Editing Potential gradient

{{short description|Local rate of change in potential with respect to displacement}}
{{refimprove|date=March 2025}}
In [[physics]], [[chemistry]] and [[biology]], a '''potential gradient''' is the local [[Rate (mathematics)|rate of change]] of the [[potential]] with respect to [[Displacement (geometry)|displacement]], i.e. spatial [[derivative]], or [[gradient]]. This quantity frequently occurs in equations of physical processes because it leads to some form of [[flux]].

==Definition==

===One dimension===

The simplest definition 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>

:<math> F = \frac{\phi_2-\phi_1}{x_2-x_1} = \frac{\Delta \phi}{\Delta x}\,\!</math>

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]]:

:<math> F = \frac{{\rm d} \phi}{{\rm d} x}.\,\!</math>

The direction of the electric potential gradient is from <math>x_1</math> to <math>x_2</math>.

===Three dimensions===

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:

:<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>

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|∇}},

:<math> \mathbf{F} = \nabla \phi.\,\!</math>

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

This expression represents a 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>

Using [[Stokes' theorem]], this is equivalently stated as

:<math> \nabla\times\mathbf{F} = \boldsymbol{0} \,\!</math>

meaning the [[Curl (mathematics)|curl]], denoted ∇×, of the vector field vanishes.

==Physics==

===Newtonian gravitation===

In the case of the [[gravitational field#classical mechanics|gravitational field]] {{math|'''g'''}}, which can be shown to be conservative,<ref>Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, {{ISBN|978-0-470-01460-8}}</ref> it is equal to the gradient in [[gravitational potential]] {{math|Φ}}:

:<math>\mathbf{g} = - \nabla \Phi. \,\!</math>

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

===Electromagnetism===
{{main|Maxwell's equations|Mathematical descriptions of the electromagnetic field}}
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]]:

:<math>\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t} = \boldsymbol{0} \,,</math>

which implies {{math|'''E'''}} is the gradient of the electric potential {{math|''V''}}, 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>

:<math>- \mathbf{E} = \nabla V. \,\!</math>

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>

:<math>- \mathbf{E} = \nabla V + \frac{\partial \mathbf{A}}{\partial t}\,\!</math>

where {{math|'''A'''}} is the electromagnetic [[vector potential]]. This last potential expression in fact reduces Faraday's law to an identity.

===Fluid mechanics===

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:

:<math> \boldsymbol{\omega} = \nabla\times\mathbf{v} = \boldsymbol{0}.</math>

This allows the [[velocity potential]] to be defined simply as:

:<math> \mathbf{v} = \nabla\phi</math>

==Chemistry==

{{main|Electrode potentials}}

In an [[Electrochemistry|electrochemical]] [[half-cell]], at the interface between the [[electrolyte]] (an [[ion]]ic [[Solution (chemistry)|solution]]) and the [[metal]] [[electrode]], the standard [[electric potential difference]] is:<ref>Physical chemistry, P.W. Atkins, Oxford University Press, 1978, {{ISBN|0-19-855148-7}}</ref>

:<math>\Delta \phi_{(M,M^{+z})} = \Delta \phi_{(M,M^{+z})}^{\ominus} + \frac{RT}{zeN_\text{A}}\ln a_{M^{+z}} \,\!</math>

where ''R'' = [[gas constant]], ''T'' = [[temperature]] of solution, ''z'' = [[Valence (chemistry)|valency]] of the metal, ''e'' = [[elementary charge]], ''N''<sub>A</sub> = [[Avogadro constant]], and ''a''<sub>M<sup>+z</sup></sub> is the [[Activity (chemistry)|activity]] of the ions in solution. Quantities with superscript ⊖ denote the measurement is taken under [[Standard 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}}

==Biology==
{{expand section | with = and authoritative, source-derived definition and explanation of this subject | small = no | date = March 2025}}
<!--This phenomenon is notrestricted to potential differences across the exterior-interior boundary, as suggested. And the net difference in electric charge across any membrane is just that (the net difference in charge). The potential is something that arises because of that difference ("difference gives rise..." language). For goodness sake, all of life exists because of the existence of potential gradients across stable barriers in living systems. Please correct and expand this.-->
In [[biology]], a potential gradient is the net difference in [[electric charge]] across a [[cell membrane]].{{dubious|date=March 2025}}{{cn|date=March 2025}}

==Non-uniqueness of potentials==

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|&nabla;}} 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 potentials, and is exploited in [[classical field theory]] and also [[gauge field theory]].

Absolute values of potentials are not physically observable, only gradients and path-dependent potential differences are. However, the [[Aharonov–Bohm effect]] is a [[quantum mechanics|quantum mechanical]] effect which illustrates that non-zero [[electromagnetic potential]]s along a closed loop (even when the {{math|'''E'''}} and {{math|'''B'''}} fields are zero everywhere in the region) lead to changes in the phase of the [[wave function]] of an electrically [[charged particle]] in the region, so the potentials appear to have measurable significance.

==Potential theory==

[[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:

:<math>\nabla\cdot\mathbf{F}= X \rho</math>

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 {{math|[[Gravitational constant|''G'']]}}, {{math|[[Vacuum permittivity|''ε''<sub>0</sub>]]}}, {{math|[[Vacuum permeability|''μ''<sub>0</sub>]]}} and other numerical factors).

Scalar potential gradients lead to [[Poisson's equation]]:

:<math>\nabla\cdot (\nabla\phi)= X \rho \quad \Rightarrow \quad \nabla^2 \phi = X \rho</math>

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.

==See also==
*[[Tensors in curvilinear coordinates]]

==References==

{{reflist}}

[[Category:Concepts in physics]]
[[Category:Spatial gradient]]