Potential gradient

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 natural sciences because it corresponds to some form of flux.

Definition

Elementary algebra/calculus

Fundamentally - the expression for a potential gradient F in one dimension takes the form:

$F = \frac{\delta \phi}{\delta x}\,\!$

where ϕ is some type of potential, and x is displacement (not distance), in the x direction. In the limit of infinitesimal displacements, the ratio of differences becomes a ratio of differentials:

$F = \frac{{\rm d} \phi}{{\rm d} x}\,\!$

In three dimensions, the resultant potential gradient is the sum of the potential gradients in each direction, in Cartesian coordinates:

$\bold{F} = \bold{e}_x\frac{\partial \phi}{\partial x} + \bold{e}_y\frac{\partial \phi}{\partial y} + \bold{e}_z\frac{\partial \phi}{\partial z}\,\!$

where e_x, e_y, e_z are unit vectors in the x, y, z directions, which can be compactly and neatly written in terms of the gradient operator ∇,

$\bold{F} = \nabla \phi\,\!$

Vector calculus

The mathematical nature of a potential gradient arises from vector calculus, which directly has application to physical flows, fluxes and gradients over space. For any conservative vector field F, there exists a scalar field ϕ, such that the gradient ∇ of the scalar field is equal to the vector field;

$\bold{F} = \nabla\phi\,\!$

using Stoke's theorem, this is equivalently stated as

$\nabla\times\bold{F} = \boldsymbol{0} \,\!$

meaning the curl ∇× of the vector field vanishes.

In physical problems, the scalar field is the potential, and the vector field is a force field, or flux/current density describing the flow some property.

Physics

In the case of the gravitational field g, which can be shown to be conservative, it is equal to the gradient in gravitational potential Φ:

$\mathbf{g} = - \nabla \Phi \,\!$

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

In electrostatics, the electrostatic (not dynamic) field has identical properties to the gravitational field; it is the gradient of the electric potential

$- \mathbf{E} = \nabla V \,\!$

The electrodynamic field has a non-zero curl, (see Faraday's law of induction), which implies the electric field cannot be only equal to the gradient in electric potential, a time-dependent term must be added;

$- \mathbf{E} = \nabla V + \frac{\partial \bold{A}}{\partial t}\,\!$

where A is the electromagnetic vector potential.

Chemistry

Main article: Electrode potentials

In an Electrochemical half-cell, at the interface between the electrolyte (an ionic solution) and the metal electrode, the standard electric potential difference is;

$\Delta \phi_{(M,M^{+z})} = \Delta \phi_{(M,M^{+z})}^{\ominus} + \frac{RT}{zeN_A}\ln a_{M^{+z}} \,\!$

where R = gas constant, T = temperature of solution, z = valency of the metal, e = elementary charge, N_A = Avagadro's constant, and a_M+z is the activity of the ions in solution. Quantities with superscript o 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 term interface.

Biology

In biology, potential gradients is the net difference in electric charge across a cell membrane.

External links

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