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Potential gradientFrom Wikipedia, the free encyclopedia

In phy sics, chemistr y and biology, a potential gradient is the local rate of change of the potential withrespect to displacement, i.e. spatial derivative, or gradient. This quantity frequently occurs in equations of physical processes because it leads to some form of flux. In electrical engineering it refers specifically toelectric potential gradient, which is equal to the electric field.

Contents

1 Definition1.1 One dimension1.2 Three dimensions

2 Physics2.1 Newtonian gravitation

2.2 Electromagnetism2.3 Fluid mechanics

3 Chemistry4 Biology5 Non-uniqueness of potentials6 P otential theo ry7 S ee also8 R eferences9 E xternal links

Definitio n

One dimens ion

The simplest definitions for a potential gradient F , in one dimension, is the following: [1]

where ( x) is some type of scalar potential and x is displacement (not distance), in the x direction, thesubscr ipts la bel two d iffe rent po sitions x1, x2, and potentia ls at those points , 1 = ( x1), 2 = ( x2). Inthe limit of infinitesimal displacements, the ratio of differences becomes a ratio of differentials:

Three dimensions

In three dimensions, Cartesian coordinates make it clear that the resultant potential gradient is the sum of the potential gradients in each direction:
http://en.wikipedia.org/wiki/Three_dimensional_spacehttp://en.wikipedia.org/wiki/Cartesian_coordinateshttp://en.wikipedia.org/wiki/Cartesian_coordinateshttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Chemistryhttp://en.wikipedia.org/wiki/Biologyhttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Potentialhttp://en.wikipedia.org/wiki/Potentialhttp://en.wikipedia.org/wiki/Potentialhttp://en.wikipedia.org/wiki/Cartesian_coordinateshttp://en.wikipedia.org/wiki/Three_dimensional_spacehttp://en.wikipedia.org/wiki/Differential_of_a_functionhttp://en.wikipedia.org/wiki/Infinitesimalhttp://en.wikipedia.org/wiki/Distancehttp://en.wikipedia.org/wiki/Displacement_(vector)http://en.wikipedia.org/wiki/Scalar_potentialhttp://en.wikipedia.org/wiki/Electric_fieldhttp://en.wikipedia.org/wiki/Electric_potentialhttp://en.wikipedia.org/wiki/Fluxhttp://en.wikipedia.org/wiki/Potentialhttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Biologyhttp://en.wikipedia.org/wiki/Chemistryhttp://en.wikipedia.org/wiki/Physics


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where ex, ey, ez are unit vectors in the x, y, z directions. This can be compactly written in terms of thegradient operator ,

although this final form holds in any curvilinear coordinate system, not just Cartesian.

This expression represents the significant feature of any conservative vector field F , namely F has acorresponding potential .[2]

Using Stoke's theorem, this is equivalently stated as

meaning the curl, denoted

, of the vector field vanishes.In physics, conservative force fields have corresponding potentials.

Physics

Newtonian gravitation

In the case of the gravitational field g , which can be shown to be conservative [citation needed ], it is equal to

the gradient in gravitational potential :

There are opposite signs between gravitational field and potential, because as the potential gradient andfield are opposite in direction, as the potential increases, the gravitational field strength decreases and viceversa.

Electromagnetism

Main articles: Maxwell's equations and Mathematical descriptions of the electromagnetic field

In electrostatics, the electric field E is independent of time t , so there is no induction of a time-dependentmagnetic field B by Faraday's law of induction:

which implies E is the gradient of the electric potential , identical to the classical gravitational field: [3]

In electrodynamics, the E field is time dependent and induces a time-dependent B field also (again byFaraday's law), so the curl of E is not zero like before, which implies the electric field is no longer thegradient of electric potential, a time-dependent term must be added; [4]
http://en.wikipedia.org/wiki/Electrodynamicshttp://en.wikipedia.org/wiki/Electric_potentialhttp://en.wikipedia.org/wiki/Faraday%27s_law_of_inductionhttp://en.wikipedia.org/wiki/Magnetic_fieldhttp://en.wikipedia.org/wiki/Electric_fieldhttp://en.wikipedia.org/wiki/Electrostaticshttp://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_fieldhttp://en.wikipedia.org/wiki/Maxwell%27s_equationshttp://en.wikipedia.org/wiki/Gravitational_potentialhttp://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttp://en.wikipedia.org/wiki/Gravitational_field#classical_mechanicshttp://en.wikipedia.org/wiki/Curl_(mathematics)http://en.wikipedia.org/wiki/Stoke%27s_theoremhttp://en.wikipedia.org/wiki/Conservative_vector_fieldhttp://en.wikipedia.org/wiki/Curvilinear_coordinate_systemhttp://en.wikipedia.org/wiki/Operator_(mathematics)http://en.wikipedia.org/wiki/Gradienthttp://en.wikipedia.org/wiki/Unit_vector


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where A is the electromagnetic vector potential. This last potential expression in fact reduces Faraday'slaw to an identity.

Fluid mechanics

In fluid mechanics, the velocity field v describes the fluid motion. An irrotational flow means the velocityfield is conservative, or equivalently the vorticity pseudovector field is zero:

This allows the velocity potential to be defined simply as:

Chemistry Main article: Electrode potentials

In an Electrochemical half-cell, at the interface between the electrolyte (an ionic solution) and the metalelectrode, the standard electric potential difference is; [5]

where R = gas constant, T = temperature of solution, z = valency of the metal, e = elementary charge, N A =Avogadro's constant, and a M+z is the activity of the ions in solution. Quantities with superscript o denotethe measurement is taken under standard conditions. The potential gradient is relatively abrupt, since thereis an almost definite boundary between the metal and solution, hence the interface term.

Biology

In biology, potential gradients is the net difference in electric charge across a cell membrane.

Non-uniqueness of potentials

Since gradients in potentials correspond to physical fields, it makes no difference if a constant is added on(it is erased by the gradient operator which includes partial differentiation). This means there is no wayto tell what the "absolute value" of the potential "is" - the zero value of potential is completely arbitraryand can be chosen anywhere by convenience (even "at infinity"). This idea also applies to vectorpotentials, and is exploited in classical field theory and also gauge field theory.

Absolute values of potentials are not physically observable, only gradients are. However, the AharonovBohm effect is a quantum mechanical effect which illustrates that non-zero electromagnetic potentials(even when the E and B fields are zero) lead to changes in the phase of the wavefunction of an electricallycharged particle, so the potentials appear to have measurable significance.
http://en.wikipedia.org/wiki/Charged_particlehttp://en.wikipedia.org/wiki/Wavefunctionhttp://en.wikipedia.org/wiki/Electromagnetic_potentialhttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effecthttp://en.wikipedia.org/wiki/Gauge_field_theoryhttp://en.wikipedia.org/wiki/Classical_field_theoryhttp://en.wikipedia.org/wiki/Vector_potentialhttp://en.wikipedia.org/wiki/Partial_differentiationhttp://en.wikipedia.org/wiki/Field_(physics)http://en.wikipedia.org/wiki/Cell_membranehttp://en.wikipedia.org/wiki/Electric_chargehttp://en.wikipedia.org/wiki/Biologyhttp://en.wikipedia.org/wiki/Standard_conditions_for_temperature_and_pressurehttp://en.wikipedia.org/wiki/Activity_(chemistry)http://en.wikipedia.org/wiki/Avogadro_constanthttp://en.wikipedia.org/wiki/Elementary_chargehttp://en.wikipedia.org/wiki/Valence_(chemistry)http://en.wikipedia.org/wiki/Temperaturehttp://en.wikipedia.org/wiki/Gas_constanthttp://en.wikipedia.org/wiki/Electric_potential_differencehttp://en.wikipedia.org/wiki/Standard_conditions_for_temperature_and_pressurehttp://en.wikipedia.org/wiki/Electrodehttp://en.wikipedia.org/wiki/Metalhttp://en.wikipedia.org/wiki/Solutionhttp://en.wikipedia.org/wiki/Ionhttp://en.wikipedia.org/wiki/Electrolytehttp://en.wikipedia.org/wiki/Half-cellhttp://en.wikipedia.org/wiki/Electrochemistryhttp://en.wikipedia.org/wiki/Electrode_potentialshttp://en.wikipedia.org/wiki/Velocity_potentialhttp://en.wikipedia.org/wiki/Pseudovectorhttp://en.wikipedia.org/wiki/Vorticityhttp://en.wikipedia.org/wiki/Irrotational_flowhttp://en.wikipedia.org/wiki/Velocity_fieldhttp://en.wikipedia.org/wiki/Fluid_mechanicshttp://en.wikipedia.org/wiki/Vector_potential


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Poten tial theory

Field equ ations, such as Gauss's laws for electricity, for magnetism, and for gravity, can be written in theform:

where is the electric charge density, monopole density (should they exist), or mass density and X is aconstant ( in terms of physical constants G , 0, 0 and other numerical factors).

Scalar po tential gradients lead to Poisson's equation:

A genera l theory of potentials has been developed to solve this equation for the potential, the gradient of that soluti on gives the physical field, solving the field equation.

See al so

Ve ctor potentialElectromagnetic four-potentialTe nsors in curvilinear coordinates

Refer ences

1. ^ E ssential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, ISBN 0-719 5-3382-1

2. ^ V ector Analysis (2nd Edition), M.R. Spiegel, S. Lipcshutz, D. Spellman, Schaums Outlines, McGraw Hill(USA), 2009, ISBN 978-0-07-161545-7

3. ^ E lectromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008,ISBN 978-0-471-92712-9

4. ^ I ntroduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007,ISBN 81-7758-293-3

5. ^ P hysical chemistry, P.W. Atkins, Oxford University Press, 1978, ISBN 0-19-855148-7

Exter nal links

[1] (http://niuhep.physics.niu.edu/~willis/phys251/chapter_19_day_2.html)

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