Physical quantity

A physical quantity is a quantifiable property of a physical object, process or state. Their value ( value size ) is specified as the product of a numerical value ( the measure ) and one unit.

Vector quantities are indicated by size value and direction. The term physical size in the present understanding was introduced by Julius Wallot and sat down slowly from 1930. This led to a conceptually clear distinction of quantity equations, numerical value equations and tailored size equations.

One size equation is the mathematical representation of a physical law, the states and their changes describes in a physical system. It represents the force while the correlation between various physical quantities usually by means of a formula label for each of these sizes dar. size equations are independent of units of the physical quantities contained in them.

Those physical quantities, which are defined as the basis of a multivariable system, called basic variables.

• 2.1 Formula and symbols
• 2.2 formatting
• 2.3 Faulty sizes
• 2.4 Examples for the identification of additional information

• 3.1 Size equations
• 3.2 dimensional analysis
• 3.3 Arithmetic operations
• 3.4 numerical value equations

• 4.1 Multivariable Systems
• 4.2 International System of Units

• 5.1 ratio and ratio sizes
• 5.2 Field and energy quantities
• 5.3 Status and process variables

Basics

A comparison of two things always requires a criteria by which the comparison takes place ( point of comparison ). This must be a feature ( or property), the two things is to own. As a physical quantity identifies a feature if this has a value such that the ratio of two characteristic values ​​is a real number factor (ratio size). A comparison on the basis of a quantity is thus quantifiable. The comparison process to determine the number of factors is called measurement. The measurability of a feature, ie to specify the unique and reproducible measurement procedure for a comparison is equivalent to the definition of a physical quantity.

All of the features of an object fall into two classes, physical quantities and all the rest. As the name suggests, the physics is concerned exclusively with the former class. The physics establish general relationships between variable values ​​, ie relationships that apply to all vehicles of this size. As the carrier is referred to here all the objects that have the size considered as a characteristic. Physical relationships are thus independent of the specific nature of a carrier.

The following paragraph deals with individual terms that are used in connection with sizes.

Size type

For example, width, height and length of a cube, the diameter of a tube, wingspan of a bird, wavelength etc. all sizes of Size type "Length". They can be compared with all the length of a meter rod. Whether also the amount of precipitation, expressed as a volume / surface, is considered similar to this, the approach is left to the user, although it is easy to measure with the ruler.

Other examples seem clearer. Although consumption data of motor vehicles in " liters per 100 kilometers " have the dimension of an area, hardly anyone will consider such usage data and areas on similar sizes look, although it is possible.

Size value

The value of a physical quantity (size value) in accordance with generally widespread opinion, the product of a number and the physical unit that is assigned to the respective size style. The ratio of two variable values ​​of similar sizes is a real number.

Very much more careful this context " spelling physical equations " of the DIN standard 1313 within the German standard work in the first issue presented in November 1931 with the occurring in the physical equations symbols can be calculated as if they the physical "size", that is meant named numbers. They are then advantageously conceived as a symbolic "products" from the numerical values ​​( numerical values ​​), and the units in accordance with the equation

Refers to a difference by a factor of 10 between the values ​​of the same magnitude as a magnitude. So magnitudes correspond to a ratio of.

There are a number of sizes, more established size values ​​immutable. This is called a constant of nature, universal constant or even physical constant (examples: speed of light in vacuum, electron charge, Planck's constant, the fine structure constant).

Numerical value and unit

It is expedient to determine the ratio of a size value to the value of the like, fixed and well-defined benchmark. The benchmark value is called unit or short unit, the measured ratio as a measure or numerical value. The size value can then be represented as a product of numerical value and unit ( see also section notation). The numerical value is a real number depending on the definition of the size - in some sizes limited to non-negative values ​​- or complex; with some dimensionless quantities such as some quantum numbers it is always integer.

The definition of a unit subject to human caprice. One possibility is the choice of a particular object, - a so-called Normals - as a carrier of the size, the size value is used as a unit. Another possibility is to take a calculated size value, for which, however, a suitable physical connection to other size values ​​must be known (see also section size equations). A third possibility is to use the value of a physical constant as a unit, if one exists for the desired size.

Theoretically, it is sufficient to define a single unit for size type. For historical reasons, but a variety of different units for the same size style has often formed. These differ as all similar size values ​​merely a pure numerical factor.

Scalars, vectors, and higher-order tensors

Certain physical quantities have an orientation in physical space, so that the measured quantity value depends on the direction of measurement. For example, the speed of a vehicle is typically directed along a road; the measured velocity is perpendicular to this zero. In general, the reference can represent any physical quantity to the space as a tensor. One case is different:

• 0 -th order tensors or scalars: These are all variables that have no directional dependence, that is, are determined solely by their size value.
• 1 -th order tensors or vectors: These are all variables which by their size value (amount) and its direction are fully determined.
• Tensors 2nd- level: These are sizes that are determined by their size value in two independent directions, described mathematically by approximately a square shape. An illustrative example is the cause-and- mediation - effect principle ( three- finger rule ), such as when the cause is in one direction and the effect in a different direction. Those are the mediating effect of " generalized susceptibilities ". The mediating system of " looking size values ​​" is then a tensor 2-nd stage.
• Tensors -th order: sizes that are determined by their size value in independent directions and are consequently described by an n-ary form

The tensor character of a variable is separated from her size value. Vectors, for example, can be represented mathematically as the product of magnitude value (magnitude ) and direction vector with magnitude one. Tensors of the second stage are, for example, the so-called inertia tensor, one occurring in the mechanics of rotating objects size, or the so-called stress tensor in the theory of elasticity, from which the name " tensor " is due, EElast ( elastic energy ) where the penultimate size the stress tensor and the strain tensor describes the so-called last.

Higher-order tensors are mathematical objects of differential geometry.

A physical size is in this case also invariant under coordinate transformations. Just as the system size values ​​is independent of the unit, the respective direction is independent of the choice of coordinate system.

Tensors have a characteristic level for their behavior under point reflection. Thus a scalar size of an object does not change when one reflects that object to a point. A vector- size, such as the speed shown by the reflection point on the other hand in the opposite direction. Some sizes behave while in rotation and translation as tensors, however, differ from under point reflection. Such variables are called Pseudotensoren. In the pseudoscalar size value changes its sign. In pseudo vectors such as the angular momentum, the direction does not rotate about by a point reflection of the object.

Spelling

The following comments are based on the national and international regulations of standardization organizations and professional societies (eg DIN 1338, ISO 31/XI, Recommendations of the International Union of Pure and Applied Physics ( IUPAP ) ).

Formula and symbols

A physical quantity is allocated in mathematical equations, a character that is called symbols. This is in principle arbitrary, but there are a number of conventions (eg SI, DIN 1304 ÖNORM A 6438, A 6401 ÖNORM, etc.) to describe certain sizes. It is often taken as symbols of the initial letter of the Latin name of a variable. Also letters of the Greek alphabet are often used. Usually there is a formula character only of a single letter, which can be provided for further distinction with an index.

For units, there are standardized characters that are called unit characters. They usually consist of one or more Latin characters or less commonly, a special character such as a degree sign or Greek letters as the Ω for the Ohm unit. For units that are named after people, the first letter of the unit character is usually capitalized.

The indication of the size value is always the product of a numerical value and unit. If you want to specify only the number value, is given to the symbols in braces. If you only want to specify the unit, as is given to the symbols in square brackets. Formally, a size value can therefore be written as follows:

Since the value depends on the chosen unit, the sole representation of the character formula in curly brackets is not unique. Therefore is common for the caption of tables and coordinate axes the presentation "G / [ G] " (eg "m / kg " ) or " G [G ] " (eg "m in kg "). The representation of units in square brackets (eg "m [ kg] " ) or in parentheses ( eg "m (kg) " ), however, does not meet the standard DIN 1313 and is in the recommendations to SI unit system is not recommended.

Since the units used are dependent on the system of units, the unit system must be specified:

Formatting

The formatting is governed by DIN 1338. Thus, the symbols will be written in italics, while the unit symbol is written with an upright font, to distinguish it from symbols. For example, " m" is the symbol for the quantity "mass " and " m" is the unit symbol for the unit "meter".

A space is written between the measured value and the unit symbol. An exception to this rule is in the degree of characters that are written without space directly behind the measure ( " an angle of 180 ° " ), provided that no other unit symbols follow ( " the outside temperature is 23 ° C"). In brief this purpose recommends a narrow space, which should also be protected before any newline, so that numerical value and unit are not separated.

In formulas, vectors are often characterized by a special notation. There are different conventions. Are usual vector arrows above the letter (), bold ( ) or dashes under the symbols (). For tensors of higher levels of uppercase letters in a sans serif font (), Gothic type ( ) or double underscore ( ) are used. Which spelling is selected also depends on whether written by hand or machine, as can not be features such as bold or serifs with a manuscript faithfully reflect.

There is the language and the tray depending on the different traditions upright, italic in connection with formulas. In more modern literature, however, the Convention has established itself, not only size icons, but everything is changeable, italics will be taken; Unit characters or Explanatory other hand, is set upright. So symbols and variable indices appear in italics. example:

Faulty sizes

In faulted size values ​​of the numerical value is specified with its measurement uncertainty or - depending on the circumstances - with its error limits, see also Deviation. This usually happens by adding terms to a "±" after the erroneous value, followed by the error value (where parentheses are required if a unit follows, so they refer to both values ​​). But also short forms like a stapled error indication or bold the uncertain digit in the numerical value are common.

The number of decimal places to be specified uncertain of the numerical value depends on the error value. If it starts with a 1 or 2, the two places are quoted, otherwise just one. If necessary, the value is as usual to round, see DIN 1333; MPE is, however, always rounded up.

Examples for the identification of additional information

Additional names or information may not in the size value of a physical quantity (ie neither the unit nor the numerical value ) appear or be added to this principle, as it would be absurd; they may be expressed only in the naming or designation of the physical quantity, ie in symbols.

For example, you can add the symbols commonly used for the frequency in the correct notation with a subscript as to point out that a revolution frequency (speed ) is meant:

It can also be a separate, clearly defined symbols are used. In order to dispense with the double index in the example above in favor of easier reading, for example, you could possibly easily remembered the symbol of " the rotational frequency, the number of revolutions " to introduce and write:

Without further explanation could be found in the rule, for example,

Use, as the symbols for the two special cases of the height and width of a length dimension commonly are common.

In practice it is not always a clean distinction between size value or unit of a physical quantity on the one hand and mere additional information on the other hand takes place, so that it comes to mixing. The rotation number listed is a common example of this. " Revolution " there is no unity, but merely describes the frequency -inducing process in more detail. Not allowed, but frequently occurring, therefore, is about

Other examples of frequently occurring false writing or ways of speaking are:

• Neutron flux density:

• Mass concentration of lead:

• Through a coil caused magnetic field strength:

Link between physical quantities

Size equations

F = 750 N = 750 kg · m/s2 = m · a because 1 N ( Newton = 1 ) = 1 kg · m/s2

The representation of natural laws and technical contexts in mathematical equations called size equations. The symbols a size equation have the meaning of physical quantities, provided they are not meant as symbols for mathematical functions or operators. Size equations are independent of the choice of units.

Size equations linking different physical quantities and their magnitudes together. For evaluation is necessary to replace the symbols by the product of numerical value and unit. The units used are irrelevant.

Dimensional analysis

The validity of a quantity equation is refutable by dimensional analysis: standing on both sides of the equation different dimensions, it is false.

Arithmetic operations

For physical quantities are not all arithmetic operations, which would be possible with sheer numbers make sense. It has been found that a small number of arithmetic operations sufficient to describe all known natural phenomena.

• Addition and subtraction are possible only between quantities of the same size type. The dimension and thus the unit of size ( n ) remains unchanged, the numerical values ​​are added together.

• Multiplication and division are unrestricted. The two variables are multiplied by their numerical values ​​multiplied and the product of the units is formed. For division same applies. The result is one that is as a rule to a different size type as the two factors, unless one of the factors is simply a dimensionless number.

• Powers can thus be formed as well. This is true for both positive and negative integer and fractional exponent ( ie including fractures and roots).

• Transcendental functions such as etc. are only defined for pure numbers as an argument. They can therefore only be applied to dimensionless quantities. The function value is also a dimensionless number.

• The differential of a quantity is of the same size type as the size itself differential and integral calculus is unrestricted.

An item is misrepresented, if this arithmetic operations would be carried out in nonsensical ways. The corresponding control is performed in the dimensional analysis to check the existence of a yet unknown law.

Numerical value equations

With

In numerical value equations, the symbols only have the meaning of numerical values ​​. These equations are therefore dependent upon the choice of units and only useful if they are also known. The use of variable values ​​in other units can easily lead to errors. It is therefore advisable to always perform calculations with size equations and evaluate them in the last step.

Formulas in historical texts, " rules of thumb " and empirical formulas are usually given in the form of numerical value equations. In some cases, the units are to be used with in the equation. The case sometimes encountered using square brackets around the unit characters, such as instead is non-standard. DIN 1313:1998-12, Section 4.3 looks symbols before in braces or the sizes of the respective desired unit division. One then obtains a so-called tailored size equation.

Size and unit systems

Size systems

Every discipline of science and technology uses a limited set of physical quantities which are laws of nature are linked. If you select from these variables a few basic sizes from, so that all other of the considered area can be represented as products of powers of the base sizes, then make all sizes together a sizing system that also provided no basis size of the other base can be depicted. The displayable from the basic sizes Sizes are called derived quantities, the product of powers is called dimension product. What sizes are selected for the base, is basically arbitrary and mostly done for practical reasons. The number of basic variables determines the degree of size system. For example, the international sizing system with its seven base quantities is a multivariable system of the seventh degree.

International System of Units

One required for each size a unit to specify size values. Therefore, each system size corresponds to a system of units of the same degree, which is composed analogously from independent base units and derived from these displayable units. The derived units are represented from the base units by products of powers - in contrast to size systems, possibly supplemented by a numerical factor. We call the unit system as coherent if all units without this additional factor may be formed. In such systems, all size equations can be interpreted as a numerical value equations and accordingly evaluated quickly.

The used in almost all countries of the world International System of Units (SI) is a coherent system of units of the seventh degree, which is based on the international sizing system; However, the International sizing system was developed later than the SI. The SI also defines standardized prefixes for units of measurement, but the multiples or parts of a SI unit itself so formed are not part of the actual unit system, as this would contradict the consistency. For example, an imaginary unit system, the basic units of centimeters () and second ( ) and the unit derived meters per second () comprises non-coherent: Because it takes a numerical factor () for the formation of this system.

(See Other competing systems of units below in the section consequences. )

Special sizes

Quotient and ratio sizes

The quotient of two quantities is a new size. Such a quantity is called a ratio size (or aspect ratio ) when the output variables of the same size type are, otherwise as the quotient size. More generally, the quotient size is defined in the DIN standard 1313 of December 1998; after which only requires that the fraction of numerator and denominator size size is constant. From April 1978 to November 1998, however, the DIN had the concept of size ratio of specific recommended in the standard edition of April 1978 only for fractions of two sizes of different dimension and of a size ratio ( a ratio size) only requires that the output quantities of the same dimension, but not necessarily same size type are. ( For example, the electric current and the magnetic flux of the same dimension but different size type. )

Frequently Quotient quantities are colloquially paraphrased incorrectly. For example, a designation of the drive speed as " distance traveled per unit of time " is factually incorrect, since the definition of a quantity independent of possible units. Taking away those names literally, this inevitably led to different values ​​depending on the size of the used unit. Corrections would have to therefore " distance traveled per past time" or say " way each time."

If two variables refer to a property of the same object, is called the quotient size also related size. Here, the denominator is the size of reference, while the numerator is the center of gravity in the naming. In particular, it refers to a related quantity as ...

• Specifically ... if it refers to the mass.
• Molar ... if it refers to the amount of substance.
• ... Density, as it relates to the volume ( or area density on the surface, and a length density of the length).

Ratio sizes are always dimensionless. They can occur as arguments of transcendental functions according to the above calculation rules. The name of a relative size usually includes an adjective such as relative or normalized or it ends on number or value. Examples include the Reynolds number and drag coefficient.

Different ratio sizes are rarely the same size type, sometimes therefore the unit characters for better separation when specifying their size value not trimmed. Frequently ratio values ​​are given in units of %, ‰ or ppm. A special position have ratio units, if they are the ratio of identical units. These are always 1 and hence idempotent, that is, they can be multiplied by itself any number of times without changing its value. Some idempotent ratio units bear a special name, such as the angle unit radians ( rad). In coherent systems of units are the ratio units is always 1, ie idempotent.

Idempotent ratio units are interesting because you can just multiply the numbers here. If one says, for example, that 30 % of the Earth 's land area and are the continent Asia represents 30 % of the land area, one can not conclude that 900 % of the earth's surface are from the continent Asia covered because % is not idempotent, ie, % 2 is not the same as % is. If you say now, however, that a proportion of the earth's surface land area is 0.3 and the Asian continent occupies a proportion of 0.3 of the land area, it can be concluded that the Earth's surface 0.09 from the continent Asia are covered, because we here unit 1 have, which is idempotent.

Field and energy quantities

Field sizes are used to describe physical fields. The square of a field size is proportional to its energetic state which is detected via a power size in linear systems. Without having to know the exact law, it follows directly that the ratio of two power quantities is equal to the square ratio of the corresponding field sizes. It is irrelevant whether the energy quantities sizes of Size type or energy -related quantities, such as power (energy per time ) and intensity ( energy per unit time and area) belong. Energy quantities therefore are also known as output sizes.

In many technical areas, the logarithmic ratios of particular interest. Such variables are called the level or degree. If, during the formation of the natural logarithm is used, it indicates this by the auxiliary unit Napier (Np ), it is the decadic logarithm, one uses the auxiliary unit Bel (B) or more often her tenth, the decibel (dB).

State and process variables

Especially in thermodynamics, a distinction between state and process variables.

State variables are physical quantities that represent a property of a system state. It is further distinguished between extensive and intensive variables. Extensive quantities such as mass and amount of substance to double its size value in doubling system, intensive variables such as temperature and pressure remain constant. Also common is the distinction between substance intrinsic and native state variables.

Process variables, however, describe a process, namely the transition between system states. Among them include in particular the sizes of "work" () and " heat " (). In order to bring its character as a pure process variables expressed, they are in many places only given as differentials, where they are often not preceded by, but one or đ.

Consequences

The consequences especially for the Maßsystemproblematik. There are still different measurement systems side by side:

• Cgs system

• MKSA system

• High energy system (see in Planck units)

In the measurement system specified see, for example, the basic equations of electrodynamics, the so-called Maxwell's equations, a formula slightly different from; but the laws of physics are invariant to such changes as mentioned. In particular, you can always convert to another by a system of measurement, even though the correlations used therein may be slightly more complicated than the conversion from meters to centimeters.