Potential gradient

Template:Short description Template:Refimprove In physics, chemistry and biology, a potential gradient is the local 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.

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, Template:ISBN</ref>

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

where Template:Math is some type of scalar potential and Template:Math is displacement (not distance) in the Template:Math direction, the subscripts label two different positions Template:Math, and potentials at those points, Template:Math. In the limit of infinitesimal displacements, the ratio of differences becomes a ratio of 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 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 Template:Math are unit vectors in the Template:Math directions. This can be compactly written in terms of the gradient operator Template: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 Template:Math, namely Template:Math has a corresponding potential Template:Math.<ref>Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipcshutz, D. Spellman, Schaum’s Outlines, McGraw Hill (USA), 2009, Template:ISBN</ref>

Using Stokes' theorem, this is equivalently stated as

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

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

Physics

Newtonian gravitation

In the case of the gravitational field Template:Math, which can be shown to be conservative,<ref>Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, Template:ISBN</ref> it is equal to the gradient in gravitational potential Template:Math:

<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

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} In electrostatics, the electric field Template:Math is independent of time Template:Math, so there is no induction of a time-dependent magnetic field Template:Math by Faraday's law of induction:

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

which implies Template:Math is the gradient of the electric potential Template:Math, identical to the classical gravitational field:<ref>Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, Template:ISBN</ref>

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

In electrodynamics, the Template:Math field is time dependent and induces a time-dependent Template:Math field also (again by Faraday's law), so the curl of Template:Math 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, Template:ISBN</ref>

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

where Template:Math 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 Template:Math describes the fluid motion. An irrotational flow means the velocity field is conservative, or equivalently the vorticity pseudovector field Template:Math 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

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

In an electrochemical half-cell, at the interface between the electrolyte (an ionic solution) and the metal electrode, the standard electric potential difference is:<ref>Physical chemistry, P.W. Atkins, Oxford University Press, 1978, Template:ISBN</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 = valency of the metal, e = elementary charge, N_A = Avogadro constant, and a_M^+z is the activity of the ions in solution. Quantities with superscript ⊖ denote the measurement is taken under standard conditions. The potential gradient is relatively abrupt, since there is an almost definite boundary between the metal and solution, hence the interface term.Template:Clarify

Biology

Template:Expand section In biology, a potential gradient is the net difference in electric charge across a cell membrane.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= Template:Fix }}Template:Cn

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 Template: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 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 mechanical effect which illustrates that non-zero electromagnetic potentials along a closed loop (even when the Template:Math and Template:Math 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 equations, such as Gauss's laws for electricity, for magnetism, and for gravity, can be written in the form:

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

where Template:Math is the electric charge density, monopole density (should they exist), or mass density and Template:Math is a constant (in terms of physical constants Template:Math, Template:Math, Template:Math 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 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