# Relating the conservation of Momentum & Energy to a wire accelerated by the Lorentz force?

Consider a simple and classical set-up of a round loop with current($I_1$)that is parallel to a wire that has current flowing as well ($I_2$).

In this "ideal" system, the wire & loop are fixed, such that the Lorentz force is cancelled. We can considered this as a steady state, where the magnetic field $B$ is at it's maximum. And current flowing in both (separate) conductors are at their maximum as well (to ignore self-inductance). If we allow the wire to accelerate(i.e set free...) as such:

How does this change or relate to:

1) Newton's 3rd law(Even in the steady state, I believe there is a Lorentz force between the two conductors).

2) Due to Newton's 3rd law, the conservation of momentum within $B$...?

3) The conservation of energy($E$) of the system?

The Lorentz force is acting on the two conductors, yet the momentum($p$) is conserved within $B$ OR caused by it? I'm using this example to understand how Momentum & Energy are conserved and contribute to the explanation of the dynamics of this system, could someone please explain how?

Use the right hand rule and you will find that $I_2$ produces a magnetic field out of the page on its left and into the page on its right. Without any integration or math, if you use the left hand rule, you will see that the left semicircle of the $I_1$ wire experiences a radially outwards force at all points. By symmetry, the net force on it is to the left. Similarly, the right semicircle of the $I_1$ wire experiences a radially inwards force at all points. Again, the net force is to the left. Add these two left forces and they will equal the rightwards force on the $I_2$ wire. Voila! Newton's third law.

If you release the system, the loop will accelerate to the left and the wire to the right (the magnitudes of forces on both are the same) The net momentum initially was $0$.

We know that $\vec{F} = \frac{d\vec{p}}{dt}$

$\vec{F_1} = -\vec{F_2}$

$\Rightarrow \frac{d\vec{p_1}}{dt}= -\frac{d\vec{p_2}}{dt}$

$\Rightarrow \vec{p_1}=-\vec{p_2}$

$\Rightarrow \vec{p_1}+\vec{p_2}=0$

Momentum is conserved.

The energy of the system is also conserved as at any point the sum of the kinetic energy and the magnetic potential energy will be constant.