In physics, Planck units are physical units of measurement originally proposed by Max Planck. They form a system of natural units because they are defined exclusively in terms of the following universal dimensionful physical constants — the units are natural because the numerical values of these five universal constants become 1 when expressed in units of this system. A black hole concept drawing by NASA. Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the science of the natural world dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. ... Antonym of psychical. ... // Introduction The definition, agreement and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day. ... Max Planck This article is about Planck, the German physicist. ... In science, a physical constant is a physical quantity whose numerical value does not change. ... The definition, agreement and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day. ... Natural is defined as of or relating to nature; this applies to both definitions of nature: essence (ones true nature) and the untouched world (force of nature). Natural is often used meaning good, healthy, or belonging to human nature. This use can be questioned, as many freely growing plants...

Constant	Symbol	Dimension
speed of light in vacuum		L T-1
Gravitational constant		M-1L3T-2
"reduced Planck's constant" or Dirac's constant	where is Planck's constant	ML2T-1
Coulomb force constant	where is the permittivity in vacuum	Q-2 M L3 T-2
Boltzmann constant		ML2T-2Θ-1

The Planck units are often semi-humorously referred to by physicists as "God's units". They eliminate anthropocentric arbitrariness from the system of units: some physicists believe that an extra-terrestrial intelligence might be expected to use the same system. Cherenkov effect in a swimming pool nuclear reactor. ... In general English usage, length (symbols: l, L) is but one particular instance of distance – an objects length is how long the object is – but in the physical sciences and engineering, the word length is in some contexts used synonymously with distance. Height is vertical distance; width (or breadth... A watch Attempting to understand time has long been a prime occupation for philosophers, scientists and artists. ... According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. ... Plancks constant, denoted h, is a physical constant that is used to describe the sizes of quanta. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... In physics, Coulombs law is an inverse-square law indicating the magnitude and direction of electrostatic force that one stationary, electrically charged object of small dimensions (ideally, a point source) exerts on another. ... Permittivity is an intensive physical quantity that describes how an electric field affects and is affected by a medium. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interactions. ... The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... Temperature is the physical property of a system which underlies the common notions of hot and cold; the material with the higher temperature is said to be hotter. ... God is the term used to denote the Supreme Being ascribed by monotheistic religions to be the creator, ruler and/or the sum total of, existence. ... Anthropocentrism (Greek άνθρωπος, anthropos, man, human being, κέντρον, kentron, center) Anthropocentrism is the knowledge, the theory, the belief, the creed, the “religion” et al, which states that human beings are and have to be (for human-beings) the most important and respected and venerated beings. ...

Natural units can help physicists reframe questions. Perhaps Frank Wilczek said it best : Frank Wilczek at Harvard University Frank Wilczek (born May 15, 1951) is an American physicist of Polish and Italian origin. ...

...We see that the question [posed] is not, "Why is gravity so feeble?" but rather, "Why is the proton's mass so small?" For in Natural (Planck) Units, the strength of gravity simply is what it is, a primary quantity, while the proton's mass is the tiny number [1/(13 quintillion)]...(June 2001 Physics Today)

The strength of gravity is simply what it is and the strength of the electromagnetic force simply is what it is. The electromagnetic force operates on a different physical quantity (electric charge) than gravity (mass) so it cannot be compared directly to gravity. To note that gravity is an extremely weak force is, from the point-of-view of natural units, like comparing apples to oranges. It is true that the electrostatic repulsive force between two protons (alone in free space) greatly exceeds the gravitational attractive force between the same two protons, and that is because the charge on the protons are approximately a natural unit of charge but the mass of the protons are far, far less than the natural unit of mass. A fundamental interaction is a mechanism by which particles interact with each other, and which cannot be explained by another more fundamental interaction. ...

Natural units have the advantage of simplifying many equations in physics by removing conversion factors. For this reason, they are popular in quantum gravity research. A black hole concept drawing by NASA. Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the science of the natural world dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. ... Quantum gravity is the field of theoretical physics attempting to unify the theory of quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ...

Nondimensionalization of some physical equations by conversion to Planck units

	Common form with dimensional conversion factor	Nondimensionalized form
Newton's Law of universal gravitation
Schrödinger's equation
Particle energy with the wave function's radian frequency
Einstein's famous mass to energy equation
The field equation for General relativity
Thermal energy per particle per degree of freedom
Coulomb's law
Maxwell's equations	Nondimensionalization refers to the partial or full removal of units from a mathematical equation by a suitable substitution of variables. ... It has been suggested that gravitation be merged into this article or section. ... In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. ... For other topics related to Einstein see Einstein (disambiguation). ... A simple introduction to this subject is provided in Special relativity for beginners Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. SR theory is based on the previous works of... In physics, the Einstein field equation or Einstein equation is a differential equation in Einsteins theory of general relativity. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ... This article is in need of attention. ... The phrase degrees of freedom is used in three different branches of science: in physics and physical chemistry, in mechanical and aerospace engineering, and in statistics. ... In physics, Coulombs law is an inverse-square law indicating the magnitude and direction of electrostatic force that one stationary, electrically charged object of small dimensions (ideally, a point source) exerts on another. ... Maxwells equations (sometimes called the Maxwell equations) are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ...	(The factors remain because the Coulomb force constant is normalized rather than the permittivity of free space .)

Base Planck units

By constraining the numerical values of the above 5 fundamental constants to be 1, then 5 base units for time, length, mass, charge, and temperature are defined.

Name	Dimension	Expression	Approx. SI equivalent measure
Planck time	Time (T)		5.39121 × 10-44 s
Planck length	Length (L)		1.61624 × 10-35 m
Planck mass	Mass (M)		2.17645 × 10-8 kg
Planck charge	Electric charge (Q)		1.8755459 × 10-18 C
Planck temperature	Temperature (Θ)		1.41679 × 1032 K

The International System of Units (abbreviated SI from the French language name Système International dUnités) is the modern form of the metric system. ... The Planck time is the natural unit of time, denoted by tP. It is considered the smallest possible measurement of time. ... A watch Attempting to understand time has long been a prime occupation for philosophers, scientists and artists. ... To help compare different orders of magnitudes this page lists times between 10-44s and 10-43s. ... Look up second in Wiktionary, the free dictionary. ... This article may not be written in the formal tone expected of an encyclopedia entry. ... In general English usage, length (symbols: l, L) is but one particular instance of distance – an objects length is how long the object is – but in the physical sciences and engineering, the word length is in some contexts used synonymously with distance. Height is vertical distance; width (or breadth... (Redirected from 1 E-35 m) Categories: Orders of magnitude (length) ... The metre (Commonwealth English) or meter (American English) (symbol: m) is the SI base unit of length. ... The Planck mass is the natural unit of mass, denoted by mP. It is the mass for which the Schwarzschild radius is equal to the Compton length divided by π. ≈ 1. ... Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. ... (Redirected from 1 E 8 kg) Categories: Orders of magnitude (mass) ... The international prototype, made of platinum-iridium, which is kept at the BIPM under conditions specified by the 1st CGPM in 1889. ... In physics, the Planck charge is the natural unit of electric charge, denoted by . ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interactions. ... The coulomb (symbol: C) is the SI unit of electric charge. ... The Planck temperature, named after German physicist Max Planck, is the natural unit of temperature, denoted by TP. The Planck units, in general, represent limits of quantum mechanics. ... Temperature is the physical property of a system which underlies the common notions of hot and cold; the material with the higher temperature is said to be hotter. ... To help compare different orders of magnitude this page lists temperatures above 1030 kelvins. ... The kelvin (symbol: K) is the SI unit of temperature, and is one of the seven SI base units. ...

Derived Planck units

As in other systems of units, the following units of physical quantity are defined in terms of the base Planck units.

Name	Dimension	Expression	Approx. SI equivalent measure
Planck energy	Energy (ML2T-2)		1.9561 × 109 J
Planck force	Force (MLT-2)		1.21027 × 1044 N
Planck power	Power (ML2T-3)		3.62831 × 1052 W
Planck density	Density (ML-3)		5.15500 × 1096 kg/m3
Planck angular frequency	Frequency (T-1)		1.85487 × 1043 s-1
Planck pressure	Pressure (ML-1T-2)		4.63309 × 10113 Pa
Planck current	Electric current (QT-1)		3.4789 × 1025 A
Planck voltage	Voltage (ML2T-2Q-1)		1.04295 × 1027 V
Planck impedance	Resistance (ML2T-1Q-2)		2.99792458 × 101 Ω

The Planck energy is the natural unit of energy, denoted by EP. 1. ... To help compare different orders of magnitude we list here energies between 109 joules (a gigajoule, symbol GJ) and 1010 joules. ... The joule (symbol: J) is the SI unit of energy, or work. ... Planck Force A derived Planck unit equated to the Planck Energy (also derived) divided by the Planck length. ... In physics, a force is an external cause responsible for any change of a physical system. ... The newton (symbol: N) is the SI unit of force. ... The Planck energy divided by the Planck time is the Planck power, equal to about 3. ... // Mechanical power In physics, power (symbol: P) is the amount of work W done per unit of time t. ... The watt (symbol: W) is the SI derived unit of power. ... The Planck density is the natural unit of density, denoted by ρP. ρP = Planck mass / (Planck length)3 = ≈ 5. ... Density (symbol: ρ - Greek: rho) is a measure of mass per unit of volume. ... Kilogram per cubic metre is the SI measure of density and is represented as kg/m³, where kg stands for kilogram and m³ stands for cubic metre. ... Sine waves of various frequencies; the lower waves have higher frequencies than those above. ... Angular frequency is a measure of how fast an object is rotating In physics (specifically mechanics and electrical engineering), angular frequency ω (also called angular speed) is a scalar measure of rotation rate. ... Pressure (symbol: p) is the force per unit area acting on a surface in a direction perpendicular to that surface. ... The pascal (symbol Pa) is the SI unit of pressure. ... 1 - conductors, Fp - Planck force, lp - Planck leght, Ip - Planck current. ... In electricity, current refers to electric current, which is the flow of electric charge. ... The ampere (symbol: A) is the SI base unit of electrical current equal to one coulomb per second. ... This article may be too technical for most readers to understand. ... The volt (symbol: V) is the SI derived unit of electric potential difference. ... Electrical resistance is a measure of the degree to which an electrical component opposes the passage of current. ... Ohm may refer to: The scientist Georg Ohm. ...

Discussion

At the "Planck scales" in length, time, density, or temperature, one must consider both the effects of quantum mechanics and general relativity. Unfortunately this requires a theory of quantum gravity which does not yet exist. Fig. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ... Quantum gravity is the field of theoretical physics attempting to unify the theory of quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ...

Most of the Planck units are either too small or too large for practical use, unless prefixed with large powers of ten. They also suffer from uncertainties in the measurement of some of the constants on which they are based, especially of the gravitational constant (which has an uncertainty of 1 in 7000). According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ...

The Planck charge was not originally defined or proposed by Planck. It is a definition of unit charge that is a natural extension of how the other Planck units were defined and is referred to by physicists in some publications. It may be of interest to note that the elementary charge, measured in terms of the Planck charge, comes out to be The elementary charge (symbol e or sometimes q) is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron. ...

where is the fine-structure constant The fine-structure constant or Sommerfeld fine-structure constant, usually denoted , is the fundamental physical constant characterizing the strength of the electromagnetic interaction. ...

.

The dimensionless fine-structure constant can be thought of as taking on the value that it does because of the amount of charge, measured in natural units (Planck charge), that electrons, protons, and other charged particles happen to have been assigned by nature. Because the electromagnetic force between two particles is proportional to the product of the charges of each particle (each which would, in Planck units, be proportional to ), the strength of the electromagnetic force relative to other fundamental forces is proportional to . Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, composed of the electric field and the magnetic field. ...

The Planck impedance comes out to be the Characteristic impedance of free space scaled down by meaning that, in terms of Planck units, . This factor comes from the fact that it is the Coulomb Force Constant in Coulomb's law that is normalized to 1, as is done in the cgs system of units, rather than the permittivity of free space . This, and the fact that the gravitational constant is normalized (rather than or or ) could be considered to be an arbitrary definition and perhaps a non-optimal one from the perspective of defining the most natural physical units as the choice for Planck units. The characteristic impedance of vacuum or characteristic impedance of free space (Z0) is a physical constant, the characteristic impedance of electromagnetic radiation in vacuum, defined by: where: = magnetic constant = electric constant = speed of light In SI units, the value is exactly expressed by: = 1. ... In physics, Coulombs law is an inverse-square law indicating the magnitude and direction of electrostatic force that one stationary, electrically charged object of small dimensions (ideally, a point source) exerts on another. ... CGS is an acronym for centimetre-gram-second. ... Permittivity is an intensive physical quantity that describes how an electric field affects and is affected by a medium. ... According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ...

An increasingly common convention in the literature of particle physics and cosmology is to use reduced Planck units in which (so called because the Planck mass is reduced by in these units). These units have the advantage of removing a factor of from the Einstein equation, Einstein-Hilbert action, Friedmann equation, and the Poisson equation for gravitation, at the expense of introducing one into Newton's law of universal gravitation. Another convention which is occasionally seen is to set , which sets the coefficient of R in the Einstein-Hilbert action to unity. Still, another convention sets so that the dimensionful constants in the gravitoelectromagnetic (GEM) counterparts to Maxwell's equations are eliminated. The GEM equations are of the same form as Maxwell's equations (and the Lorentz force equation) of electromagnetic interaction with mass (or mass density) replacing charge (or charge density) and replacing the permittivity and are applicable in weak gravitational fields or reasonably flat space-time. Like electromagnetic radiation, gravitational radiation propagates at the speed of and has characteristic impedance of free space which becomes unity if units are judiciously defined so that and . Particles erupt from the collision point of two relativistic (100GeV) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... Cosmology, as a branch of astrophysics, is the study of the large-scale structure of the universe and is concerned with fundamental questions about its formation and evolution. ... The Planck mass is the natural unit of mass, denoted by mP. It is the mass for which the Schwarzschild radius is equal to the Compton length divided by π. ≈ 1. ... For other topics related to Einstein see Einstein (disambig) In physics, the Einstein field equation or the Einstein equation is a tensor equation in the theory of gravitation. ... In general relativity, Einsteins field equations can be derived from an action principle starting from the Einstein-Hilbert action: where g is the (pseudo)Riemannian metric, R is the Ricci scalar, n is the number of spacetime dimensions and k is a constant which depends on the units chosen... The Friedman equations relate various cosmological parameters within the context of general relativity. ... Poissons equation is the partial differential equation: Or alternately: or i. ... It has been suggested that this article or section be merged into Gravity. ... In theoretical physics, notably general relativity, gravitomagnetism or gravitoelectromagnetism (GEM) describes effects expected from the motion of gravitational charges (i. ... Maxwells equations (sometimes called the Maxwell equations) are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ... In radio communications, characteristic impedance (acoustic impedance or sound impedance) of a uniform transmission line is the impedance of a circuit that, when connected to the output terminals of a line of arbitrary length, causes the line to appear infinitely long. ...

Planck units and the invariant scaling of nature

Some theoreticians and experimentalists have conjectured that some physical "constants" might actually change over time, a proposition that introduces many difficult questions. A few such questions that are relevant here might be: How would such a change make a noticeable operational difference in physical measurement or, more basically, our perception of reality? If some physical constant had changed, would we even notice it? How would physical reality be different? Which changed constants would result in a meaningful and measureable difference?

Referring to Duff Comment on time-variation of fundamental constants and Duff, Okun, and Veneziano Trialogue on the number of fundamental constants (The operationally indistinguishable world of Mr. Tompkins), if all physical quantities (masses and other properties of particles) were expressed in terms of Planck units, those quantities would be dimensionless numbers (mass divided by the Planck mass, length divided by the Planck length, etc.) and the only quantities that we ultimately measure in physical experiments or in our perception of reality are dimensionless numbers. When one commonly measures a length with a ruler or tape-measure, that person is actually counting tick marks on a given standard or is measuring the length relative to that given standard, which is a dimensionless value. It is no different for physical experiments, as all physical quantities are measured relative to some other like dimensioned values. The eponymous character of Mr. ...

We can notice a difference if some dimensionless physical quantity such as or the proton/electron mass ratio changes (atomic structures would change) but if all dimensionless physical quantities remained constant (this includes all possible ratios of identically dimensioned physical quantity), we could not tell if a dimensionful quantity, such as the speed of light, c, has changed. And, indeed, the Tompkins concept becomes meaningless in our existence if a dimensionful quantity such as c has changed, even drastically. A variable speed of light (VSL) is the concept that the speed of light may not be constant over time. ...

If the speed of light c, were somehow suddenly cut in half and changed to c/2, (but with all dimensionless physical quantities continuing to remain constant), then the Planck Length would increase by a factor of from the point-of-view of some unaffected "god-like" observer on the outside. But then the size of atoms (approximately the Bohr radius) are related to the Planck length by an unchanging dimensionless constant: In the Bohr model of the structure of an atom, put forward by Niels Bohr in 1913, electrons orbit a central nucleus. ...

Then atoms would be bigger (in one dimension) by , each of us would be taller by , and so would our meter sticks be taller (and wider and thicker) by a factor of and we would not know the difference. Our perception of distance and lengths relative to the Planck length is, by axiom, an unchanging dimensionless constant.

Our clocks would tick slower by a factor of (from the point-of-view of this unaffected "god-like" observer) because the Planck time has increased by but we would not know the difference (our perception of durations of time relative to the Planck time is, by axiom, an unchanging dimensionless constant). This hypothetical god-like observer on the outside might observe that light now travels at half the speed that it used to (as well as all other observed velocities) but it would still travel 299792458 of our new meters in the time elapsed by one of our new seconds. We would not notice any difference. Cherenkov effect in a swimming pool nuclear reactor. ...

This in one sense contradicts George Gamow in Mr. Tompkins who suggests that if a dimensionful universal constant such as c changed, we would easily notice the difference; however, as noted, the disagreement is better thought of as the ambiguity in the phrase "changing a physical constant", when one does not specify whether one does so keeping all other dimensionless constants the same, or does so keeping all other dimensionful constants the same. The latter is a somewhat confusing possibility since most of our unit definitions are related to the outcomes of physical experiments which themselves depend on the constants, the only exception being the kilogram. Gamow does not address this subtlety; the thought experiments he conducts in his popular works assume the latter. George Gamow (pronounced GAM-off) (March 4, 1904 – August 19, 1968) , born Georgiy Antonovich Gamow (Георгий Антонович Гамов) was a Ukrainian born physicist and cosmologist. ... The eponymous character of Mr. ... The international prototype, made of platinum-iridium, which is kept at the BIPM under conditions specified by the 1st CGPM in 1889. ...

Max Planck's discovery of the natural units

Max Planck first listed his set of units (and gave values for them remarkably close to those used today) in May of 1899 in a paper presented to the Prussian Academy of Sciences. Max Planck: 'Über irreversible Strahlungsvorgänge'. Sitzungsberichte der Preußischen Akademie der Wissenschaften, vol. 5, p. 479 (1899) Max Planck This article is about Planck, the German physicist. ... This article is about the month of May. ... 1899 was a common year starting on Sunday (see link for calendar). ... 1899 was a common year starting on Sunday (see link for calendar). ...

At the time he presented the units, quantum mechanics had not been invented. He had not yet discovered the theory of black-body radiation (first published December 1900) in which the Planck's Constant made its first appearance and for which Planck was later awarded the Nobel prize. The relevant parts of Planck's 1899 paper leave some confusion as to how he managed to come up with the units of time, length, mass, temperature etc. which today we define using Dirac's Constant and motivate by references to quantum physics before things like and quantum physics were known. Here's a quote from the 1899 paper that gives an idea of how Planck thought about the set of units. As the temperature decreases, the peak of the black body radiation curve moves to lower intensities and longer wavelengths. ... Look up December in Wiktionary, the free dictionary Template:DecemberCalendar2006 December is the twelfth and last month of the year in the Gregorian Calendar and one of seven Gregorian months with the length of 31 days. ... 1900 (MCM) is a common year starting on Monday. ... 1899 was a common year starting on Sunday (see link for calendar). ...

...ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Kulturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können...

...These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as "natural units"...

See also

• atomic units
• geometrized units
• dimensional analysis
• physical constants
• George Stoney

Atomic units (au) are a convenient system of units of measurement used in atomic physics, particularly for describing the properties of electrons. ... // Introduction In physics, especially in the general theory of relativity, geometrized units or geometric units constitute a physical unit system in which all physical quantities are identified with geometric quantities such as areas, lengths, dimensionless numbers, path curvatures, or sectional curvatures. ... Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ... In science, a physical constant is a physical quantity whose numerical value does not change. ... George Johnstone Stoney (1826-1911) was an Irish physicist. ...

External link

• The NIST website(National Institute of Standards and Technology) is a convenient source of data on the commonly recognized constants, including Planck units.
• Planck's original paper

As a non-regulatory agency of the United States Department of Commerce’s Technology Administration, the National Institute of Standards (NIST) develops and promotes measurement, standards, and technology to enhance productivity, facilitate trade, and improve the quality of life. ... Max Planck This article is about Planck, the German physicist. ... In physics, Planck units are physical units of measurement originally proposed by Max Planck. ... The Planck time is the natural unit of time, denoted by tP. It is considered the smallest possible measurement of time. ... This article may not be written in the formal tone expected of an encyclopedia entry. ... The Planck mass is the natural unit of mass, denoted by mP. It is the mass for which the Schwarzschild radius is equal to the Compton length divided by Ï€. ≈ 1. ... In physics, the Planck charge is the natural unit of electric charge, denoted by . ... The Planck temperature, named after German physicist Max Planck, is the natural unit of temperature, denoted by TP. The Planck units, in general, represent limits of quantum mechanics. ... The Planck energy is the natural unit of energy, denoted by EP. 1. ... Planck Force A derived Planck unit equated to the Planck Energy (also derived) divided by the Planck length. ... The Planck energy divided by the Planck time is the Planck power, equal to about 3. ... The Planck density is the natural unit of density, denoted by ρP. ρP = Planck mass / (Planck length)3 = ≈ 5. ... 1 - conductors, Fp - Planck force, lp - Planck leght, Ip - Planck current. ...