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 in equations of physical processes because it leads to some form of flux. In electrical engineering it refers specifically to electric potential gradient, which is equal to the electric field.

The simplest definition for a potential gradient F in one dimension is the following:

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:

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

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

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

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