Energy of Magnetic field by current in a long bar

[zhanhai]

Obviously, the energy density is proportional to the square of B, which is proportional to 1/r (radius).

Thus, the EM energy as the space integral of the energy density is infinite.

This would suggest that a current through a very long bar cannot be realized. May I understand such a result as that current flowing through a very long bar can not be truly realized without a returning line for forming a loop?

 

[mfb]

There is a huge difference between "very long" and "infinitely long" as the divergence is very slow (logarithmic in distance). For "very long", you still have a return path somewhere which will remove the divergence.

A current over a finite distance without a return path would accumulate charge somewhere which stops the current flow at some point.