Does constant DC current in curved wire produce EM wave?

[renormalize]

I'm not sure whether Poynting vector field and Poynting theorem actually makes sense only in case $E$ and $B$ are solutions of the EM wave equation.

As pointed out by Feynman, even static EM fields give rise to Poynting vector fields that represent flows of EM energy that circulate continuously (but without radiating). See the Wikipedia entry Static Fields and its reference to the Feynman lectures.

 

[cianfa72]

See the Wikipedia entry Static Fields and its reference to the Feynman lectures.

From Wikipedia

While the circulating energy flow may seem unphysical, its existence is necessary to maintain conservation of angular momentum. The momentum of an electromagnetic wave in free space is equal to its power divided by c, the speed of light. Therefore, the circular flow of electromagnetic energy implies an angular momentum

Here it talks of momentum of electromagnetic wave in free space (bold is mine). Does it mean that $E$ and $B$ are actually solution of EM wave equation ?

 

[renormalize]

From Wikipedia

Here it talks of momentum of electromagnetic wave in free space (bold is mine). Does it mean that $E$ and $B$ are actually solution of EM wave equation ?

Yes, but since the fields are static, the EM "wave" satisfies a "wave-equation" with vanishing time derivatives, i.e., Laplace's equation: $\nabla^{2}\vec{E}=\nabla^{2}\vec{B}=0$.

 

[cianfa72]

Yes, but since the fields are static, the EM "wave" satisfies a "wave-equation" with vanishing time derivatives, i.e., Laplace's equation: $\nabla^{2}\vec{E}=\nabla^{2}\vec{B}=0$.

There is therefore a flow of EM energy, however it is "confined" in the space surrounding the DC circuit: there is not a flow of EM energy that radiates/propagates out from there.