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| In [[physics]], [[chemistry]] and [[biology]], a '''potential gradient''' is the local [[derivative|rate of change]] of the [[potential]] with respect 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 to [[electric potential]] gradient, which is equal to the [[electric field]].
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| ==Definition==
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| ===One dimension===
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| The simplest definitions for a potential gradient ''F'', in one dimension, is the following:<ref>Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, ISBN 0-7195-3382-1</ref> | |
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| :<math> F = \frac{\phi_2-\phi_1}{x_2-x_1} = \frac{\Delta \phi}{\Delta x}\,\!</math>
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| where {{math|''ϕ''(''x'')}} is some type of [[scalar potential]] and {{math|''x''}} is [[Displacement (vector)|displacement]] (not [[distance]]), in the {{math|''x''}} direction, the subscripts label two different positions {{math|''x''<sub>1</sub>, ''x''<sub>2</sub>}}, and potentials at those points, {{math|''ϕ''<sub>1</sub> {{=}} ''ϕ''(''x''<sub>1</sub>), ''ϕ''<sub>2</sub> {{=}} ''ϕ''(''x''<sub>2</sub>)}}. In the limit of [[infinitesimal]] displacements, the ratio of differences becomes a ratio of [[differential of a function|differentials]]:
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| :<math> F = \frac{{\rm d} \phi}{{\rm d} x}.\,\!</math>
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| ===Three dimensions===
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| In [[three dimensional space|three dimensions]], [[Cartesian coordinates]] make it clear that the resultant potential gradient is the sum of the potential gradients in each direction:
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| :<math> \mathbf{F} = \mathbf{e}_x\frac{\partial \phi}{\partial x} + \mathbf{e}_y\frac{\partial \phi}{\partial y} + \mathbf{e}_z\frac{\partial \phi}{\partial z}\,\!</math>
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| where {{math|'''e'''<sub>x</sub>, '''e'''<sub>y</sub>, '''e'''<sub>z</sub>}} are [[unit vector]]s in the {{math|''x, y, z''}} directions. This can be compactly written in terms of the [[gradient]] [[operator (mathematics)|operator]] {{math|∇}},
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| :<math> \mathbf{F} = \nabla \phi.\,\!</math>
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| although this final form holds in any [[curvilinear coordinate system]], not just Cartesian.
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| This expression represents the significant feature of any [[conservative vector field]] {{math|'''F'''}}, namely {{math|'''F'''}} has a corresponding potential {{math|''ϕ''}}.<ref>Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipcshutz, D. Spellman, Schaum’s Outlines, McGraw Hill (USA), 2009, ISBN 978-0-07-161545-7</ref>
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| Using [[Stoke's theorem]], this is equivalently stated as
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| :<math> \nabla\times\mathbf{F} = \boldsymbol{0} \,\!</math>
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| meaning the [[Curl (mathematics)|curl]], denoted ∇×, of the vector field vanishes.
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| In physics, conservative force fields have corresponding potentials.
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| ==Physics==
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| ===Newtonian gravitation===
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| In the case of the [[gravitational field#classical mechanics|gravitational field]] {{math|'''g'''}}, which can be shown to be conservative{{Citation needed|date=November 2013}}, it is equal to the gradient in [[gravitational potential]] {{math|Φ}}:
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| :<math>\mathbf{g} = - \nabla \Phi. \,\!</math>
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| There are opposite signs between gravitational field and potential, because as the potential gradient and field are opposite in direction, as the potential increases, the gravitational field strength decreases and vice versa.
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| ===Electromagnetism===
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| {{main|Maxwell's equations|Mathematical descriptions of the electromagnetic field}}
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| In [[electrostatics]], the [[electric field]] {{math|'''E'''}} is independent of time {{math|''t''}}, so there is no induction of a time-dependent [[magnetic field]] {{math|'''B'''}} by [[Faraday's law of induction]]:
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| :<math>\nabla\times\mathbf{E} = \frac{\partial\mathbf{B}}{\partial t} = \boldsymbol{0} \,,</math>
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| which implies {{math|'''E'''}} is the gradient of the [[electric potential]] {{math|''ϕ''}}, identical to the classical gravitational field:<ref>Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9</ref>
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| :<math>- \mathbf{E} = \nabla V. \,\!</math>
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| In [[electrodynamics]], the {{math|'''E'''}} field is time dependent and induces a time-dependent {{math|'''B'''}} field also (again by Faraday's law), so the curl of {{math|'''E'''}} is not zero like before, which implies the electric field is no longer the gradient of electric potential, a time-dependent term must be added;<ref>Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3</ref>
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| :<math>- \mathbf{E} = \nabla V + \frac{\partial \mathbf{A}}{\partial t}\,\!</math>
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| where {{math|'''A'''}} is the electromagnetic [[vector potential]]. This last potential expression in fact reduces Faraday's law to an identity.
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| ===Fluid mechanics===
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| In [[fluid mechanics]], the [[velocity field]] {{math|'''v'''}} describes the fluid motion. An [[irrotational flow]] means the velocity field is conservative, or equivalently the [[vorticity]] [[pseudovector]] field {{math|'''ω'''}} is zero:
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| :<math> \boldsymbol{\omega} = \nabla\times\mathbf{v} = \boldsymbol{0}.</math>
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| This allows the [[velocity potential]] to be defined simply as:
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| :<math> \mathbf{v} = \nabla\phi</math>
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| ==Chemistry==
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| {{main|Electrode potentials}}
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| In an [[Electrochemistry|Electrochemical]] [[half-cell]], at the interface between the [[electrolyte]] (an [[ion]]ic [[solution]]) and the [[metal]] [[electrode]], the [[Standard conditions for temperature and pressure|standard]] [[electric potential difference]] is;<ref>Physical chemistry, P.W. Atkins, Oxford University Press, 1978, ISBN 0-19-855148-7</ref>
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| :<math>\Delta \phi_{(M,M^{+z})} = \Delta \phi_{(M,M^{+z})}^{\ominus} + \frac{RT}{zeN_A}\ln a_{M^{+z}} \,\!</math>
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| where ''R'' = [[gas constant]], ''T'' = [[temperature]] of solution, ''z'' = [[Valence (chemistry)|valency]] of the metal, ''e'' = [[elementary charge]], ''N<sub>A</sub>'' = [[Avogadro constant|Avogadro's constant]], and ''a''<sub>M</sub>+z is the [[Activity (chemistry)|activity]] of the ions in solution. Quantities with superscript <s>o</s> denote the measurement is taken under [[Standard conditions for temperature and pressure|standard conditions]]. The potential gradient is relatively abrupt, since there is an almost definite boundary between the metal and solution, hence the interface term.<!---What is this sentence trying to say!??--->{{clarify|date=March 2013}}
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| ==Biology==
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| In [[biology]], potential gradients is the net difference in [[electric charge]] across a [[cell membrane]].
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| ==Non-uniqueness of potentials==
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| Since gradients in potentials correspond to [[Field (physics)|physical field]]s, it makes no difference if a constant is added on (it is erased by the gradient operator {{math|∇}} which includes [[partial differentiation]]). This means there is no way to tell what the "absolute value" of the potential "is" - the zero value of potential is completely arbitrary and can be chosen anywhere by convenience (even "at infinity"). This idea also applies to [[vector potential]]s, and is exploited in [[classical field theory]] and also [[gauge field theory]].
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| Absolute values of potentials are not physically observable, only gradients are. However, the [[Aharonov–Bohm effect]] is a [[quantum mechanics|quantum mechanical]] effect which illustrates that non-zero [[electromagnetic potential]]s (even when the {{math|'''E'''}} and {{math|'''B'''}} fields are zero) lead to changes in the phase of the [[wavefunction]] of an electrically [[charged particle]], so the potentials appear to have measurable significance.
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| ==Potential theory==
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| [[Field equation]]s, such as Gauss's laws [[Gauss's law|for electricity]], [[Gauss's law for magnetism|for magnetism]], and [[Gauss's law for gravity|for gravity]], can be written in the form:
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| :<math>\nabla\cdot\mathbf{F}= X \rho</math> | |
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| where {{math|''ρ''}} is the electric [[charge density]], [[magnetic monopole|monopole]] density (should they exist), or [[mass density]] and {{math|''X''}} is a constant (in terms of [[physical constant]]s [[Gravitational constant|{{math|''G''}}]], [[Vacuum permittivity|{{math|''ε''<sub>0</sub>}}]], [[Vacuum permeability|{{math|''μ''<sub>0</sub>}}]] and other numerical factors).
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| Scalar potential gradients lead to [[Poisson's equation]]:
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| :<math>\nabla\cdot (\nabla\phi)= X \rho \quad \Rightarrow \quad \nabla^2 \phi = X \rho</math>
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| A general [[potential theory|theory of potentials]] has been developed to solve this equation for the potential, the gradient of that solution gives the physical field, solving the field equation.
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| ==See also==
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| *[[Vector potential]]
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| *[[Electromagnetic four-potential]]
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| *[[Tensors in curvilinear coordinates]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| * [http://niuhep.physics.niu.edu/~willis/phys251/chapter_19_day_2.html]
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| [[Category:Concepts in physics]]
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| [[Category:Vector calculus]]
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| [[pl:Gradient potencjału]]
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