# Expression of the force derived from this potential

• Andrei0408
In summary, the conversation is about finding the appropriate equations and guidelines for calculating the force applied by a potential function to a test point charge. The provided link explains that the force can be calculated by finding the gradient of the potential function, and in this case, the function does not depend on certain variables which simplifies the calculation.
Andrei0408
Homework Statement
Knowing that the potential is U = - α / r , where α is a positive constant and r the distance from the
potential field source, find the expression of the force deriving from this potential; Give examples of forces that derive from such a potential.
Relevant Equations
Not sure
I just don't know what equations I should use, or what exactly I need to do. I just need some guidelines, thank you!

It is a well known fact that given the potential function ##U(r,\theta,\phi)## of a field , the force that the field applies to a test point charge (or test point mass or whatever is the field subject)## q## is given by $$\vec{F}=-q\nabla U$$.

So all you have to do is calculate the gradient ##\nabla U## of the function ##U##. I assume from the context that the calculation must be done in spherical coordinates. So it will be $$\nabla U=\frac{\partial U}{\partial r}\hat r+\frac{1}{r}\frac{\partial U}{\partial \theta}\hat\theta+\frac{1}{r\sin\theta}\frac{\partial U}{\partial \phi}\hat\phi$$

Also note that in your case the function U does not depend on ##\theta## and ##\phi## so the above formula for the gradient of U simplifies a lot.

Andrei0408 and etotheipi

## 1. What is the expression of the force derived from this potential?

The expression of the force derived from a potential is given by the negative gradient of the potential function. In mathematical terms, it can be written as F = -∇V, where F is the force vector and V is the potential function.

## 2. How is the force derived from a potential related to the motion of a particle?

The force derived from a potential is responsible for the motion of a particle. It determines the direction and magnitude of the force acting on the particle, which in turn affects its acceleration and ultimately its motion.

## 3. Can the force derived from a potential be positive or negative?

Yes, the force derived from a potential can be either positive or negative. A positive force indicates that the particle is being pushed in the direction of the force, while a negative force indicates that the particle is being pulled in the opposite direction.

## 4. What is the relationship between the force derived from a potential and the potential energy of a system?

The force derived from a potential is directly related to the potential energy of a system. As the potential energy increases, the magnitude of the force also increases. This means that a higher potential energy corresponds to a stronger force acting on the particle.

## 5. How does the force derived from a potential affect the stability of a system?

The force derived from a potential plays a crucial role in determining the stability of a system. A system is considered stable if the force derived from the potential is a restoring force, meaning it brings the system back to its equilibrium position. On the other hand, if the force derived from the potential is a destabilizing force, it can cause the system to become unstable and undergo chaotic motion.

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