Understanding the Relationship Between Force and Potential: Proving F = -dv/dx

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SUMMARY

The relationship between force and potential is established through the equation F = -dV/dx, where F represents force, V denotes electric potential, and x is distance. This relationship arises from the definition of the electric field E, which is expressed as E = -dΦ/dx, leading to the conclusion that F = qE = -q dΦ/dx. In three-dimensional space, the gradient operator replaces the one-dimensional derivative, maintaining the validity of the equation. Additionally, the concept of potential in quantum mechanics raises questions about the implications of a particle having zero potential (V=0).

PREREQUISITES
  • Understanding of electric fields and their mathematical representation
  • Familiarity with the concepts of force and potential energy
  • Basic knowledge of calculus, particularly derivatives
  • Awareness of quantum mechanics principles related to potential
NEXT STEPS
  • Study the derivation of the electric field from electrostatic potential
  • Explore the implications of force and potential in classical mechanics
  • Learn about the gradient operator in three-dimensional calculus
  • Investigate the role of potential in quantum mechanics and its physical interpretations
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Students and professionals in physics, particularly those focused on electromagnetism and quantum mechanics, as well as educators seeking to clarify the relationship between force and potential energy.

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how can we prove this relation F= -dv/dx
could some one explain what we mean by the force equal to the change it potential per distance and
 
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sciboudy said:
how can we prove this relation F= -dv/dx
could some one explain what we mean by the force equal to the change it potential per distance and

Your question is ambiguously presented, because it appears as if "v" is velocity, rather than "V" as in electrical potential difference.

You should know that the electric field E is E = -d\Phi /dx in 1-dimension. Since F=qE, then F=qE= -q d\Phi /dx, where \Phi is the electrostatic potential. But q \Phi is V, the potential difference. Thus, F=- dV/dx.

In 3D, the derivative in 1D is replaced by the grad operator.

Zz.
 
If you accept that energy (work done) is basically Force x distance then:
Force x distance = change in energy
F x dx = dE so F = dE/dx
 
OK thank you sir ZApperz and thank you truesearch
now i have another question based in the meaning of potential in the quantum mechanics
what is the potential for example whe i say a particle have V=0 ? is that mean the particle will never stop any time ?? or what
 

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