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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.
The simplest definition for a potential gradient F in one dimension is the following:
where ?(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, ?1 = ?(x1), ?2 = ?(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.
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.
which implies E is the gradient of the electric potential V, identical to the classical gravitational field:
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.
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.