Here's an interesting paradox.
The problem is this. Lets say that you have a charged capacitor. There are no wires attached to it, and the charge on the capacitor is not changing with time.
Around this capacitor, there is this superconducting coil that carries a current.Look at the figure below.
in | | out
The two vertical lines denote the capacitor
, in stands for the wire going in, and 'out' stands for the wire coming out. Lets assume that the other two sides of the loop, which complete the circuit
are very far away.
Now its easy to see that in between the plates of the capacitor there is a uniform electric field, and in the same region of space there is a magnetic field(due to the current in the wire), which is perpendicular to this electric field.
So these crossed Electric and Magnetic fields have a linear momentum, proportional to E X B. Thus the whole system has momentum.
On the other hand, if a system has linear momentum, its center of mass must move. The center of mass here is at rest, and so the system cannot have momentum!