# Conservation of momentum on moving charge

The forces of attraction or repulsion between two stationary charges are the action and reaction forces between them. So when we move one of the charges then the other charge also gets affected. But the force goes in the speed of light. So when one charge is moved apart the force between them decreases. But the other charge can't feel it instantly. So in this time both two charges don't attract other in the same force. So action force isn't equal to reaction force. Thus conservation of momentum is violated. How can this happen?

• Is this actually a question? – DrSheldon Oct 25 '18 at 7:00
• @Dr Sheldon Sorry. Edited now. – Theoretical Oct 25 '18 at 7:08

You are actually completely correct about this. There are many situations with electromagnetism (EM) that violate the conservation of mechanical momentum. Note the use of the word “mechanical” in mechanical momentum.

It turns out that the EM field also contains momentum. In EM the total momentum is conserved including the momentum of the field as well as the mechanical momentum. In CGS units the momentum density of the field is $$\frac{1}{4\pi c}E\times B$$.

So whenever the mechanical momentum is decreased you will find that the field momentum increases, and vice versa. The mechanical momentum is not transferred instantly, as you correctly suspect in your question. The field carries momentum at a maximum speed of c, consistent with the usual universal speed limit.

The charges would either fall on each other or diverge to infinity, if they were a closed system ( classically, quantum mechanically they could be in the energy levels of an atom). It means that they are kept stationary by an external force, and thus the system is not closed. Momentum conservation holds for closed isolated systems. In addition , moving the holder of one of them, introduces acceleration, i.e.dp/dt and so the system is not closed and needs a different analysis too, as accelerated charges radiate.

There is no problem with momentum conservation.