How Is the Minimal Distance Between a Proton and a Nucleus Determined?

[Alexstre]

Homework Statement

A proton is fired at a nucleus containing Z protons and N neutrons, with a kinetic energy K. Show that the minimal distance r_0 = [(Ze^2) / 4pi*epsilon_0] * (1/K)

Homework Equations

E=(q/4pi*epsilon_0) * (r-r' / |r-r'|^3)

The Attempt at a Solution

I know that at the minimal distance, the kinetic energy will have become potential energy and that will be "pushed" by the electromagnetic force of the protons in the nucleus. So whenever the proton is stable (ie. not moving), the forces applied to the proton cancel each other out.

I know (think) that the energy applied by the nucleus is something similar to:
E= (q/4pi*epsilon_0) * (r/r3)
and that E_k = 1/2 mv^2

From there I should find the forces and form an equation where they cancel each other out and solve for r_0. I'm just not sure how to proceed to this step.

Thanks!

 

[diazona]

So whenever the proton is stable (ie. not moving), the forces applied to the proton cancel each other out.

Not quite. Think about Newton's second law. What must be true about the object's motion, according to Newton's second law, if (and only if) the forces applied to the object cancel out?

Also, you're on the right track, thinking of energy, but the formula you gave is for electric field, not energy. So that's not the formula you should be using.

 

[rl.bhat]

Potential energy stored in a system of two charges is

V = k*q1*q2/r.

Equate it to the kinetic energy to find r