Q:

Alignment of a system in equilibrium and its potential energy

Consider a flexible free conductor which is placed near a strong long bar magnet. How will the the conductor arrange itself if current is passed through it? I asked my professor: he gave me a hint that always a system has tendency to be in a state such that its potential energy is minimum. Why is this so?
PS. I feel the conductor will wrap itself around the wire. But I don't have any justification for it. Please tell me if I am correct. Also if yes, what is the reason?

A:

This addresses the broader question of potential energy minima, not the specific situation of a conductor/magnet/current.
You often see statements that lower potential energy states are preferred in some way. Personally I think there are a few issues with such statements, especially for systems in which only conservative forces act.
Nonetheless, I think the following will help shed some light.
Potential energy is always associated with some type of conservative force. The relationship between the conservative force $F$ and potential energy $U$ is
$$\vec F = -\vec \nabla U(\vec r).$$
In one dimension, this can be simplified to
$$F = -\frac{dU}{dx}.$$
This equation amounts to a definition of potential energy. All of the statements about classical systems preferring lower potential energy have some basis in this definition.
To gain some insight into this equation, imagine a particle that is part of a one-dimensional system with potential energy. There is some potential energy $U$ of the system when the particle is at position $x$. The spatial derivative $dU/dx$ tells you whether potential energy increases or decreases as you move in the positive $x$ direction. Imagine for our system that $dU/dx >0$, meaning as you move in the +x direction, potential energy increases and as you move in the negative x-direction, potential energy decreases. So now if we consider the force $F=-dU/dx$ on the particle, we'll find that when $dU/dx >0$, the force $F<0$, meaning it points in the negative direction, which is toward lower potential energy.
You can convince yourself that if instead $dU/dx < 0$, the force would still point toward lower potential energy.
So statements about systems tending to minimize potential energy can alternatively be stated as forces on objects point toward lower potential energy. The reason this is true is because of the relationship $\vec F = -\vec \nabla U$. This relationship itself is simply a definition of potential energy. If physicists had put a positive sign rather than a negative sign there, we'd be saying that potential energy tends to increase rather than decrease.

