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.
where ϕ(x) is some type of scalar potential and x is displacement (not distance) in the x direction, the subscripts label two different positions x1, x2, and potentials at those points, ϕ1 = ϕ(x1), ϕ2 = ϕ(x2). In the limit of infinitesimal displacements, the ratio of differences becomes a ratio of differentials:
The direction of the electric potential gradient is from x1 to x2.
although this final form holds in any curvilinear coordinate system, not just Cartesian.
This expression represents a significant feature of any conservative vector field F, namely F has a corresponding potential ϕ.
Using Stokes' theorem, this is equivalently stated as
meaning the curl, denoted ∇×, of the vector field vanishes.
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.
In electrodynamics, the E field is time dependent and induces a time-dependent B field also (again by Faraday's law), so the curl of 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:
where A is the electromagnetic vector potential. This last potential expression in fact reduces Faraday's law to an identity.
where R = gas constant, T = temperature of solution, z = valency of the metal, e = elementary charge, NA = Avogadro constant, and aM+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.
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 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 E and 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.
Scalar potential gradients lead to Poisson's equation:
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.
- Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, ISBN 0-7195-3382-1
- Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipcshutz, D. Spellman, Schaum’s Outlines, McGraw Hill (USA), 2009, ISBN 978-0-07-161545-7
- Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 978-0-470-01460-8
- Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9
- Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3
- Physical chemistry, P.W. Atkins, Oxford University Press, 1978, ISBN 0-19-855148-7