What is the relationship between force and potential in particle interactions?

In summary, the conversation discusses the confusion about finding force vectors F_12 and F_21 from an interaction potential between two repelling particles. It is clarified that the x-component of F_12 is given by -du/dx and the components for F_21 are the negative of F_12. The interaction potential is a scalar function and the force on particle 1 is found using the gradient of the potential, while the force on particle 2 is the negative of that. Newton's 3rd Law holds due to the potential only depending on the relative vector between the particles.
  • #1
Tim667
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TL;DR Summary
Slight confusion about vectors from a potential
Suppose I have some interaction potential, u(r), between two repelling particles. We will name them particles 1 and 2.

I want to find the force vectors F_12 and F_21. Would I be correct in saying that the x-component of F_12 would be given by -du/dx, y-component -du/dy etc? And to find the components for the other force vector, would this simply be the negative of the first vector?

So F_12 would be given by (-du/dx, -du/dy, -du/dz) and F_21= (du/dx, du/dy, du/dz)?

Thank you
 
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  • #2
Tim667 said:
Summary: Slight confusion about vectors from a potential

Suppose I have some potential, u(r), between two repelling particles. We will name them particles 1 and 2.

I want to find the force vectors F_12 and F_21. Would I be correct in saying that the x-component of F_12 would be given by -du/dx, y-component -du/dy etc? And to find the components for the other force vector, would this simply be the negative of the first vector?

So F_12 would be given by (-du/dx, -du/dy, -du/dz) and F_21= (du/dx, du/dy, du/dz)?

Thank you
You need to be careful. Normally with a potential you have a particle moving under the influence of an external force. In this case technically you have a different potential for each particle.
 
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  • #3
PeroK said:
You need to be careful. Normally with a potential you have a particle moving under the influence of an external force. In this case technically you have a different potential for each particle.
I see, I should specify that this is an interaction potential between the two particles
 
  • #4
Tim667 said:
I see, I should specify that this is an interaction potential between the two particles
Okay. Of course. As long as you define ##r## the right way round, what you have looks right. I must be getting tired.
 
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  • #5
PeroK said:
Okay. Of course. As long as you define ##r## the right way round, what you have looks right. I must be getting tired.
That's okay. I think I was confused because potentials are usually scalar functions, and I wasn't sure you could get vectors from them
 
  • #6
Tim667 said:
That's okay. I think I was confused because potentials are usually scalar functions, and I wasn't sure you could get vectors from them
Yes, the gradient takes a scalar function of position and generates a vector function of position.
 
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  • #7
An interactian potential usually takes the form ##V(\vec{x}_1-\vec{x}_2)##. The force on particle 1 is
$$\vec{F}_{12}=-\vec{\nabla}_1 V(\vec{x}_1-\vec{x}_2)$$
and on particle 2
$$\vec{F}_{21}=-\vec{\nabla}_2 V(\vec{x}_1-\vec{x}_2)=-\vec{F}_{12}.$$
Newton's 3rd Law holds, because the potential only depends on the relative vector ##\vec{r}=\vec{x}_1-\vec{x}_2##.
 
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1. What is the force-potential relationship?

The force-potential relationship, also known as the potential energy curve, is a mathematical representation of the relationship between the force acting on an object and its corresponding potential energy. It describes how the potential energy of an object changes as its position or configuration changes.

2. How is the force-potential relationship calculated?

The force-potential relationship is calculated using the equation F = -dU/dx, where F is the force, U is the potential energy, and x is the position or configuration. This equation is derived from the fundamental principle of energy conservation, which states that the total energy of a system remains constant.

3. What is the significance of the force-potential relationship?

The force-potential relationship is significant because it helps us understand and predict the behavior of physical systems. It allows us to determine the forces acting on an object at any given point and to calculate its potential energy. This relationship is essential in many areas of science, including physics, chemistry, and engineering.

4. How does the force-potential relationship affect motion?

The force-potential relationship affects motion by determining the forces that act on an object and how they change as the object moves. For example, if the force-potential relationship is linear, the object will experience a constant force and undergo uniform motion. If the relationship is quadratic, the object will experience a changing force and undergo accelerated motion.

5. Can the force-potential relationship be applied to all physical systems?

Yes, the force-potential relationship can be applied to all physical systems, as long as the system is conservative. This means that the total energy of the system remains constant, and there is no energy loss due to friction or other non-conservative forces. In non-conservative systems, the force-potential relationship may not hold, and other factors must be considered to describe the behavior of the system.

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