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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).
To derive Darken's second equation the gradient in Gibb's chemical potential is analyzed. The gradient in potential energy, denoted by F 2 , is the force which causes atoms to diffuse. [ 1 ] To begin, the flux J is equated to the product of the differential of the gradient and the mobility B , which is defined as the diffusing atom's velocity ...
Siméon Denis Poisson. Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics.For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate the corresponding electrostatic or gravitational (force) field.
Just as with the internal energy version of the fundamental equation, the chain rule can be used on the above equations to find k+2 equations of state with respect to the particular potential. If Φ is a thermodynamic potential, then the fundamental equation may be expressed as:
In other words, the strain energy density function can be expressed uniquely in terms of the principal stretches or in terms of the invariants of the left Cauchy–Green deformation tensor or right Cauchy–Green deformation tensor and we have: For isotropic materials,
where: is the rate of change of the energy density in the volume. ∇•S is the energy flow out of the volume, given by the divergence of the Poynting vector S. J•E is the rate at which the fields do work on charges in the volume (J is the current density corresponding to the motion of charge, E is the electric field, and • is the dot product).
The potential relative to a distant point on the Earth is highest at the point where current enters the ground, and declines with distance from the source. Ground potential rise is a concern in the design of electrical substations because the high potential may be a hazard to people or equipment.
Therefore, the potential energy drop is equal to the work done to the bed and banks, which is the stream power. We know that change in potential energy over change in time is given by the equation: Δ P E Δ t = m g Δ z Δ t {\displaystyle {\frac {\Delta PE}{\Delta t}}=mg{\frac {\Delta z}{\Delta t}}}