Moving charge deflected by a loop of wire

[TimeBender2004]

Homework Statement

A charge q with mass m has a velocity v far away. A small metallic loop of area A and
resistance R is fixed and oriented such, that it and the vector ~v are in the same plane. The
impact parameter of the charge with respect to the loop is h. Assuming that h is much
larger than the size of the loop, there is no gravity and no air resistance, find the velocity
(magnitude and direction) of the charge after a very long time.

Relevant Equations

Magnetic field produced by a moving charge, Faraday's law, Magnetic field by a magnetic dipole, Lorentz Force

I know that the magnitude of the velocity can't change because the only force acting on the charge will be the Lorentz force which does not do any mechanical work.
The tricky thing is finding the final angle of the particle. If we try to just imagine how the particle would move, then as it approaches the loop of wire, the magnetic field from the loop generated by the changing flux will cause the charge to deflect away until the flux begins to decrease which will then cause the charge to begin deflecting in the opposite direction. However, depending on the orientation of the trajectory at that instance, the only two reasonable solutions would be that the charge ends up deflecting with some angle, or the charge ends up with the same direction.
I tried deriving the equations of motion but ended up with just a mess as shown in my work. If I use the Lagrangian or Hamiltonian, then I need to find some potential function from the Lorentz force which depends on the velocity of the charge.

 

[kuruman]

I suggest that, before you start writing Lagrangians and Hamiltonians, you back off and look at what's happening here from a distance, literally. Here are things to consider.

• The charge is very far away from the loop and most likely remains far away. What does this suggest about the interaction between the charge and the loop? Can you devise a simple model based on the "far away" approximation?
• Can you sketch the particle trajectory to guide your thinking? Note that the induced dipole moment is perpendicular to the plane of the motion which makes the Lorentz force in the plane. The impact parameter $h$ is by definition the distance of closest approach.
• Can you rewrite the expression for the magnetic field at the charge due to the induced current$I_{ind}$ in the loop at the position of the charge? The one you have does not take into account that the loop is in the plane of the motion.
• How sure are you that only the direction of the particle's velocity changes but not its speed? We are told that the loop has resistance $R$ which means that there are $~I_{ind}^2R~$ losses dissipated as heat. Where does the heat energy come from if not from the kinetic energy of the charge?
• What, if anything, is conserved?

Disclaimer: I have not solved the problem. The above is a list of what I would consider in order to put the problem into perspective.