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Terms like kinetic energy, and even gravitational potential energy (from a Newtonian standpoint) are a little bit easier to understand because they are exist in everyday life, but when you get E&M (I've only studied classical E&M) you have this idea of a field, the field has energy and you need to calculate it? How does a field have energy?

Consider a conducting bar, moving on rails with a large resistor at one end bathed in a strong, uniform magnetic field. As the bar moves, the magnetic flux increases, inducing a current in the circuit and thus energy is lost in the resistor as heat, so the bar must be slowing down (it never actually stops even though it moves a finite distance, as i've calculated). I can ascertain what happens in this situation because I know about E&M, but it bothers me that I don't feel like I understand what is actually happening. If you showed me that and I didn't know physics, I would say the bar moves an infinite distance. How can some invisible field (with energy) stop a real, moving object? This is the whole idea of transfer of this energy which I don't know what it is?

Is there a good model of a physical interpretation of this phenomenon? Is it just a mathematical construct?