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The simplest definition for a potential gradient F in one dimension is the following: [1] = = 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 x 1, x 2, and potentials at those points, ϕ 1 = ϕ(x 1), ϕ 2 = ϕ(x 2).
The vector potential admitted by a solenoidal field is not unique. If is a vector potential for , then so is +, where is any continuously differentiable scalar function. . This follows from the fact that the curl of the gradient is ze
In neurobiology, the length constant (λ) is a mathematical constant used to quantify the distance that a graded electric potential will travel along a neurite via passive electrical conduction. The greater the value of the length constant, the further the potential will travel.
M. C. Escher's lithograph print Ascending and Descending illustrates a non-conservative vector field, impossibly made to appear to be the gradient of the varying height above ground (gravitational potential) as one moves along the staircase. The force field experienced by the one moving on the staircase is non-conservative in that one can ...
where D is the diffusion coefficient for the electron in the considered medium, n is the number of electrons per unit volume (i.e. number density), q is the magnitude of charge of an electron, μ is electron mobility in the medium, and E = −dΦ/dx (Φ potential difference) is the electric field as the potential gradient of the electric potential.
Scalar potential is not determined by the vector field alone: indeed, the gradient of a function is unaffected if a constant is added to it. If V is defined in terms of the line integral, the ambiguity of V reflects the freedom in the choice of the reference point r 0 .
Position vectors r and r′ used in the calculation. The starting point is Maxwell's equations in the potential formulation using the Lorenz gauge: =, = where φ(r, t) is the electric potential and A(r, t) is the magnetic vector potential, for an arbitrary source of charge density ρ(r, t) and current density J(r, t), and is the D'Alembert operator. [2]
The electric potential and the magnetic vector potential together form a four-vector, so that the two kinds of potential are mixed under Lorentz transformations. Practically, the electric potential is a continuous function in all space, because a spatial derivative of a discontinuous electric potential yields an electric field of impossibly ...