Maxwell justified the rule based on the fact that the area on a "pV" diagram corresponds to mechanical work, saying that work done on the system in going from "c" to "b" should equal work released on going from "a" to "b". This is because the change in free energy "A"("T","V") equals the work done during a reversible process, and, as a state variable, the free energy must be path-independent. In particular, the value of "A" at point "b" should be the same regardless of whether the path taken is from left or right across the horizontal isobar, or follows the original Van der Waals isotherm.


This derivation is not entirely rigorous, since it requires a reversible path through a region of thermodynamic instability, while "b" is unstable. Nevertheless, modern derivations from chemical potential reach the same conclusion, and it remains a necessary modification to the Van der Waals and to any other analytic equation of state.


The Maxwell equal area rule can also be derived from an assumption of equal chemical potential "μ" of coexisting liquid and vapour phases. On the isotherm shown in the above plot, points "a" and "c" are the only pair of points which fulfill the equilibrium condition of having equal pressure, temperature and chemical potential. It follows that systems with volumes intermediate between these two points will consist of a mixture of the pure liquid and gas with specific volumes equal to the pure liquid and gas phases at points "a" and "c".


The Van der Waals equation may be solved for "VG" and "VL" as functions of the temperature and the vapor pressure "pV". Since:


where "A" is the Helmholtz free energy, it follows that the equal area rule can be expressed as:


Since the gas and liquid volumes are functions of "pV" and "T" only, this equation is then solved numerically to obtain "pV" as a function of temperature (and number of particles "N"), which may then be used to determine the gas and liquid volumes.


A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown


in the accompanying figure. One sees that the two locii meet at the critical point (1,1,1) smoothly. An isotherm of the Van der Waals fluid taken at "T r" = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal.


We reiterate that the extensive volume "V"  is related to the volume per particle "v=V/N"  where "N = nN"A  is the number of particles in the system.


The equation of state does not give us all the thermodynamic parameters of the system. We can take the equation for the Helmholtz energy "A"


From the equation derived above for ln"Q", we find


Where Φ is an undetermined constant, which may be taken from the Sackur–Tetrode equation for an ideal gas to be:


This equation expresses "A"  in terms of its natural variables "V"  and "T" , and therefore gives us all thermodynamic information about the system. The mechanical equation of state was already derived above


The entropy equation of state yields the entropy ("S" )


from which we can calculate the internal energy


Similar equations can be written for the other thermodynamic potential and the chemical potential, but expressing any potential as a function of pressure "p"  will require the solution of a third-order polynomial, which yields a complicated expression. Therefore, expressing the enthalpy and the Gibbs energy as functions of their natural variables will be complicated.


Although the material constant "a" and "b" in the usual form of the Van der Waals equation differs for every single fluid considered, the equation can be recast into an invariant form applicable to "all" fluids.


Defining the following reduced variables ("fR", "fC" are the reduced and critical variable versions of "f", respectively),


The first form of the Van der Waals equation of state given above can be recast in the following reduced form:


This equation is "invariant" for all fluids; that is, the same reduced form equation of state applies, no matter what "a" and "b" may be for the particular fluid.


This invariance may also be understood in terms of the principle of corresponding states. If two fluids have the same reduced pressure, reduced volume, and reduced temperature, we say that their states are corresponding. The states of two fluids may be corresponding even if their measured pressure, volume, and temperature are very different. If the two fluids' states are corresponding, they exist in the same regime of the reduced form equation of state. Therefore, they will respond to changes in roughly the same way, even though their measurable physical characteristics may differ significantly.


The Van der Waals equation is a cubic equation of state; in the reduced formulation the cubic equation is:


At the critical temperature, where formula_66 we get as expected


For "TR" < 1, there are 3 values for "vR".


For "TR" > 1, there is 1 real value for "vR".


The solution of this equation for the case where there are three separate roots may be found at Maxwell construction


The equation is also usable as a PVT equation for compressible fluids (e.g. polymers). In this case specific volume changes are small and it can be written in a simplified form:


where "p" is the pressure, "V" is specific volume, "T" is the temperature and "A, B, C" are parameters.


The Murnaghan equation of state is a relationship between the volume of a body and the pressure to which it is subjected. This is one of many state equations that have been used in earth sciences and shock physics to model the behavior of matter under conditions of high pressure. It owes its name to Francis D. Murnaghan who proposed it in 1944 to reflect material behavior under a pressure range as wide as possible to reflect an experimentally established fact: the more a solid is compressed, the more difficult it is to compress further.


The Murnaghan equation is derived, under certain assumptions, from the equations of continuum mechanics. It involves two adjustable parameters: the modulus of incompressibility "K"0 and its first derivative with respect to the pressure, "K"'0, both measured at ambient pressure. In general, these coefficients are determined by a regression on experimentally obtained values of volume "V" as a function of the pressure "P". These experimental data can be obtained by X-ray diffraction or by shock tests. Regression can also be performed on the values of the energy as a function of the volume obtained from ab-initio and molecular dynamics calculations.


The Murnaghan equation of state is typically expressed as:


If the reduction in volume under compression is low, i.e., for "V"/"V"0 greater than about 90%, the Murnaghan equation can model experimental data with satisfactory accuracy. Moreover, unlike many proposed equations of state, it gives an explicit expression of the volume as a function of pressure "V"("P"). But its range of validity is limited and physical interpretation inadequate. However, this equation of state continues to be widely used in models of solid explosives. Of more elaborate equations of state, the most used in earth physics is the Birch–Murnaghan equation of state. In shock physics of metals and alloys, another widely used equation of state is the Mie–Grüneisen equation of state.


The study of the internal structure of the earth through the knowledge of the mechanical properties of the constituents of the inner layers of the planet involves extreme conditions; the pressure can be counted in hundreds of gigapascal and temperatures in thousands of degrees. The study of the properties of matter under these conditions can be done experimentally through devices such as diamond anvil cell for static pressures, or by subjecting the material to shock waves. It also gave rise to theoretical work to determine the equation of state, that is to say the relations among the different parameters that define in this case the state of matter: the volume (or density), temperature and pressure.


Dozens of equations have been proposed by various authors. These are empirical relationships, the quality and relevance depend on the use made of it and can be judged by different criteria: the number of independent parameters that are involved, the physical meaning that can be assigned to these parameters, the quality of the experimental data, and the consistency of theoretical assumptions that underlie their ability to extrapolate the behavior of solids at high compression.


Generally, at constant temperature, the bulk modulus is defined by:


The easiest way to get an equation of state linking "P" and "V" is to assume that "K" is constant, that is to say, independent of pressure and deformation of the solid, then we simply find the Hooke's law. In this case, the volume decreases exponentially with pressure. This is not a satisfactory result because it is experimentally established that as a solid is compressed, it becomes more difficult to compress. To go further, we must take into account the variations of the elastic properties of the solid with compression.


The assumption Murnaghan is to assume that the bulk modulus is a linear function of pressure :


Murnaghan equation is the result of the integration of the differential equation:


We can also express the volume depending on the pressure:


This simplified presentation is however criticized by Poirier as lacking rigor. The same relationship can be shown in a different way from the fact that the incompressibility of the product of the modulus and the thermal expansion coefficient is not dependent on the pressure for a given material. This equation of state is also a general case of the older Polytrope relation which also has a constant power relation.


In some circumstances, particularly in connection with ab initio calculations, the expression of the energy as a function of the volume will be preferred, which can be obtained by integrating the above equation according to the relationship "P" = −"dE"/"dV" . It can be written to "K"'0 different from 3,


Despite its simplicity, the Murnaghan equation is able to reproduce the experimental data for a range of pressures that can be quite large, on the order of "K"0/2. It also remains satisfactory as the ratio "V"/"V"0 remains above about 90%. In this range, the Murnaghan equation has an advantage compared to other equations of state if one wants to express the volume as a function of pressure.


Nevertheless, other equations may provide better results and several theoretical and experimental studies show that the Murnaghan equation is unsatisfactory for many problems. Thus, to the extent that the ratio "V"/"V"0 becomes very low, the theory predicts that "K"' goes to 5/3, which is the Thomas–Fermi limit. However, in the Murnaghan equation, "K"' is constant and set to its initial value. In particular, the value "K"'0 = 5/3 becomes inconsistent with the theory under some situations. In fact, when extrapolated, the behavior predicted by the Murnaghan equation becomes quite quickly unlikely.


Regardless of this theoretical argument, experience clearly shows that "K"' decreases with pressure, or in other words that the second derivative of the incompressibility modulus "K"" is strictly negative. A second order theory based on the same principle (see next section) can account for this observation, but this approach is still unsatisfactory. Indeed, it leads to a negative bulk modulus in the limit where the pressure tends to infinity. In fact, this is an inevitable contradiction whatever polynomial expansion is chosen because there will always be a dominant term that diverges to infinity.


These important limitations have led to the abandonment of the Murnaghan equation, which W. Holzapfel calls "a useful mathematical form without any physical justification". In practice, the analysis of compression data is done by using more sophisticated equations of state. The most commonly used within the science community is the Birch–Murnaghan equation, second or third order in the quality of data collected.


Finally, a very general limitation of this type of equation of state is their inability to take into account the phase transitions induced by the pressure and temperature of melting, but also multiple solid-solid transitions that can cause abrupt changes in the density and bulk modulus based on the pressure.


In practice, the Murnaghan equation is used to perform a regression on a data set, where one gets the values of the coefficients "K"0 and "K"'0. These coefficients obtained, and knowing the value of the volume to ambient conditions, then we are in principle able to calculate the volume, density and bulk modulus for any pressure.


The data set is mostly a series of volume measurements for different values of applied pressure, obtained mostly by X-ray diffraction. It is also possible to work on theoretical data, calculating the energy for different values of volume by ab initio methods, and then regressing these results. This gives a theoretical value of the modulus of elasticity which can be compared to experimental results.


The following table lists some of the results of different materials, with the sole purpose of illustrating some numerical analyses that have been made using the Murnaghan equation, without prejudice to the quality of the models obtained. Given the criticisms that have been made in the previous section on the physical meaning of the Murnaghan equation, these results should be considered with caution.


To improve the models or avoid criticism outlined above, several generalizations of the Murnaghan equation have been proposed. They usually consist in dropping a simplifying assumption and adding another adjustable parameter. This can improve the qualities of refinement, but also lead to complicated expressions. The question of the physical meaning of these additional parameters is also raised.


A possible strategy is to include an additional term "P"2 in the previous development, requiring that formula_7. Solving this differential equation gives the equation of the second-order Murnaghan:


where formula_9. Found naturally in the first order equation taking formula_10. Developments to an order greater than 2 are possible in principle, but at the cost of adding an adjustable parameter for each term.


In physics and thermodynamics, an equation of state is a thermodynamic equation relating state variables which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature (PVT), or internal energy. Equations of state are useful in describing the properties of fluids, mixtures of fluids, solids, and the interior of stars.


At present, there is no single equation of state that accurately predicts the properties of all substances under all conditions. An example of an equation of state correlates densities of gases and liquids to temperatures and pressures, known as the ideal gas law, which is roughly accurate for weakly polar gases at low pressures and moderate temperatures. This equation becomes increasingly inaccurate at higher pressures and lower temperatures, and fails to predict condensation from a gas to a liquid.


Another common use is in modeling the interior of stars, including neutron stars, dense matter (quark–gluon plasmas) and radiation fields. A related concept is the perfect fluid equation of state used in cosmology.


Equations of state can also describe solids, including the transition of solids from one crystalline state to another.


In a practical context, equations of state are instrumental for PVT calculations in process engineering problems, such as petroleum gas/liquid equilibrium calculations. A successful PVT model based on a fitted equation of state can be helpful to determine the state of the flow regime, the parameters for handling the reservoir fluids, and pipe sizing.


Measurements of equation-of-state parameters, especially at high pressures, can be made using lasers.


Boyle's Law was perhaps the first expression of an equation of state. In 1662, the Irish physicist and chemist Robert Boyle performed a series of experiments employing a J-shaped glass tube, which was sealed on one end. Mercury was added to the tube, trapping a fixed quantity of air in the short, sealed end of the tube. Then the volume of gas was measured as additional mercury was added to the tube. The pressure of the gas could be determined by the difference between the mercury level in the short end of the tube and that in the long, open end. Through these experiments, Boyle noted that the gas volume varied inversely with the pressure. In mathematical form, this can be stated as:


The above relationship has also been attributed to Edme Mariotte and is sometimes referred to as Mariotte's law. However, Mariotte's work was not published until 1676.


Charles's law or Law of Charles and Gay-Lussac (1787).


In 1787 the French physicist Jacques Charles found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to roughly the same extent over the same 80-kelvin interval. Later, in 1802, Joseph Louis Gay-Lussac published results of similar experiments, indicating a linear relationship between volume and temperature (Charles's Law):


Dalton's Law of partial pressure states that the pressure of a mixture of gases is equal to the sum of the pressures of all of the constituent gases alone.


Mathematically, this can be represented for "n" species as:


In 1834, Émile Clapeyron combined Boyle's Law and Charles' law into the first statement of the "ideal gas law". Initially, the law was formulated as "pVm" = "R"("TC" + 267) (with temperature expressed in degrees Celsius), where "R" is the gas constant. However, later work revealed that the number should actually be closer to 273.2, and then the Celsius scale was defined with 0°C = 273.15K, giving:


Van der Waals equation of state (1873).


In 1873, J. D. van der Waals introduced the first equation of state derived by the assumption of a finite volume occupied by the constituent molecules. His new formula revolutionized the study of equations of state, and was most famously continued via the Redlich–Kwong equation of state and the Soave modification of Redlich-Kwong.


General form of an equation of state.


For a given amount of substance contained in a system, the temperature, volume, and pressure are not independent quantities; they are connected by a relationship of the general form


An equation used to model this relationship is called an equation of state. In the following sections major equations of state are described, and the variables used here are defined as follows. Any consistent set of units may be used, although SI units are preferred. Absolute temperature refers to use of the Kelvin (K) or Rankine (°R) temperature scales, with zero being absolute zero.


The classical ideal gas law may be written


In the form shown above, the equation of state is thus


If the calorically perfect gas approximation is used, then the ideal gas law may also be expressed as follows


where formula_19 is the density, formula_20 is the adiabatic index (ratio of specific heats), formula_21 is the internal energy per unit mass (the "specific internal energy"), formula_22 is the specific heat at constant volume, and formula_23 is the specific heat at constant pressure.


Since for atomic and molecular gases, the classical ideal gas law is well suited in most cases, let us describe the equation of state for elementary particles with mass formula_24 and spin formula_25 that takes into account of quantum effects. In the following, the upper sign will always correspond to Fermi-Dirac statistics and the lower sign to Bose–Einstein statistics. The equation of state of such gases with formula_26 particles occupying a volume formula_27 with temperature formula_28 and pressure formula_29 is given by


where formula_31 is the Boltzmann constant and formula_32 the chemical potential is given by the following implicit function


In the limiting case where formula_34, this equation of state will reduce to that of the classical ideal gas. It can be shown that the above equation of state in the limit formula_34 reduces to


With a fixed number density formula_37, decreasing the temperature causes in Fermi gas, a increase in the value for pressure from its classical value implying an effective repulsion between particles (this is an apparent repulsion due to quantum exchange effects not because of actual interactions between particles since in ideal gas, interactional forces are neglected) and in Bose gas, a decrease in pressure from its classical value implying an effective attraction.


Cubic equations of state are called such because they can be rewritten as a cubic function of formula_38.


The Van der Waals equation of state may be written:


where formula_38 is molar volume. The substance-specific constants formula_41 and formula_42 can be calculated from the critical properties formula_43, formula_44, and formula_45 (noting that formula_45 is the molar volume at the critical point) as:


Proposed in 1873, the van der Waals equation of state was one of the first to perform markedly better than the ideal gas law. In this landmark equation formula_41 is called the attraction parameter and formula_42 the repulsion parameter or the effective molecular volume. While the equation is definitely superior to the ideal gas law and does predict the formation of a liquid phase, the agreement with experimental data is limited for conditions where the liquid forms. While the van der Waals equation is commonly referenced in text-books and papers for historical reasons, it is now obsolete. Other modern equations of only slightly greater complexity are much more accurate.


The van der Waals equation may be considered as the ideal gas law, "improved" due to two independent reasons:


With the reduced state variables, i.e. formula_61, formula_62 and formula_63, the reduced form of the Van der Waals equation can be formulated:


The benefit of this form is that for given formula_65 and formula_66, the reduced volume of the liquid and gas can be calculated directly using Cardano's method for the reduced cubic form:


For formula_68 and formula_69, the system is in a state of vapor–liquid equilibrium. The reduced cubic equation of state yields in that case 3 solutions. The largest and the lowest solution are the gas and liquid reduced volume.


Introduced in 1949, the Redlich-Kwong equation of state was a considerable improvement over other equations of the time. It is still of interest primarily due to its relatively simple form. While superior to the van der Waals equation of state, it performs poorly with respect to the liquid phase and thus cannot be used for accurately calculating vapor–liquid equilibria. However, it can be used in conjunction with separate liquid-phase correlations for this purpose.


The Redlich-Kwong equation is adequate for calculation of gas phase properties when the ratio of the pressure to the critical pressure (reduced pressure) is less than about one-half of the ratio of the temperature to the critical temperature (reduced temperature):


Where "ω" is the acentric factor for the species.


This formulation for formula_77 is due to Graboski and Daubert. The original formulation from Soave is:


We can also write it in the polynomial form, with:


where formula_83 is the universal gas constant and "Z"="PV"/("RT") is the compressibility factor.


In 1972 G. Soave replaced the 1/ term of the Redlich-Kwong equation with a function "α"("T","ω") involving the temperature and the acentric factor (the resulting equation is also known as the Soave-Redlich-Kwong equation of state; SRK EOS). The "α" function was devised to fit the vapor pressure data of hydrocarbons and the equation does fairly well for these materials.


Note especially that this replacement changes the definition of "a" slightly, as the formula_44 is now to the second power.


Volume translation of Peneloux et al. (1982).


The SRK EOS may be written as


where formula_77 and other parts of the SRK EOS is defined in the SRK EOS section.


A downside of the SRK EOS, and other cubic EOS, is that the liquid molar volume is significantly less accurate than the gas molar volume. Peneloux et alios (1982) proposed a simple correction for this by introducing a volume translation


where formula_89 is an additional fluid component parameter that translates the molar volume slightly. On the liquid branch of the EOS, a small change in molar volume corresponds to a large change in pressure. On the gas branch of the EOS, a small change in molar volume corresponds to a much smaller change in pressure than for the liquid branch. Thus, the perturbation of the molar gas volume is small. Unfortunately, there are two versions that occur in science and industry.


In the first version only formula_90 is translated,


In the second version both formula_90 and formula_93 are translated, or the translation of formula_90 is followed by a renaming of the composite parameter . This gives


The c-parameter of a fluid mixture is calculated by


The c-parameter of the individual fluid components in a petroleum gas and oil can be estimated by the correlation


where the Rackett compressibility factor formula_98 can be estimated by


A nice feature with the volume translation method of Peneloux et al. (1982) is that it does not affect the vapor-liquid equilibrium calculations. This method of volume translation can also be applied to other cubic EOSs if the c-parameter correlation is adjusted to match the selected EOS.


where formula_104 is the acentric factor of the species, formula_83 is the universal gas constant and formula_106 is compressibility factor.


The Peng–Robinson equation of state (PR EOS) was developed in 1976 at The University of Alberta by Ding-Yu Peng and Donald Robinson in order to satisfy the following goals:


For the most part the Peng–Robinson equation exhibits performance similar to the Soave equation, although it is generally superior in predicting the liquid densities of many materials, especially nonpolar ones. The departure functions of the Peng–Robinson equation are given on a separate article.


The analytic values of its characteristic constants are:


A modification to the attraction term in the Peng–Robinson equation of state published by Stryjek and Vera in 1986 (PRSV) significantly improved the model's accuracy by introducing an adjustable pure component parameter and by modifying the polynomial fit of the acentric factor.


where formula_111 is an adjustable pure component parameter. Stryjek and Vera published pure component parameters for many compounds of industrial interest in their original journal article. At reduced temperatures above 0.7, they recommend to set formula_112 and simply use formula_113. For alcohols and water the value of formula_114 may be used up to the critical temperature and set to zero at higher temperatures.


A subsequent modification published in 1986 (PRSV2) further improved the model's accuracy by introducing two additional pure component parameters to the previous attraction term modification.


where formula_111, formula_117, and formula_118 are adjustable pure component parameters.


PRSV2 is particularly advantageous for VLE calculations. While PRSV1 does offer an advantage over the Peng–Robinson model for describing thermodynamic behavior, it is still not accurate enough, in general, for phase equilibrium calculations. The highly non-linear behavior of phase-equilibrium calculation methods tends to amplify what would otherwise be acceptably small errors. It is therefore recommended that PRSV2 be used for equilibrium calculations when applying these models to a design. However, once the equilibrium state has been determined, the phase specific thermodynamic values at equilibrium may be determined by one of several simpler models with a reasonable degree of accuracy.


One thing to note is that in the PRSV equation, the parameter fit is done in a particular temperature range which is usually below the critical temperature. Above the critical temperature, the PRSV alpha function tends to diverge and become arbitrarily large instead of tending towards 0. Because of this, alternate equations for alpha should be employed above the critical point. This is especially important for systems containing hydrogen which is often found at temperatures far above its critical point. Several alternate formulations have been proposed. Some well known ones are by Twu et al or by Mathias and Copeman.


Babalola modified the Peng–Robinson Equation of state as:


The attractive force parameter ‘a’, which was considered to be a constant with respect to pressure in Peng–Robinson EOS. The modification, in which parameter ‘a’ was treated as a variable with respect to pressure for multicomponent multi-phase high density reservoir systems was to improve accuracy in the prediction of properties of complex reservoir fluids for PVT modeling. The variation was represented with a linear equation where a1 and a2 represent the slope and the intercept respectively of the straight line obtained when values of parameter ‘a’ are plotted against pressure.


This modification increases the accuracy of Peng–Robinson equation of state for heavier fluids particularly at pressure ranges (>30MPa) and eliminates the need for tuning the original Peng-Robinson equation of state. Values for a


The Elliott, Suresh, and Donohue (ESD) equation of state was proposed in 1990. The equation seeks to correct a shortcoming in the Peng–Robinson EOS in that there was an inaccuracy in the van der Waals repulsive term. The EOS accounts for the effect of the shape of a non-polar molecule and can be extended to polymers with the addition of an extra term (not shown). The EOS itself was developed through modeling computer simulations and should capture the essential physics of the size, shape, and hydrogen bonding.


The characteristic size parameter is related to the shape parameter formula_89 through


Noting the relationships between Boltzmann's constant and the Universal gas constant, and observing that the number of molecules can be expressed in terms of Avogadro's number and the molar mass, the reduced number density formula_127 can be expressed in terms of the molar volume as


The shape parameter formula_138 appearing in the Attraction term and the term formula_139 are given by


where formula_142 is the depth of the square-well potential and is given by


The model can be extended to associating components and mixtures of nonassociating components. Details are in the paper by J.R. Elliott, Jr. "et al." (1990).


The Cubic-Plus-Association (CPA) equation of state combines the Soave-Redlich-Kwong equation with an association term from Wertheim theory. The development of the equation began in 1995 as a research project that was funded by Shell, and in 1996 an article was published which presented the CPA equation of state.


In the association term formula_153 is the mole fraction of molecules not bonded at site A.


where "a" is associated with the interaction between molecules and "b" takes into account the finite size of the molecules, similar to the Van der Waals equation.


Although usually not the most convenient equation of state, the virial equation is important because it can be derived directly from statistical mechanics. This equation is also called the Kamerlingh Onnes equation. If appropriate assumptions are made about the mathematical form of intermolecular forces, theoretical expressions can be developed for each of the coefficients. "A" is the first virial coefficient, which has a constant value of 1 and makes the statement that when volume is large, all fluids behave like ideal gases. The second virial coefficient "B" corresponds to interactions between pairs of molecules, "C" to triplets, and so on. Accuracy can be increased indefinitely by considering higher order terms. The coefficients "B", "C", "D", etc. are functions of temperature only.


One of the most accurate equations of state is that from Benedict-Webb-Rubin-Starling shown next. It was very close to a virial equation of state. If the exponential term in it is expanded to two Taylor terms, a virial equation can be derived:


Note that in this virial equation, the fourth and fifth virial terms are zero. The second virial coefficient is monotonically decreasing as temperature is lowered. The third virial coefficient is monotonically increasing as temperature is lowered.


Values of the various parameters for 15 substances can be found in


The Lee-Kesler equation of state is based on the corresponding states principle, and is a modification of the BWR equation of state.


Multiparameter equations of state (MEOS) can be used to represent pure fluids with high accuracy, in both the liquid and gaseous states. MEOS's represent the Helmholtz function of the fluid as the sum of ideal gas and residual terms. Both terms are explicit in reduced temperature and reduced density - thus:


The reduced density and temperature are typically, though not always, the critical values for the pure fluid.


Other thermodynamic functions can be derived from the MEOS by using appropriate derivatives of the Helmholtz function; hence, because integration of the MEOS is not required, there are few restrictions as to the functional form of the ideal or residual terms. Typical MEOS use upwards of 50 fluid specific parameters, but are able to represent the fluid's properties with high accuracy. MEOS are available currently for about 50 of the most common industrial fluids including refrigerants. The IAPWS95 reference equation of state for water is also an MEOS. Mixture models for MEOS exist, as well.


One example of such an equation of state is the form proposed by Span and Wagner.


This is a somewhat simpler form that is intended to be used more in technical applications. Reference equations of state require a higher accuracy and use a more complicated form with more terms.


When considering water under very high pressures, in situations such as underwater nuclear explosions, sonic shock lithotripsy, and sonoluminescence, the stiffened equation of state is often used:


where formula_164 is the internal energy per unit mass, formula_165 is an empirically determined constant typically taken to be about 6.1, and formula_166 is another constant, representing the molecular attraction between water molecules. The magnitude of the correction is about 2 gigapascals (20,000 atmospheres).


The equation is stated in this form because the speed of sound in water is given by formula_167.


Thus water behaves as though it is an ideal gas that is "already" under about 20,000 atmospheres (2 GPa) pressure, and explains why water is commonly assumed to be incompressible: when the external pressure changes from 1 atmosphere to 2 atmospheres (100 kPa to 200 kPa), the water behaves as an ideal gas would when changing from 20,001 to 20,002 atmospheres (2000.1 MPa to 2000.2 MPa).


This equation mispredicts the specific heat capacity of water but few simple alternatives are available for severely nonisentropic processes such as strong shocks.


An ultrarelativistic fluid has equation of state


where formula_29 is the pressure, formula_170 is the mass density, and formula_171 is the speed of sound.


The equation of state for an ideal Bose gas is


where α is an exponent specific to the system (e.g. in the absence of a potential field, α = 3/2), "z" is exp("μ"/"kT") where "μ" is the chemical potential, Li is the polylogarithm, ζ is the Riemann zeta function, and "T""c" is the critical temperature at which a Bose–Einstein condensate begins to form.


Jones–Wilkins–Lee equation of state for explosives (JWL equation).


The equation of state from Jones–Wilkins–Lee is used to describe the detonation products of explosives.


The ratio formula_174 is defined by using formula_175 = density of the explosive (solid part) and formula_176 = density of the detonation products. The parameters formula_177, formula_178, formula_179, formula_180 and formula_181 are given by several references. In addition, the initial density (solid part) formula_182, speed of detonation formula_183, Chapman–Jouguet pressure formula_184 and the chemical energy of the explosive formula_185 are given in such references. These parameters are obtained by fitting the JWL-EOS to experimental results. Typical parameters for some explosives are listed in the table below.


Equations of state for solids and liquids.


The Anton-Schmidt equation is an empirical equation of state for crystalline solids, e.g. for pure metals or intermetallic compounds.


Quantum mechanical investigations of intermetallic compounds show that the dependency of the pressure under isotropic deformation can be described empirically by


Integration of formula_2 leads to equation of the state for the total energy. The energy formula_3 required to compress a solid to volume formula_4 is


The fitting parameters formula_7 and formula_8 are related to material properties, where


However, the fitting parameter formula_14 does not reproduce the total energy of the free atoms.


The total energy equation is used to determine elastic and thermal material constants in quantum chemical simulation packages.


In differential geometry and gauge theory, the Nahm equations are a system of ordinary differential equations introduced by Werner Nahm in the context of the "Nahm transform" – an alternative to Ward's twistor construction of monopoles. The Nahm equations are formally analogous to the algebraic equations in the ADHM construction of instantons, where finite order matrices are replaced by differential operators.


Deep study of the Nahm equations was carried out by Nigel Hitchin and Simon Donaldson. Conceptually, the equations arise in the process of infinite-dimensional hyperkähler reduction. Among their many applications we can mention: Hitchin's construction of monopoles, where this approach is critical for establishing nonsingularity of monopole solutions; Donaldson's description of the moduli space of monopoles; and the existence of hyperkähler structure on coadjoint orbits of complex semisimple Lie groups, proved by Peter Kronheimer, Olivier Biquard, and A.G. Kovalev.


Let "T"1("z"),"T"2("z"), "T"3("z") be three matrix-valued meromorphic functions of a complex variable "z". The Nahm equations are a system of matrix differential equations


together with certain analyticity properties, reality conditions, and boundary conditions. The three equations can be written concisely using the Levi-Civita symbol, in the form


More generally, instead of considering "N" by "N" matrices, one can consider Nahm's equations with values in a Lie algebra g.


The variable "z" is restricted to the open interval (0,2), and the following conditions are imposed:


The Nahm equations can be written in the Lax form as follows. Set


then the system of Nahm equations is equivalent to the Lax equation


As an immediate corollary, we obtain that the spectrum of the matrix "A" does not depend on "z". Therefore, the characteristic equation


which determines the so-called spectral curve in the twistor space "TP"1, is invariant under the flow in "z".


In mathematical physics, the quantum KZ equations or quantum Knizhnik–Zamolodchikov equations or qKZ equations are the analogue for quantum affine algebras of the Knizhnik–Zamolodchikov equations for affine Kac–Moody algebras. They are a consistent system of difference equations satisfied by the "N"-point functions, the vacuum expectations of products of primary fields. In the limit as the deformation parameter "q" approaches 1, the "N"-point functions of the quantum affine algebra tend to those of the affine Kac–Moody algebra and the difference equations become partial differential equations. The quantum KZ equations have been used to study exactly solved models in quantum statistical mechanics.


In mathematical physics, the Gordon decomposition (named after Walter Gordon) of the Dirac current is a splitting of the charge or particle-number current into a part that arises from the motion of the center of mass of the particles and a part that arises from gradients of the spin density. It makes explicit use of the Dirac equation and so it applies only to "on-shell" solutions of the Dirac equation.


For any solution formula_1 of the massive Dirac equation,


the Lorentz covariant number-current formula_3 may be expressed as


is the spinor generator of Lorentz transformations.


The corresponding momentum-space version for plane wave solutions formula_6 and formula_7 obeying


One sees that from Dirac's equation that


and, from the conjugate of Dirac's equation,


From Dirac algebra, one may show that Dirac matrices satisfy


which amounts to just the Gordon decomposition, after some algebra.


The second, spin-dependent, part of the current coupled to the photon field, formula_17 yields, up to an ignorable total divergence,


that is, an effective Pauli moment term, formula_19.


This decomposition of the current into a particle number-flux (first term) and bound spin contribution (second term) requires formula_20.


If one assumed that the given solution has energy formula_21 so that formula_22, one might obtain a decomposition that is valid for both massive and massless cases.


Using the Dirac equation again, one finds that


where formula_28 is the vector of Pauli matrices.


With the particle-number density identified with formula_29, and for a near plane-wave


solution of finite extent, one may interpret the first term in the decomposition as the current formula_30, due to particles moving at speed formula_31.


The second term, formula_32 is the current due to the gradients in the intrinsic magnetic moment density. The magnetic moment itself is found by integrating by parts to show that


For a single massive particle in its rest frame, where formula_34, the magnetic moment reduces to


where formula_36 and formula_37 is the Dirac value of the gyromagnetic ratio.


For a single massless particle obeying the right-handed Weyl equation, the spin-1/2 is locked to the direction formula_38 of its kinetic momentum and the magnetic moment becomes


For the both massive and massless cases, one also has an expression for the momentum density as part of the symmetric Belinfante–Rosenfeld stress–energy tensor


Using the Dirac equation one may evaluate formula_41 to find the energy density to be formula_42, and the momentum density,


If one used the non-symmetric canonical energy-momentum tensor


one would not find the bound spin-momentum contribution.


By an integration by parts one finds that the spin contribution to the total angular momentum is


This is what is expected, so the division by 2 in the spin contribution to the momentum density is necessary. The absence of a division by 2 in the formula for the current reflects the formula_37 gyromagnetic ratio of the electron. In other words, a spin-density gradient is twice as effective at making an electric current as it is at contributing to the linear momentum.


Motivated by the Riemann–Silberstein vector form of Maxwell's equations, Michael Berry uses the Gordon strategy to obtain gauge-invariant expressions for the intrinsic spin angular-momentum density for solutions to Maxwell's equations.


He assumes that the solutions are monochromatic and uses the phasor expressions formula_47, formula_48. The time average of the Poynting vector momentum density is then given by


We have used Maxwell's equations in passing from the first to the second and third lines, and in expression such as formula_52 the scalar product is between the fields so that the vector character is determined by the formula_53.


and for a fluid with intrinsic angular momentum density formula_55 we have


these identities suggest that the spin density can be identified as either


The two decompositions coincide when the field is paraxial. They also coincide when the field is a pure helicity state – i.e. when formula_59 where the helicity formula_60 takes the values formula_61 for light that is right or left circularly polarized respectively. In other cases they may differ.


In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. In the context of quantum field theory (QFT), the equations determine the dynamics of quantum fields.


The solutions to the equations, universally denoted as or (Greek psi), are referred to as "wave functions" in the context of RQM, and "fields" in the context of QFT. The equations themselves are called "wave equations" or "field equations", because they have the mathematical form of a wave equation or are generated from a Lagrangian density and the field-theoretic Euler–Lagrange equations (see classical field theory for background).


In the Schrödinger picture, the wave function or field is the solution to the Schrödinger equation;


one of the postulates of quantum mechanics. All relativistic wave equations can be constructed by specifying various forms of the Hamiltonian operator "Ĥ" describing the quantum system. Alternatively, Feynman's path integral formulation uses a Lagrangian rather than a Hamiltonian operator.


More generally – the modern formalism behind relativistic wave equations is Lorentz group theory, wherein the spin of the particle has a correspondence with the representations of the Lorentz group.


Late 1920s: Relativistic quantum mechanics of spin-0 and spin- particles.


A description of quantum mechanical systems which could account for "relativistic" effects was sought for by many theoretical physicists; from the late 1920s to the mid-1940s. The first basis for relativistic quantum mechanics, i.e. special relativity applied with quantum mechanics together, was found by all those who discovered what is frequently called the Klein–Gordon equation:


by inserting the energy operator and momentum operator into the relativistic energy–momentum relation:


The solutions to () are scalar fields. The KG equation is undesirable due to its prediction of "negative" energies and probabilities, as a result of the quadratic nature of () – inevitable in a relativistic theory. This equation was initially proposed by Schrödinger, and he discarded it for such reasons, only to realize a few months later that its non-relativistic limit (what is now called the Schrödinger equation) was still of importance. Nevertheless, – () is applicable to spin-0 bosons.


Neither the non-relativistic nor relativistic equations found by Schrödinger could predict the fine structure in the Hydrogen spectral series. The mysterious underlying property was "spin". The first two-dimensional "spin matrices" (better known as the Pauli matrices) were introduced by Pauli in the Pauli equation; the Schrödinger equation with a non-relativistic Hamiltonian including an extra term for particles in magnetic fields, but this was "phenomenological". Weyl found a relativistic equation in terms of the Pauli matrices; the Weyl equation, for "massless" spin- fermions. The problem was resolved by Dirac in the late 1920s, when he furthered the application of equation () to the electron – by various manipulations he factorized the equation into the form:


and one of these factors is the Dirac equation (see below), upon inserting the energy and momentum operators. For the first time, this introduced new four-dimensional spin matrices and in a relativistic wave equation, and explained the fine structure of hydrogen. The solutions to () are multi-component spinor fields, and each component satisfies (). A remarkable result of spinor solutions is that half of the components describe a particle while the other half describe an antiparticle; in this case the electron and positron. The Dirac equation is now known to apply for all massive spin- fermions. In the non-relativistic limit, the Pauli equation is recovered, while the massless case results in the Weyl equation.


Although a landmark in quantum theory, the Dirac equation is only true for spin- fermions, and still predicts negative energy solutions, which caused controversy at the time (in particular – not all physicists were comfortable with the "Dirac sea" of negative energy states).


1930s–1960s: Relativistic quantum mechanics of higher-spin particles.


The natural problem became clear: to generalize the Dirac equation to particles with "any spin"; both fermions and bosons, and in the same equations their antiparticles (possible because of the spinor formalism introduced by Dirac in his equation, and then-recent developments in spinor calculus by van der Waerden in 1929), and ideally with positive energy solutions.


This was introduced and solved by Majorana in 1932, by a deviated approach to Dirac. Majorana considered one "root" of ():


where is a spinor field now with infinitely many components, irreducible to a finite number of tensors or spinors, to remove the indeterminacy in sign. The matrices and are infinite-dimensional matrices, related to infinitesimal Lorentz transformations. He did not demand that each component of to satisfy equation (), instead he regenerated the equation using a Lorentz-invariant action, via the principle of least action, and application of Lorentz group theory.


Majorana produced other important contributions that were unpublished, including wave equations of various dimensions (5, 6, and 16). They were anticipated later (in a more involved way) by de Broglie (1934), and Duffin, Kemmer, and Petiau (around 1938–1939) see Duffin–Kemmer–Petiau algebra. The Dirac–Fierz–Pauli formalism was more sophisticated than Majorana’s, as spinors were new mathematical tools in the early twentieth century, although Majorana’s paper of 1932 was difficult to fully understand; it took Pauli and Wigner some time to understand it, around 1940.


Dirac in 1936, and Fierz and Pauli in 1939, built equations from irreducible spinors and , symmetric in all indices, for a massive particle of spin for integer (see Van der Waerden notation for the meaning of the dotted indices):


where is the momentum as a covariant spinor operator. For , the equations reduce to the coupled Dirac equations and and together transform as the original Dirac spinor. Eliminating either or shows that and each fulfill ().


In 1941, Rarita and Schwinger focussed on spin- particles and derived the Rarita–Schwinger equation, including a Lagrangian to generate it, and later generalized the equations analogous to spin for integer . In 1945, Pauli suggested Majorana's 1932 paper to Bhabha, who returned to the general ideas introduced by Majorana in 1932. Bhabha and Lubanski proposed a completely general set of equations by replacing the mass terms in () and () by an arbitrary constant, subject to a set of conditions which the wave functions must obey.


Finally, in the year 1948 (the same year as Feynman's path integral formulation was cast), Bargmann and Wigner formulated the general equation for massive particles which could have any spin, by considering the Dirac equation with a totally symmetric finite-component spinor, and using Lorentz group theory (as Majorana did): the Bargmann–Wigner equations. In the early 1960s, a reformulation of the Bargmann–Wigner equations was made by H. Joos and Steven Weinberg, the Joos–Weinberg equation. Various theorists at this time did further research in relativistic Hamiltonians for higher spin particles.


The relativistic description of spin particles has been a difficult problem in quantum theory. It is still an area of the present-day research because the problem is only partially solved; including interactions in the equations is problematic, and paradoxical predictions (even from the Dirac equation) are still present.


The following equations have solutions which satisfy the superposition principle, that is, the wave functions are additive.


Throughout, the standard conventions of tensor index notation and Feynman slash notation are used, including Greek indices which take the values 1, 2, 3 for the spatial components and 0 for the timelike component of the indexed quantities. The wave functions are denoted ", and are the components of the four-gradient operator.


In matrix equations, the Pauli matrices are denoted by " in which , where is the identity matrix:


and the other matrices have their usual representations. The expression


is a matrix operator which acts on 2-component spinor fields.


The gamma matrices are denoted by "", in which again , and there are a number of representations to select from. The matrix is "not" necessarily the identity matrix. The expression


is a matrix operator which acts on 4-component spinor fields.


Note that terms such as "" scalar multiply an identity matrix of the relevant dimension, the common sizes are or , and are "conventionally" not written for simplicity.


The Duffin–Kemmer–Petiau equation is an alternative equation for spin-0 and spin-1 particles:


Start with the standard special relativity (SR) 4-vectors


Note that each 4-vector is related to another by a Lorentz scalar:


Now, just apply the standard Lorentz scalar product rule to each one:


The last equation is a fundamental quantum relation.


When applied to a Lorentz scalar field formula_21, one gets the Klein–Gordon equation, the most basic of the quantum relativistic wave equations.


The Schrödinger equation is the low-velocity limiting case ("v" « "c") of the Klein–Gordon equation.


When the relation is applied to a four-vector field formula_25 instead of a Lorentz scalar field formula_21, then one gets the Proca equation (in Lorenz gauge):


If the rest mass term is set to zero (light-like particles), then this gives the free Maxwell equation (in Lorenz gauge)


Under a proper orthochronous Lorentz transformation in Minkowski space, all one-particle quantum states of spin with spin z-component locally transform under some representation of the Lorentz group:


where is some finite-dimensional representation, i.e. a matrix. Here is thought of as a column vector containing components with the allowed values of . The quantum numbers and as well as other labels, continuous or discrete, representing other quantum numbers are suppressed. One value of may occur more than once depending on the representation. Representations with several possible values for are considered below.


The irreducible representations are labeled by a pair of half-integers or integers . From these all other representations can be built up using a variety of standard methods, like taking tensor products and direct sums. In particular, space-time itself constitutes a 4-vector representation so that . To put this into context; Dirac spinors transform under the representation. In general, the representation space has subspaces that under the subgroup of spatial rotations, SO(3), transform irreducibly like objects of spin "j", where each allowed value:


occurs exactly once. In general, "tensor products of irreducible representations" are reducible; they decompose as direct sums of irreducible representations.


The representations and can each separately represent particles of spin . A state or quantum field in such a representation would satisfy no field equation except the Klein–Gordon equation.


There are equations which have solutions that do not satisfy the superposition principle.


In physics, the acoustic wave equation governs the propagation of acoustic waves through a material medium. The form of the equation is a second order partial differential equation. The equation describes the evolution of acoustic pressure formula_1 or particle velocity u as a function of position x and time formula_2. A simplified form of the equation describes acoustic waves in only one spatial dimension, while a more general form describes waves in three dimensions.


For lossy media, more intricate models need to be applied in order to take into account frequency-dependent attenuation and phase speed. Such models include acoustic wave equations that incorporate fractional derivative terms, see also the acoustic attenuation article or the survey paper.


The wave equation describing sound in one dimension (position formula_3) is


where formula_1 is the acoustic pressure (the local decoration the from the ambient pressure), and where formula_6 is the speed of sound.same


Provided that the speed formula_6 is a constant, not dependent on frequency (the dispersionless case), then the most general solution is


where formula_9 and formula_10 are any two twice-differentiable functions. This may be pictured as the superposition of two waveforms of arbitrary profile, one (formula_9) travelling up the x-axis and the other (formula_10) down the x-axis at the speed formula_6. The particular case of a sinusoidal wave travelling in one direction is obtained by choosing either formula_9 or formula_10 to be a sinusoid, and the other to be zero, giving


where formula_17 is the angular frequency of the wave and formula_18 is its wave number.


The derivation of the wave equation involves three steps: derivation of the equation of state, the linearized one-dimensional continuity equation, and the linearized one-dimensional force equation.


The equation of state (ideal gas law)


In an adiabatic process, pressure "P" as a function of density formula_20 can be linearized to


where "C" is some constant. Breaking the pressure and density into their mean and total components and noting that formula_22:


The adiabatic bulk modulus for a fluid is defined as


Condensation, "s", is defined as the change in density for a given ambient fluid density.


The continuity equation (conservation of mass) in one dimension is


Where "u" is the flow velocity of the fluid.


Again the equation must be linearized and the variables split into mean and variable components.


Rearranging and noting that ambient density changes with neither time nor position and that the condensation multiplied by the velocity is a very small number:


Euler's Force equation (conservation of momentum) is the last needed component. In one dimension the equation is:


where formula_33 represents the convective, substantial or material derivative, which is the derivative at a point moving along with the medium rather than at a fixed point.


Rearranging and neglecting small terms, the resultant equation becomes the linearized one-dimensional Euler Equation:


Taking the time derivative of the continuity equation and the spatial derivative of the force equation results in:


Multiplying the first by formula_38, subtracting the two, and substituting the linearized equation of state,


where formula_41 is the speed of propagation.


Feynman provides a derivation of the wave equation for sound in three dimensions as


where formula_43 is the Laplace operator, formula_1 is the acoustic pressure (the local deviation from the ambient pressure), and formula_6 is the speed of sound.


A similar looking wave equation but for the vector field particle velocity is given by


In some situations, it is more convenient to solve the wave equation for an abstract scalar field velocity potential which has the form


and then derive the physical quantities particle velocity and acoustic pressure by the equations (or definition, in the case of particle velocity):


The following solutions are obtained by separation of variables in different coordinate systems. They are phasor solutions, that is they have an implicit time-dependence factor of formula_50 where formula_51 is the angular frequency. The explicit time dependence is given by


where the asymptotic approximations to the Hankel functions, when formula_56, are


Depending on the chosen Fourier convention, one of these represents an outward travelling wave and the other a nonphysical inward travelling wave. The inward travelling solution wave is only nonphysical because of the singularity that occurs at r=0; inward travelling waves do exist.


Sound pressure or acoustic pressure is the local pressure deviation from the ambient (average or equilibrium) atmospheric pressure, caused by a sound wave. In air, sound pressure can be measured using a microphone, and in water with a hydrophone. The SI unit of sound pressure is the pascal (Pa).


A sound wave in a transmission medium causes a deviation (sound pressure, a "dynamic" pressure) in the local ambient pressure, a "static" pressure.


Sound pressure, denoted "p", is defined by


In a sound wave, the complementary variable to sound pressure is the particle velocity. Together, they determine the sound intensity of the wave.


"Sound intensity", denoted I and measured in W·m−2 in SI units, is defined by


"Acoustic impedance", denoted "Z" and measured in Pa·m−3·s in SI units, is defined by


"Specific acoustic impedance", denoted "z" and measured in Pa·m−1·s in SI units, is defined by


The "particle displacement" of a "progressive sine wave" is given by


It follows that the particle velocity and the sound pressure along the direction of propagation of the sound wave "x" are given by


Taking the Laplace transforms of "v" and "p" with respect to time yields


Since formula_18, the amplitude of the specific acoustic impedance is given by


Consequently, the amplitude of the particle displacement is related to that of the acoustic velocity and the sound pressure by


When measuring the sound pressure created by a sound source, it is important to measure the distance from the object as well, since the sound pressure of a "spherical" sound wave decreases as 1/"r" from the centre of the sphere (and not as 1/"r"2, like the sound intensity):


If the sound pressure "p"1 is measured at a distance "r"1 from the centre of the sphere, the sound pressure "p"2 at another position "r"2 can be calculated:


The inverse-proportional law for sound pressure comes from the inverse-square law for sound intensity:


The sound pressure may vary in direction from the centre of the sphere as well, so measurements at different angles may be necessary, depending on the situation. An obvious example of a sound source whose spherical sound wave varies in level in different directions is a bullhorn.


Sound pressure level (SPL) or acoustic pressure level is a logarithmic measure of the effective pressure of a sound relative to a reference value.


Sound pressure level, denoted "L""p" and measured in dB, is defined by


The commonly used reference sound pressure in air is


which is often considered as the threshold of human hearing (roughly the sound of a mosquito flying 3 m away). The proper notations for sound pressure level using this reference are or , but the suffix notations , , dBSPL, or dBSPL are very common, even if they are not accepted by the SI.


Most sound-level measurements will be made relative to this reference, meaning will equal an SPL of . In other media, such as underwater, a reference level of is used. These references are defined in ANSI S1.1-2013.


The main instrument for measuring sound levels in the environment is the sound level meter. Most sound level meters provide readings in A, C, and Z-weighted decibels and must meet international standards such as IEC 61672-2013.


The lower limit of audibility is defined as SPL of , but the upper limit is not as clearly defined. While ( or ) is the largest pressure variation an undistorted sound wave can have in Earth's atmosphere (i.e. if the thermodynamic properties of the air are disregarded, in reality the sound wave become progressively non-linear starting over 150 dB), larger sound waves can be present in other atmospheres or other media such as under water or through the Earth.


Ears detect changes in sound pressure. Human hearing does not have a flat spectral sensitivity (frequency response) relative to frequency versus amplitude. Humans do not perceive low- and high-frequency sounds as well as they perceive sounds between 3,000 and 4,000 Hz, as shown in the equal-loudness contour. Because the frequency response of human hearing changes with amplitude, three weightings have been established for measuring sound pressure: A, B and C. A-weighting applies to sound pressures levels up to , B-weighting applies to sound pressures levels between and , and C-weighting is for measuring sound pressure levels above .


In order to distinguish the different sound measures, a suffix is used: A-weighted sound pressure level is written either as dBA or LA. B-weighted sound pressure level is written either as dBB or LB, and C-weighted sound pressure level is written either as dBC or LC. Unweighted sound pressure level is called "linear sound pressure level" and is often written as dBL or just L. Some sound measuring instruments use the letter "Z" as an indication of linear SPL.


According to the inverse proportional law, when sound level "L""p"1 is measured at a distance "r"1, the sound level "L""p"2 at the distance "r"2 is


The formula for the sum of the sound pressure levels of "n" incoherent radiating sources is


in the formula for the sum of the sound pressure levels yields


A Dynamical Theory of the Electromagnetic Field


"A Dynamical Theory of the Electromagnetic Field" is a paper by James Clerk Maxwell on electromagnetism, published in 1865. In the paper, Maxwell derives an electromagnetic wave equation with a velocity for light in close agreement with measurements made by experiment, and deduces that light is an electromagnetic wave.


In part III of the paper, which is entitled "General Equations of the Electromagnetic Field", Maxwell formulated twenty equations which were to become known as Maxwell's equations, until this term became applied instead to a vectorized set of four equations selected in 1884, which had all appeared in "On Physical Lines of Force".


Heaviside's versions of Maxwell's equations are distinct by virtue of the fact that they are written in modern vector notation. They actually only contain one of the original eight—equation "G" (Gauss's Law). Another of Heaviside's four equations is an amalgamation of Maxwell's law of total currents (equation "A") with Ampère's circuital law (equation "C"). This amalgamation, which Maxwell himself had actually originally made at equation (112) in "On Physical Lines of Force", is the one that modifies Ampère's Circuital Law to include Maxwell's displacement current.


Eighteen of Maxwell's twenty original equations can be vectorized into six equations, labeled (A) to (F) below, each of which represents a group of three original equations in component form. The 19th and 20th of Maxwell's component equations appear as (G) and (H) below, making a total of eight vector equations. These are listed below in Maxwell's original order, designated by the letters that Maxwell assigned to them in his 1864 paper.


Maxwell did not consider completely general materials; his initial formulation used linear, isotropic, nondispersive media with permittivity "ϵ" and permeability "μ", although he also discussed the possibility of anisotropic materials.


Gauss's law for magnetism () is not included in the above list, but follows directly from equation (B) by taking divergences (because the divergence of the curl is zero).


Substituting (A) into (C) yields the familiar differential form of the .


Equation (D) implicitly contains the Lorentz force law and the differential form of Faraday's law of induction. For a "static" magnetic field, formula_11 vanishes, and the electric field becomes conservative and is given by , so that (D) reduces to


This is simply the Lorentz force law on a per-unit-charge basis — although Maxwell's equation (D) first appeared at equation (77) in "On Physical Lines of Force" in 1861, 34 years before Lorentz derived his force law, which is now usually presented as a supplement to the four "Maxwell's equations". The cross-product term in the Lorentz force law is the source of the so-called "motional emf" in electric generators (see also "Moving magnet and conductor problem"). Where there is no motion through the magnetic field — e.g., in transformers — we can drop the cross-product term, and the force per unit charge (called ) reduces to the electric field , so that Maxwell's equation (D) reduces to


Taking curls, noting that the curl of a gradient is zero, we obtain


which is the differential form of Faraday's law. Thus the three terms on the right side of equation (D) may be described, from left to right, as the motional term, the transformer term, and the conservative term.


In deriving the electromagnetic wave equation, Maxwell considers the situation only from the rest frame of the medium, and accordingly drops the cross-product term. But he still works from equation (D), in contrast to modern textbooks which tend to work from Faraday's law (see below).


The constitutive equations (E) and (F) are now usually written in the rest frame of the medium as and .


Maxwell's equation (G), viewed in isolation as printed in the 1864 paper, at first seems to say that .  However, if we trace the signs through the previous two triplets of equations, we see that what seem to be the components of are in fact the components of . The notation used in Maxwell's later "Treatise on Electricity and Magnetism" is different, and avoids the misleading first impression.


In part VI of "A Dynamical Theory of the Electromagnetic Field", subtitled "Electromagnetic theory of light", Maxwell uses the correction to Ampère's Circuital Law made in part III of his 1862 paper, "On Physical Lines of Force", which is defined as displacement current, to derive the electromagnetic wave equation.


He obtained a wave equation with a speed in close agreement to experimental determinations of the speed of light. He commented,


Maxwell's derivation of the electromagnetic wave equation has been replaced in modern physics by a much less cumbersome method which combines the corrected version of Ampère's Circuital Law with Faraday's law of electromagnetic induction.


To obtain the electromagnetic wave equation in a vacuum using the modern method, we begin with the modern 'Heaviside' form of Maxwell's equations. Using (SI units) in a vacuum, these equations are


If we take the curl of the curl equations we obtain


where formula_20 is any vector function of space, we recover the wave equations


is the speed of light in free space.


Of this paper and Maxwell's related works, fellow physicist Richard Feynman said: "From the long view of this history of mankind – seen from, say, 10,000 years from now – there can be little doubt that the most significant event of the 19th century will be judged as Maxwell's discovery of the laws of electromagnetism."


Albert Einstein used Maxwell's equations as the starting point for his special theory of relativity, presented in "The Electrodynamics of Moving Bodies", one of Einstein's 1905 "Annus Mirabilis" papers. In it is stated:


Maxwell's equations can also be derived by extending general relativity into five physical dimensions.


Electromagnetic or magnetic induction is the production of an electromotive force across an electrical conductor in a changing magnetic field.


Michael Faraday is generally credited with the discovery of induction in 1831, and James Clerk Maxwell mathematically described it as Faraday's law of induction. Lenz's law describes the direction of the induced field. Faraday's law was later generalized to become the Maxwell–Faraday equation, one of the four Maxwell equations in his theory of electromagnetism.


Electromagnetic induction has found many applications, including electrical components such as inductors and transformers, and devices such as electric motors and generators.


Electromagnetic induction was discovered by Michael Faraday, published in 1831. It was discovered independently by Joseph Henry in 1832.


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 used Faraday's ideas as the basis of his quantitative electromagnetic theory. In Maxwell's model, 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 slightly different from Faraday's original formulation 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.


In 1834 Heinrich Lenz formulated the law named after him to describe the "flux through the circuit". Lenz's law gives the direction of the induced EMF and current resulting from electromagnetic induction.


Faraday's law of induction and Lenz's law.


Faraday's law of induction makes use of the magnetic flux ΦB through a region of space enclosed by a wire loop. The magnetic flux is defined by a surface integral:


where "dA is an element of the surface Σ enclosed by the wire loop, B is the magnetic field. The dot product B·"dA corresponds to an infinitesimal amount of magnetic flux. In more visual terms, the magnetic flux through the wire loop is proportional to the number of magnetic flux lines that pass through the loop.


When the flux through the surface changes, Faraday's law of induction says that the wire loop acquires an electromotive force (EMF). The most widespread version of this law states that the induced electromotive force in any closed circuit is equal to the rate of change of the magnetic flux enclosed by the circuit:


where formula_3 is the EMF and ΦB is the magnetic flux. The direction of the electromotive force is given by Lenz's law which states that an induced current will flow in the direction that will oppose the change which produced it. This is due to the negative sign in the previous equation. To increase the generated EMF, a common approach is to exploit flux linkage by creating a tightly wound coil of wire, composed of "N" identical turns, each with the same magnetic flux going through them. The resulting EMF is then "N" times that of one single wire.


Generating an EMF through a variation of the magnetic flux through the surface of a wire loop can be achieved in several ways:


In general, the relation between the EMF formula_5 in a wire loop encircling a surface Σ, and the electric field E in the wire is given by


where "d"ℓ is an element of contour of the surface Σ, combining this with the definition of flux


we can write the integral form of the Maxwell–Faraday equation


It is one of the four Maxwell's equations, and therefore plays a fundamental role in the theory of classical electromagnetism.


Faraday's law describes two different phenomena: the "motional EMF" generated by a magnetic force on a moving wire (see Lorentz force), and the "transformer EMF" this is generated by an electric force due to a changing magnetic field (due to the differential form of the Maxwell–Faraday equation). James Clerk Maxwell drew attention to the separate physical phenomena in 1861. This is believed to be a unique example in physics of where such a fundamental law is invoked to explain two such different phenomena.


Albert Einstein noticed that the two situations both corresponded to a relative movement between a conductor and a magnet, and the outcome was unaffected by which one was moving. This was one of the principal paths that led him to develop special relativity.


The principles of electromagnetic induction are applied in many devices and systems, including:


The EMF generated by Faraday's law of induction due to relative movement of a circuit and a magnetic field is the phenomenon underlying electrical generators. When a permanent magnet is moved relative to a conductor, or vice versa, an electromotive force is created. If the wire is connected through an electrical load, current will flow, and thus electrical energy is generated, converting the mechanical energy of motion to electrical energy. For example, the "drum generator" is based upon the figure to the bottom-right. A different implementation of this idea is the Faraday's disc, shown in simplified form on the right.


When the electric current in a loop of wire changes, the changing current creates a changing magnetic field. A second wire in reach of this magnetic field will experience this change in magnetic field as a change in its coupled magnetic flux, "d" ΦB / "d t". Therefore, an electromotive force is set up in the second loop called the induced EMF or transformer EMF. If the two ends of this loop are connected through an electrical load, current will flow.


A current clamp is a type of transformer with a split core which can be spread apart and clipped onto a wire or coil to either measure the current in it or, in reverse, to induce a voltage. Unlike conventional instruments the clamp does not make electrical contact with the conductor or require it to be disconnected during attachment of the clamp.


Faraday's law is used for measuring the flow of electrically conductive liquids and slurries. Such instruments are called magnetic flow meters. The induced voltage ℇ generated in the magnetic field "B" due to a conductive liquid moving at velocity "v" is thus given by:


where ℓ is the distance between electrodes in the magnetic flow meter.


Electrical conductors moving through a steady magnetic field, or stationary conductors within a changing magnetic field, will have circular currents induced within them by induction, called eddy currents. Eddy currents flow in closed loops in planes perpendicular to the magnetic field. They have useful applications in eddy current brakes and induction heating systems. However eddy currents induced in the metal magnetic cores of transformers and AC motors and generators are undesirable since they dissipate energy (called core losses) as heat in the resistance of the metal. Cores for these devices use a number of methods to reduce eddy currents:


Eddy currents occur when a solid metallic mass is rotated in a magnetic field, because the outer portion of the metal cuts more magnetic lines of force than the inner portion; hence the induced electromotive force is not uniform; this tends to cause electric currents between the points of greatest and least potential. Eddy currents consume a considerable amount of energy and often cause a harmful rise in temperature.


Only five laminations or plates are shown in this example, so as to show the subdivision of the eddy currents. In practical use, the number of laminations or punchings ranges from 40 to 66 per inch (16 to 26 per centimetre), and brings the eddy current loss down to about one percent. While the plates can be separated by insulation, the voltage is so low that the natural rust/oxide coating of the plates is enough to prevent current flow across the laminations.


This is a rotor approximately 20 mm in diameter from a DC motor used in a Note the laminations of the electromagnet pole pieces, used to limit parasitic inductive losses.


In this illustration, a solid copper bar conductor on a rotating armature is just passing under the tip of the pole piece N of the field magnet. Note the uneven distribution of the lines of force across the copper bar. The magnetic field is more concentrated and thus stronger on the left edge of the copper bar (a,b) while the field is weaker on the right edge (c,d). Since the two edges of the bar move with the same velocity, this difference in field strength across the bar creates whorls or current eddies within the copper bar.


High current power-frequency devices, such as electric motors, generators and transformers, use multiple small conductors in parallel to break up the eddy flows that can form within large solid conductors. The same principle is applied to transformers used at higher than power frequency, for example, those used in switch-mode power supplies and the intermediate frequency coupling transformers of radio receivers.


In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an electric field and a magnetic field experiences a force of


(in SI units). It says that the electromagnetic force on a charge is a combination of a force in the direction of the electric field proportional to the magnitude of the field and the quantity of charge, and a force at right angles to the magnetic field and the velocity of the charge, proportional to the magnitude of the field, the charge, and the velocity. Variations on this basic formula describe the magnetic force on a current-carrying wire (sometimes called Laplace force), the electromotive force in a wire loop moving through a magnetic field (an aspect of Faraday's law of induction), and the force on a moving charged particle.


Historians suggest that the law is implicit in a paper by James Clerk Maxwell, published in 1865. Hendrik Lorentz arrived at a complete derivation in 1895, identifying the contribution of the electric force a few years after Oliver Heaviside correctly identified the contribution of the magnetic force.


Lorentz force law as the definition of E and B.


In many textbook treatments of classical electromagnetism, the Lorentz force law is used as the "definition" of the electric and magnetic fields E and B. To be specific, the Lorentz force is understood to be the following empirical statement:


This is valid, even for particles approaching the speed of light (that is, magnitude of v, |v| ≈ "c"). So the two vector fields E and B are thereby defined throughout space and time, and these are called the "electric field" and "magnetic field". The fields are defined everywhere in space and time with respect to what force a test charge would receive regardless of whether a charge is present to experience the force.


As a definition of E and B, the Lorentz force is only a definition in principle because a real particle (as opposed to the hypothetical "test charge" of infinitesimally-small mass and charge) would generate its own finite E and B fields, which would alter the electromagnetic force that it experiences. In addition, if the charge experiences acceleration, as if forced into a curved trajectory, it emits radiation that causes it to lose kinetic energy. See for example Bremsstrahlung and synchrotron light. These effects occur through both a direct effect (called the radiation reaction force) and indirectly (by affecting the motion of nearby charges and currents).


The force F acting on a particle of electric charge "q" with instantaneous velocity v, due to an external electric field E and magnetic field B, is given by (in SI units):


where × is the vector cross product (all boldface quantities are vectors). In terms of Cartesian components, we have:


In general, the electric and magnetic fields are functions of the position and time. Therefore, explicitly, the Lorentz force can be written as:


in which r is the position vector of the charged particle, "t" is time, and the overdot is a time derivative.


A positively charged particle will be accelerated in the "same" linear orientation as the E field, but will curve perpendicularly to both the instantaneous velocity vector v and the B field according to the right-hand rule (in detail, if the fingers of the right hand are extended to point in the direction of v and are then curled to point in the direction of B, then the extended thumb will point in the direction of F).


The term "q"E is called the electric force, while the term "q"(v × B) is called the magnetic force. According to some definitions, the term "Lorentz force" refers specifically to the formula for the magnetic force, with the "total" electromagnetic force (including the electric force) given some other (nonstandard) name. This article will "not" follow this nomenclature: In what follows, the term "Lorentz force" will refer to the expression for the total force.


The magnetic force component of the Lorentz force manifests itself as the force that acts on a current-carrying wire in a magnetic field. In that context, it is also called the Laplace force.


The Lorentz force is a force exerted by the electromagnetic field on the charged particle, that is, it is the rate at which linear momentum is transferred from the electromagnetic field to the particle. Associated with it is the power which is the rate at which energy is transferred from the electromagnetic field to the particle. That power is


Notice that the magnetic field does not contribute to the power because the magnetic force is always perpendicular to the velocity of the particle.


For a continuous charge distribution in motion, the Lorentz force equation becomes:


where formula_9 is the force on a small piece of the charge distribution with charge formula_10. If both sides of this equation are divided by the volume of this small piece of the charge distribution formula_11, the result is:


where formula_13 is the "force density" (force per unit volume) and formula_14 is the charge density (charge per unit volume). Next, the current density corresponding to the motion of the charge continuum is


so the continuous analogue to the equation is


The total force is the volume integral over the charge distribution:


By eliminating formula_14 and formula_18, using Maxwell's equations, and manipulating using the theorems of vector calculus, this form of the equation can be used to derive the Maxwell stress tensor formula_19, in turn this can be combined with the Poynting vector formula_20 to obtain the electromagnetic stress–energy tensor T used in general relativity.


In terms of formula_19 and formula_20, another way to write the Lorentz force (per unit volume) is


where formula_24 is the speed of light and ∇· denotes the divergence of a tensor field. Rather than the amount of charge and its velocity in electric and magnetic fields, this equation relates the energy flux (flow of "energy" per unit time per unit distance) in the fields to the force exerted on a charge distribution. See Covariant formulation of classical electromagnetism for more details.


The density of power associated with the Lorentz force in a material medium is


If we separate the total charge and total current into their free and bound parts, we get that the density of the Lorentz force is


where: formula_27 is the density of free charge; formula_28 is the polarization density; formula_29 is the density of free current; and formula_30 is the magnetization density. In this way, the Lorentz force can explain the torque applied to a permanent magnet by the magnetic field. The density of the associated power is


The above-mentioned formulae use SI units which are the most common. In older cgs-Gaussian units, which are somewhat more common among some theoretical physicists as well as condensed matter experimentalists, one has instead


where "c" is the speed of light.


Although this equation looks slightly different, it is completely equivalent, since


where ε0 is the vacuum permittivity and μ0 the vacuum permeability. In practice, the subscripts "cgs" and "SI" are always omitted, and the unit system has to be assessed from context.


Trajectories of particles due to the Lorentz force.


In many cases of practical interest, the motion in a magnetic field of an electrically charged particle (such as an electron or ion in a plasma) can be treated as the superposition of a relatively fast circular motion around a point called the guiding center and a relatively slow drift of this point. The drift speeds may differ for various species depending on their charge states, masses, or temperatures, possibly resulting in electric currents or chemical separation.


In real materials the Lorentz force is inadequate to describe the collective behavior of charged particles, both in principle and as a matter of computation. The charged particles in a material medium not only respond to the E and B fields but also generate these fields. Complex transport equations must be solved to determine the time and spatial response of charges, for example, the Boltzmann equation or the Fokker–Planck equation or the Navier–Stokes equations. For example, see magnetohydrodynamics, fluid dynamics, electrohydrodynamics, superconductivity, stellar evolution. An entire physical apparatus for dealing with these matters has developed. See for example, Green–Kubo relations and Green's function (many-body theory).


When a wire carrying an electric current is placed in a magnetic field, each of the moving charges, which comprise the current, experiences the Lorentz force, and together they can create a macroscopic force on the wire (sometimes called the Laplace force). By combining the Lorentz force law above with the definition of electric current, the following equation results, in the case of a straight, stationary wire:


where is a vector whose magnitude is the length of wire, and whose direction is along the wire, aligned with the direction of conventional current charge flow .


If the wire is not straight but curved, the force on it can be computed by applying this formula to each infinitesimal segment of wire , then adding up all these forces by integration. Formally, the net force on a stationary, rigid wire carrying a steady current is


This is the net force. In addition, there will usually be torque, plus other effects if the wire is not perfectly rigid.


One application of this is Ampère's force law, which describes how two current-carrying wires can attract or repel each other, since each experiences a Lorentz force from the other's magnetic field. For more information, see the article: Ampère's force law.


The magnetic force () component of the Lorentz force is responsible for "motional" electromotive force (or "motional EMF"), the phenomenon underlying many electrical generators. When a conductor is moved through a magnetic field, the magnetic field exerts opposite forces on electrons and nuclei in the wire, and this creates the EMF. The term "motional EMF" is applied to this phenomenon, since the EMF is due to the "motion" of the wire.


In other electrical generators, the magnets move, while the conductors do not. In this case, the EMF is due to the electric force ("q"E) term in the Lorentz Force equation. The electric field in question is created by the changing magnetic field, resulting in an "induced" EMF, as described by the Maxwell–Faraday equation (one of the four modern Maxwell's equations).


Both of these EMFs, despite their apparently distinct origins, are described by the same equation, namely, the EMF is the rate of change of magnetic flux through the wire. (This is Faraday's law of induction, see below.) Einstein's special theory of relativity was partially motivated by the desire to better understand this link between the two effects. In fact, the electric and magnetic fields are different facets of the same electromagnetic field, and in moving from one inertial frame to another, the solenoidal vector field portion of the "E"-field can change in whole or in part to a "B"-field or "vice versa".


Lorentz force and Faraday's law of induction.


Given a loop of wire in a magnetic field, Faraday's law of induction states the induced electromotive force (EMF) in the wire is:


is the magnetic flux through the loop, B is the magnetic field, Σ("t") is a surface bounded by the closed contour ∂Σ("t"), at time "t", dA is an infinitesimal vector area element of Σ("t") (magnitude is the area of an infinitesimal patch of surface, direction is orthogonal to that surface patch).


The "sign" of the EMF is determined by Lenz's law. Note that this is valid for not only a "stationary" wirebut also for a "moving" wire.


From Faraday's law of induction (that is valid for a moving wire, for instance in a motor) and the Maxwell Equations, the Lorentz Force can be deduced. The reverse is also true, the Lorentz force and the Maxwell Equations can be used to derive the Faraday Law.


Let Σ("t") be the moving wire, moving together without rotation and with constant velocity v and Σ("t") be the internal surface of the wire. The EMF around the closed path ∂Σ("t") is given by:


is the electric field and dℓ is an infinitesimal vector element of the contour ∂Σ("t").


NB: Both dℓ and dA have a sign ambiguity; to get the correct sign, the right-hand rule is used, as explained in the article Kelvin–Stokes theorem.


The above result can be compared with the version of Faraday's law of induction that appears in the modern Maxwell's equations, called here the "Maxwell–Faraday equation":


The Maxwell–Faraday equation also can be written in an "integral form" using the Kelvin–Stokes theorem.


So we have, the Maxwell Faraday equation:


The two are equivalent if the wire is not moving. Using the Leibniz integral rule and that "div" B = 0, results in,


since this is valid for any wire position it implies that,


Faraday's law of induction holds whether the loop of wire is rigid and stationary, or in motion or in process of deformation, and it holds whether the magnetic field is constant in time or changing. However, there are cases where Faraday's law is either inadequate or difficult to use, and application of the underlying Lorentz force law is necessary. See inapplicability of Faraday's law.


Note that the Maxwell Faraday's equation implies that the Electric Field E is non conservative when the Magnetic Field B varies in time, and is not expressible as the gradient of a scalar field, and not subject to the gradient theorem since its rotational is not zero.


The E and B fields can be replaced by the magnetic vector potential A and (scalar) electrostatic potential "ϕ" by


where ∇ is the gradient, ∇⋅ is the divergence, ∇× is the curl.


Using an identity for the triple product this can be rewritten as,


(Notice that the coordinates and the velocity components should be treated as independent variables, so the del operator acts only on formula_50, not on formula_51; thus, there is no need of using Feynman's subscript notation in the equation above). Using the chain rule, the total derivative of formula_50 is:


With v = ẋ, we can put the equation into the convenient Euler–Lagrange form


The Lagrangian for a charged particle of mass "m" and charge "q" in an electromagnetic field equivalently describes the dynamics of the particle in terms of its "energy", rather than the force exerted on it. The classical expression is given by:


where A and "ϕ" are the potential fields as above. The quantity formula_58 can be thought as a velocity-dependent potential function. Using Lagrange's equations, the equation for the Lorentz force given above can be obtained again.


The potential energy depends on the velocity of the particle, so the force is velocity dependent, so it is not conservative.


The action is the relativistic arclength of the path of the particle in spacetime, minus the potential energy contribution, plus an extra contribution which quantum mechanically is an extra phase a charged particle gets when it is moving along a vector potential.


Using the metric signature , the Lorentz force for a charge "q" can be written in covariant form:


where "pα" is the four-momentum, defined as


"τ" the proper time of the particle, "Fαβ" the contravariant electromagnetic tensor


and "U" is the covariant 4-velocity of the particle, defined as:


The fields are transformed to a frame moving with constant relative velocity by:


where Λ"μα" is the Lorentz transformation tensor.


The component ("x"-component) of the force is


Substituting the components of the covariant electromagnetic tensor "F" yields


Using the components of covariant four-velocity yields


The calculation for , 3 (force components in the "y" and "z" directions) yields similar results, so collecting the 3 equations into one:


and since differentials in coordinate time "dt" and proper time "dτ" are related by the Lorentz factor,


This is precisely the Lorentz force law, however, it is important to note that p is the relativistic expression,


The electric and magnetic fields are dependent on the velocity of an observer, so the relativistic form of the Lorentz force law can best be exhibited starting from a coordinate-independent expression for the electromagnetic and magnetic fields formula_72, and an arbitrary time-direction, formula_73. This can be settled through Space-Time Algebra (or the geometric algebra of space-time), a type of Clifford algebra defined on a pseudo-Euclidean space, as


formula_76 is a space-time bivector (an oriented plane segment, just like a vector is an oriented line segment), which has six degrees of freedom corresponding to boosts (rotations in space-time planes) and rotations (rotations in space-space planes). The dot product with the vector formula_73 pulls a vector (in the space algebra) from the translational part, while the wedge-product creates a trivector (in the space algebra) who is dual to a vector which is the usual magnetic field vector.


The relativistic velocity is given by the (time-like) changes in a time-position vector formula_78, where


(which shows our choice for the metric) and the velocity is


The proper (invariant is an inadequate term because no transformation has been defined) form of the Lorentz force law is simply


Note that the order is important because between a bivector and a vector the dot product is anti-symmetric. Upon a spacetime split like one can obtain the velocity, and fields as above yielding the usual expression.


In the general theory of relativity the equation of motion for a particle with mass formula_81 and charge formula_82, moving in a space with metric tensor formula_83 and electromagnetic field formula_84, is given as


where formula_86 (formula_87 is taken along the trajectory), formula_88, and formula_89.


The equation can also be written as


where formula_91 is the Christoffel symbol (of the torsion-free metric connection in general relativity), or as


where formula_93 is the covariant differential in general relativity (metric, torsion-free).


The Lorentz force occurs in many devices, including:


In its manifestation as the Laplace force on an electric current in a conductor, this force occurs in many devices including:


The numbered references refer in part to the list immediately below.


In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be implied by the postulates of special relativity.