Energy creation in a moving solenoid?

The electromagnetic energy $U_{em}$ stored in the magnetic field of an ideal solenoid at rest appears for a moving observer as $U'_{em} $, a sum of magnetic and electric energies satisfying: $U'_{em} > \gamma U_{em}$. We explain this seemingly paradoxical result calculating the stresses in the solenoid structure and showing that the total energy of the solenoid transforms by the expected relativistic factor $\gamma$.