How is it possible for a magnetic field to store energy and momentum? And can deliver it to another object(charge/dipole) or field? I can't depict it. I know it occurs but how?
For example, the Lorentz force acting on a current carrying wire placed in an exterior magnetic field, how can we relate energy stored in the magnetic field(s) and momentum to those systems?
Answer
Here is how I think about this intuitively (I assume you know the mathematical formulation, but that doesn't necessarily help with the intuition; so forgive the deliberately imprecise language I am about to use).
When you have a very low magnetic field, you draw field lines far apart. As the field increases, the field lines come closer together. Now if you think of field lines as repelling each other, this is akin to compressing a gas by adding more molecules to a volume: as you add more molecules, the pressure goes up, and the energy stored in the gas goes up. Every time you add another molecule you have to work against a greater pressure, so energy stored goes as pressure squared. The same thing for magnetic fields: if you consider the field lines as repelling, then "adding another field line" to a bunch that's already there (for instance, inside your current loop) requires you to do work - the larger the field, the more work it is to add another line.
And if you allow the magnetic field to "escape" (akin to the gas expanding), the change $\frac{dB}{dt}$ will result in the induction of currents and voltages, and work being done - that's how the energy can be extracted again.
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