Exploring Kinetic Energy and Force in a 1D System

[MaxManus]

Homework Statement

We addrewss a one dimensional system. An atom moves along the x-axis with a kinetic energy K = 0.5*m*v**2. In the range -x0<x<x0 the atom enters the trap, and is affected by a magnetic fiels. The interaction with the magnetic field gives rise to a potential U(x) which we model:

for abs(x)>= x0, U(x) = U0
for abs (x) < x0, U(x) = U0*abs(x)/x0

a) Find the force F(x) on the atom from the magnetic field. Is this force conservative?

Let us also assume that the atom is charged and also subject to a constant electronic force, F0 acting in the positive x-direction

b) If the atom has kinetic energy K = 0 at x = 0, how laarge must F0 be in order for the atom to escape? And if the kinetic energy is K = U0/2 at x = 0, how large must F0 then be in order for the atom to escape.

Homework Equations

The Attempt at a Solution

a)
F = -dU/dx

F = 0 for abs(x) >= x0
F = U0/x0 for -x0<x<0
F = -U0/x0 0for 0<x<x0

The force is conservative for it is only dependent on the position

b) Here I need some help.
Is the idea that the force must give the atom larger total energy than u0?

 

[Redbelly98]

b) Here I need some help.
Is the idea that the force must give the atom larger total energy than u0?

Not sure if that would be the case or not.

The way I think about it is, how is the potential modified due to the presence of the constant electrostatic force?

 

[nlightner]

Yes, that is correct. In order for the atom to escape, the total energy must be greater than the potential energy at that point. So, in order for the atom to escape with a kinetic energy of 0, the electronic force F0 must equal or exceed U0. And for a kinetic energy of U0/2, the electronic force must be greater than U0/2.