Yukawa potential energy function

[astenroo]

Homework Statement

This is not yet an attempt at solving a problem. I just need confirmation on that I'm on the right track. So, I am supposed to derive an expression for the radial force function from the given Yukawa-function.

Homework Equations

U(r) = -(r/r0)U0 exp-(-r/r0)

The Attempt at a Solution

I'm thinking about two possibilities here. First: Derivation of said function with respect to r (normal differential). Second: Partial differentiation to solve for the gradient (although I think this is not necessary since r is a fixed value, and no coordinates are given in the context of the problem)

 

[RoyalCat]

$$\vec F = -\nabla U$$

http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates

In this case the normal derivative is the same as the partial derivative.

 

[astenroo]

$$\vec F = -\nabla U$$

http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates

In this case the normal derivative is the same as the partial derivative.

Ah true, since this function only has one variable which is r. What I am a bit concerned about is that if the system for spherical coordinates should be used in this derivation... Or they aren't needed since the radial force is dependant only on r (in this given situation)?

 

[RoyalCat]

Ah true, since this function only has one variable which is r. What I am a bit concerned about is that if the system for spherical coordinates should be used in this derivation... Or they aren't needed since the radial force is dependant only on r (in this given situation)?

It doesn't matter, you have only one axis specified in the problem, and that's the radial axis. The way to take the gradient with respect to the radial axis is $$\frac{\partial f}{\partial r}\hat r$$
It doesn't matter what the other two axes are.

 

[rwisz]

I can confirm that you are on the right track. To derive the expression for the radial force function, you can indeed use differentiation with respect to r. This will give you the derivative of the potential energy function, which can then be equated to the negative of the radial force function according to the relationship F(r) = -dU(r)/dr. Partial differentiation is not necessary in this case since the function only depends on one variable, r. Keep up the good work!