Potential associated with a conservative force field F

AI Thread Summary
The discussion centers on the relationship between conservative force fields and potential energy. It establishes that the force can be derived from potential energy using the equation F = -∇U(r), and clarifies that while force, field, and potential are distinct concepts, they are interrelated. The electric field E is linked to the potential V through E = -∇V, and when a charge q is introduced, the force is given by F = qE. The potential is defined up to an arbitrary constant, often set to zero at infinity for convenience. Overall, the mathematical framework supports the correlation between these physical quantities while emphasizing their unique characteristics.
AntonioJ
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Homework Statement
Given the potential energy, the force is obtained as F = -∇U(r). A conservative force field F is associated with a potential f by F = ∇f.
Relevant Equations
Does the first expression arise from this last one? If so, with -∇U(r), would one obtain the electric field E instead of the force F?
Given the potential energy, the force is obtained as F = -∇U(r). A conservative force field F is associated with a potential f by F = ∇f. Does the first expression arise from this last one? If so, with -∇U(r), would one obtain the electric field E instead of the force F?
 
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F=-\nabla U
is enough. I feel no necessity to introduce f of f=-U + const.
 
Force, field and potential are 3 different things. But can be correlated each other. Field like E is a space deformation (can be due to an extra charge for example) then some field like the electrostatic can be associated to potential V, E= -nabla V is correct. Then when comes another charge q in the field Coulomb law acts and F=qE. So you have U(r)= qV(r).
V is generally determined with a constant. For electrical field V=0 when r is infinite.
Mathematically this constant disappears in calculation (derivation or integration)
 
A force is a vector field, ##\vec{F}(\vec{x})##. If it's conservative, there exists a scalar potential, ##U##, then by definition
$$\vec{F}(\vec{x})=-\vec{\nabla} U(\vec{x}).$$
If ##\vec{\nabla} \times \vec{F}=0## in an open singly-connected neighborhood of a point, then there exists a potential (at least) in this neighborhood (Poincare's Lemma).

The potential is determined only up to an arbitrary additive constant. Indeed it's convenient to define it to go to 0 at infinity (if possible for the given force).
 
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