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

[sciboudy]

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

 

[ZapperZ]

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 [itex]E = -d\Phi /dx[/itex] in 1-dimension. Since F=qE, then [itex]F=qE= -q d\Phi /dx[/itex], where [itex]\Phi[/itex] is the electrostatic potential. But [itex]q \Phi [/itex] is V, the potential difference. Thus, F=- dV/dx.

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

Zz.

 

[truesearch]

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

 

[sciboudy]

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

 

[PJC01]

how we can prove this relation

The relationship between force and potential is a fundamental concept in physics and is described by the equation F = -dv/dx. This means that the force acting on an object is equal to the negative of the derivative of its potential energy with respect to its position. In simpler terms, this means that the force acting on an object is equal to the rate of change of its potential energy with respect to its position.

To prove this relationship, we can use the concept of work. Work is defined as the force applied to an object multiplied by the distance it moves in the direction of the force. In other words, work is equal to the integral of force with respect to distance. Mathematically, this can be represented as W = ∫Fdx.

Now, let's consider a system where a force F is applied to an object and it moves a distance dx. This results in a change in potential energy, which we can represent as dU. Therefore, the work done by the force can be written as dW = Fdx = dU.

Since work is also equal to the change in potential energy, we can equate dW and dU, which gives us dW = dU = Fdx. Using the chain rule, we can rewrite this equation as dU = -dv/dx * dx.

Now, if we compare this to the definition of work, we can see that dU = -dv/dx * dx is equivalent to dU = Fdx. Therefore, we can conclude that F = -dv/dx, which proves the relationship between force and potential.

In summary, we can prove the relationship F = -dv/dx by using the concept of work and equating it to the change in potential energy. This relationship is crucial in understanding the behavior of objects in various physical systems and is a fundamental concept in physics.