclear and particle physics.

Mass–energy equivalence arose from special relativity as a paradox described by the French polymath Henri Poincaré. Einstein was the first to propose the equivalence of mass and energy as a general principle and a consequence of the symmetries of space and time. The principle first appeared in "Does the inertia of a body depend upon its energy-content?", one of his annus mirabilis papers, published on 21 November 1905. The formula and its relationship to momentum, as described by the energy–momentum relation, were later developed by other physicists.

Description

Mass–energy equivalence states that all objects having mass, or massive objects, have a corresponding intrinsic energy, even when they are stationary. In the rest frame of an object, where by definition it is motionless and so has no momentum, the mass and energy are equivalent and they differ only by a constant, the speed of light squared

. In Newtonian mechanics, a motionless body has no kinetic energy, and it may or may not have other amounts of internal stored energy, like chemical energy or thermal energy, in addition to any potential energy it may have from its position in a field of force. These energies tend to be much smaller than the mass of the object multiplied by
, which is on the order of
joules for a mass of one kilogram. Due to this principle, the mass of the atoms that come out of a nuclear reaction is less than the mass of the atoms that go in, and the difference in mass shows up as heat and light with the same equivalent energy as the difference. In analyzing these explosions, Einstein's formula can be used with E as the energy released and removed, and m as the change in mass.

In relativity, all the energy that moves with an object (i.e., the energy as measured in the object's rest frame) contributes to the total mass of the body, which measures how much it resists acceleration. If an isolated box of ideal mirrors could contain light, the individually massless photons would contribute to the total mass of the box, by the amount equal to their energy divided by

. For an observer in the rest frame, removing energy is the same as removing mass and the formula
indicates how much mass is lost when energy is removed. In the same way, when any energy is added to an isolated system, the increase in the mass is equal to the added energy divided by
.

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<![CDATA[Re: Important Laws/Principles in Physics]]> 20) Pauli exclusion principle

In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated by Austrian physicist Wolfgang Pauli in 1925 for electrons, and later extended to all fermions with his spin–statistics theorem of 1940.

In the case of electrons in atoms, it can be stated as follows: it is impossible for two electrons of a poly-electron atom to have the same values of the four quantum numbers: n, the principal quantum number; ℓ, the azimuthal quantum number;

, the magnetic quantum number; and
, the spin quantum number. For example, if two electrons reside in the same orbital, then their n, ℓ, and
values are the same; therefore their
must be different, and thus the electrons must have opposite half-integer spin projections of 1/2 and -1/2.

Particles with an integer spin, or bosons, are not subject to the Pauli exclusion principle: any number of identical bosons can occupy the same quantum state, as with, for instance, photons produced by a laser or atoms in a Bose-Einstein condensate.

A more rigorous statement is that, concerning the exchange of two identical particles, the total (many-particle) wave function is antisymmetric for fermions, and symmetric for bosons. This means that if the space and spin coordinates of two identical particles are interchanged, then the total wave function changes its sign for fermions and does not change for bosons.

If two fermions were in the same state (for example the same orbital with the same spin in the same atom), interchanging them would change nothing and the total wave function would be unchanged. The only way the total wave function can both change sign as required for fermions and also remain unchanged is that this function must be zero everywhere, which means that the state cannot exist. This reasoning does not apply to bosons because the sign does not change.

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<![CDATA[Re: Important Laws/Principles in Physics]]> 19) Stefan–Boltzmann law

The Stefan–Boltzmann law describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time

(also known as the black-body radiant emittance) is directly proportional to the fourth power of the black body's thermodynamic temperature T:

.

The constant of proportionality σ, called the Stefan–Boltzmann constant, is derived from other known physical constants. Since 2019, the value of the constant is

where k is the Boltzmann constant, h is Planck's constant, and c is the speed of light in a vacuum. The radiance from a specified angle of view (watts per square metre per steradian) is given by

A body that does not absorb all incident radiation (sometimes known as a grey body) emits less total energy than a black body and is characterized by an emissivity,

:

.

has dimensions of energy flux (energy per unit time per unit area), and the SI units of measure are joules per second per square metre, or equivalently, watts per square metre. The SI unit for absolute temperature T is the kelvin.
is the emissivity of the grey body; if it is a perfect blackbody,
. In the still more general (and realistic) case, the emissivity depends on the wavelength,
.

To find the total power radiated from an object, multiply by its surface area, A:

.

Wavelength- and subwavelength-scale particles, metamaterials, and other nanostructures are not subject to ray-optical limits and may be designed to exceed the Stefan–Boltzmann law.

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<![CDATA[Re: Important Laws/Principles in Physics]]> 18) Snell's law

Snell's law (also known as Snell–Descartes law and ibn-Sahl law and the law of refraction) is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water, glass, or air. This law was named after the Dutch astronomer and mathematician Willebrord Snellius (also called Snell).

In optics, the law is used in ray tracing to compute the angles of incidence or refraction, and in experimental optics to find the refractive index of a material. The law is also satisfied in meta-materials, which allow light to be bent "backward" at a negative angle of refraction with a negative refractive index.

Snell's law states that, for a given pair of media, the ratio of the sines of the angle of incidence θ1 and angle of refraction θ2 is equal to the ratio of phase velocities (v1 / v2) in the two media, or equivalently, to the refractive indices (n2 / n1) of the two media.

The law follows from Fermat's principle of least time, which in turn follows from the propagation of light as waves.

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<![CDATA[Re: Important Laws/Principles in Physics]]> 17) Quantitative electrolysis and Faraday's laws

Quantitative aspects of electrolysis were originally developed by Michael Faraday in 1834. Faraday is also credited to have coined the terms electrolyte, electrolysis, among many others while he studied quantitative analysis of electrochemical reactions. Also he was an advocate of the law of conservation of energy.

First law

Faraday concluded after several experiments on electric current in a non-spontaneous process that the mass of the products yielded on the electrodes was proportional to the value of current supplied to the cell, the length of time the current existed, and the molar mass of the substance analyzed. In other words, the amount of a substance deposited on each electrode of an electrolytic cell is directly proportional to the quantity of electricity passed through the cell.

Below is a simplified equation of Faraday's first law:

where

m is the mass of the substance produced at the electrode (in grams),
Q is the total electric charge that passed through the solution (in coulombs),
n is the valence number of the substance as an ion in solution (electrons per ion),
M is the molar mass of the substance (in grams per mole).

Second law

Faraday devised the laws of chemical electrodeposition of metals from solutions in 1857. He formulated the second law of electrolysis stating "the amounts of bodies which are equivalent to each other in their ordinary chemical action have equal quantities of electricity naturally associated with them." In other words, the quantities of different elements deposited by a given amount of electricity are in the ratio of their chemical equivalent weights.

An important aspect of the second law of electrolysis is electroplating, which together with the first law of electrolysis has a significant number of applications in industry, as when used to protectively coat metals to avoid corrosion.

Applications

There are various extremely important electrochemical processes in both nature and industry, like the coating of objects with metals or metal oxides through electrodeposition, the addition (electroplating) or removal (electropolishing) of thin layers of metal from an object's surface, and the detection of alcohol in drunk drivers through the redox reaction of ethanol. The generation of chemical energy through photosynthesis is inherently an electrochemical process, as is production of metals like aluminum and titanium from their ores. Certain diabetes blood sugar meters measure the amount of glucose in the blood through its redox potential. In addition to established electrochemical technologies (like deep cycle lead acid batteries) there is also a wide range of new emerging technologies such as fuel cells, large format lithium-ion batteries, electrochemical reactors and super-capacitors that are becoming increasingly commercial. Electrochemistry also has important applications in the food industry, like the assessment of food/package interactions, the analysis of milk composition, the characterization and the determination of the freezing end-point of ice-cream mixes, or the determination of free acidity in olive oil.

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