This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Potential gradient" – news ·newspapers ·books ·scholar ·JSTOR(March 2025) (Learn how and when to remove this message)

Inphysics,chemistry andbiology, apotential gradient is the localrate of change of thepotential with respect todisplacement, i.e. spatialderivative, orgradient. This quantity frequently occurs in equations of physical processes because it leads to some form offlux.

The simplest definition for a potential gradientF in one dimension is the following:^[1]

${\displaystyle F=\frac{{\varphi }_2-{\varphi }_1}{x_2-x_1}=\frac{\mathrm{\Delta }\varphi }{\mathrm{\Delta }x}}$

whereϕ(x) is some type ofscalar potential andx isdisplacement (notdistance) in thex direction, the subscripts label two different positionsx₁,x₂, and potentials at those points,ϕ₁ =ϕ(x₁),ϕ₂ =ϕ(x₂). In the limit ofinfinitesimal displacements, the ratio of differences becomes a ratio ofdifferentials:

${\displaystyle F=\frac{\mathrm{d}\varphi }{\mathrm{d}x}.}$

The direction of the electric potential gradient is from${\displaystyle x_1}$ to${\displaystyle x_2}$.

Inthree dimensions,Cartesian coordinates make it clear that the resultant potential gradient is the sum of the potential gradients in each direction:

${\displaystyle \mathbf{F}={\mathbf{e}}_x\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }x}+{\mathbf{e}}_y\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }y}+{\mathbf{e}}_z\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }z}}$

wheree_x,e_y,e_z areunit vectors in thex, y, z directions. This can be compactly written in terms of thegradientoperator∇,

${\displaystyle \mathbf{F}=\mathrm{\nabla }\varphi .}$

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

This expression represents a significant feature of anyconservative vector fieldF, namelyF has a corresponding potentialϕ.^[2]

UsingStokes' theorem, this is equivalently stated as

${\displaystyle \mathrm{\nabla }\times \mathbf{F}=\mathbf{0}}$

meaning thecurl, denoted ∇×, of the vector field vanishes.

In the case of thegravitational fieldg, which can be shown to be conservative,^[3] it is equal to the gradient ingravitational potentialΦ:

${\displaystyle \mathbf{g}=-\mathrm{\nabla }\mathrm{\Phi }.}$

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.

Inelectrostatics, theelectric fieldE is independent of timet, so there is no induction of a time-dependentmagnetic fieldB byFaraday's law of induction:

${\displaystyle \mathrm{\nabla }\times \mathbf{E}=-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}=\mathbf{0},}$

which impliesE is the gradient of the electric potentialV, identical to the classical gravitational field:^[4]

${\displaystyle -\mathbf{E}=\mathrm{\nabla }V.}$

Inelectrodynamics, theE field is time dependent and induces a time-dependentB field also (again by Faraday's law), so the curl ofE 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:^[5]

${\displaystyle -\mathbf{E}=\mathrm{\nabla }V+\frac{\mathrm{\partial }\mathbf{A}}{\mathrm{\partial }t}}$

whereA is the electromagneticvector potential. This last potential expression in fact reduces Faraday's law to an identity.

Influid mechanics, thevelocity fieldv describes the fluid motion. Anirrotational flow means the velocity field is conservative, or equivalently thevorticitypseudovector fieldω is zero:

${\displaystyle \mathit{\omega }=\mathrm{\nabla }\times \mathbf{v}=\mathbf{0}.}$

This allows thevelocity potential to be defined simply as:

${\displaystyle \mathbf{v}=\mathrm{\nabla }\varphi }$

In anelectrochemicalhalf-cell, at the interface between theelectrolyte (anionicsolution) and themetalelectrode, the standardelectric potential difference is:^[6]

${\displaystyle \mathrm{\Delta }{\varphi }_{(M,M^{+z})}=\mathrm{\Delta }{\varphi }_{(M,M^{+z})}^{\ominus }+\frac{RT}{ze{N}_{\text{A}}}\ln a_{M^{+z}}}$

whereR =gas constant,T =temperature of solution,z =valency of the metal,e =elementary charge,N_A =Avogadro constant, anda_M^+z is theactivity of the ions in solution. Quantities with superscript ⊖ denote the measurement is taken understandard conditions. The potential gradient is relatively abrupt, since there is an almost definite boundary between the metal and solution, hence the interface term.^{[clarification needed]}

This sectionneeds expansion with: and authoritative, source-derived definition and explanation of this subject. You can help byadding to it.(March 2025)

Inbiology, a potential gradient is the net difference inelectric charge across acell membrane.^{[dubious –discuss][citation needed]}

Since gradients in potentials correspond tophysical fields, it makes no difference if a constant is added on (it is erased by the gradient operator∇ which includespartial 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 inclassical field theory and alsogauge field theory.

Absolute values of potentials are not physically observable, only gradients and path-dependent potential differences are. However, theAharonov–Bohm effect is aquantum mechanical effect which illustrates that non-zeroelectromagnetic potentials along a closed loop (even when theE andB fields are zero everywhere in the region) lead to changes in the phase of thewave function of an electricallycharged particle in the region, so the potentials appear to have measurable significance.

Field equations, such as Gauss's lawsfor electricity,for magnetism, andfor gravity, can be written in the form:

${\displaystyle \mathrm{\nabla }\cdot \mathbf{F}=X\rho }$

whereρ is the electriccharge density,monopole density (should they exist), ormass density andX is a constant (in terms ofphysical constantsG,ε₀,μ₀ and other numerical factors).

Scalar potential gradients lead toPoisson's equation:

${\displaystyle \mathrm{\nabla }\cdot (\mathrm{\nabla }\varphi )=X\rho \Rightarrow {\mathrm{\nabla }}^2\varphi =X\rho }$

A generaltheory 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.

• Tensors in curvilinear coordinates

• Concepts in physics
• Spatial gradient

• Articles with short description
• Short description matches Wikidata
• Articles needing additional references from March 2025
• All articles needing additional references
• Wikipedia articles needing clarification from March 2013
• Articles to be expanded from March 2025
• All articles to be expanded
• All accuracy disputes
• Articles with disputed statements from March 2025
• All articles with unsourced statements
• Articles with unsourced statements from March 2025