場會否消耗能量？

The answer above might be incorrect.

Electromagnetic field does carry energy. The energy density is proportional to E^2 + B^2. Just think of light, classically interpreted as electromagnetic wave and carries non-zero energy. So an oscillating electron (say in a wire with 0 resistance) would produce an electromagnetic wave and loss energy.

If the current along a wire (say again with 0 resistance) is constant in time, the magnetic field is also constant. So there is no energy loss during the process (recall energy density ~ B^2). This non-zero energy also explains inductance. The current "stores" energy in its magnetic field, and it takes energy to drive a current from 0 to I > 0 even when the wire is perfectly conducting.

For gravity, there is also energy stored in a gravitation field, and the energy density is proportional to the Ricci scalar (if you know what I am talking about). And an object could loss energy through gravitational wave known as gravitational radiation.

2009-11-01 11:09:22 補充：
Matter loss energy by producing a field. Oscillating charge losses energy to the EM wave it produces. Orbiting double stars loss energy slowly to the gravitational wave. Tthe energy required to push a current from 0 to I is partly stored in the magnetic field while other are loss as heat.

2009-11-01 11:09:43 補充：
Oh by the way, if the wire is in vacuum, heat is transfer through radiation, which is also an EM wave.

No. The Law of Conservation of Energy applies to a field.

Unless there is external interaction with a given field, the energy possessed by the field will be conserved.

As an exmaple, the gravitational fields of the earth or the sun are produced because of their masses. As long as mass exists, the field appears.

In your quoted example, the energy lost by the flowing electrons in a wire is because of resistance of the wire (which, microscopically, is lost due to collisions of electrons with atoms of the wire) that transform the electron energy into heat. The magnetic field associated with a moving electron is a natural property.