Electromagnetic linear momentum for a system of two moving charges

• angrystudent
In summary, radiation causes the sum of linear momentum plus electromagnetic momentum to not be conserved.

angrystudent

Homework Statement
I want to obtain the "total" EM linear momentum for a system of two moving charges
Relevant Equations
Lienard-Wiechert potentials, Lorentz force law
When you write out the equations of motion for a system of two isolated charges, you can add both of the equations and get the increase in the particles linear momentum on one side. On the other side, you get the sum of all the forces between the particles. I understand that this sum of forces could be written as the negative time derivative of the system's total electromagnetic momentum. But since the forces are quite complicated, I just can't seem to deduce the expression of this EM momentum

I guess that you could write the ##\mathbf{E}## and ##\mathbf{B}## fields due to both charges and integrate the momentum density ##\mathbf{D}\times\mathbf{B}## over all space to get the desired momentum, but I'm not too skilled with the integration of retarded functions. I have also tried to write the forces without deriving the potentials explicitly,
since
$$\mathbf{F}_{12} = -q_2\nabla_2V_{12} -q_2\partial_t(V_{12} \mathbf{v}_1/c^2) + q_2\mathbf{v}_2 \times \nabla_2 \times(V_{12} \mathbf{v}_1/c^2)$$
and
$$\mathbf{F}_{21} = -q_1\nabla_1V_{21} -q_1\partial_t(V_{21} \mathbf{v}_2/c^2) + q_1\mathbf{v}_1 \times \nabla_1\times(V_{21} \mathbf{v}_2/c^2)$$
but that doesn't seem to give an inspiring result

tl;dr I'm trying to write the sum of ##\mathbf{F}_{12}+\mathbf{F}_{21}## as a total time derivative. ##V_{12}## and ##V_{21}## refer to the Lienard-Wiechert potential

Last edited:
I just realized that electromagnetic radiation contradicts my statement. As soon as the charges get accelerated, then the sum of linear momentum plus electromagnetic momentum is not conserved anymore because the momentum flux is non-zero at infinity. My initial approach doesn't make sense due to radiation phenomena This means that the Griffiths book chapter on conservation laws gets contradicted by the chapter on radiation.

1. What is electromagnetic linear momentum?

Electromagnetic linear momentum is a property of a system that describes the amount of motion in a straight line due to the interaction of electric and magnetic fields.

2. How is electromagnetic linear momentum calculated for a system of two moving charges?

The electromagnetic linear momentum for a system of two moving charges is calculated by taking the cross product of the electric and magnetic fields at a given point in space and integrating over the entire system.

3. What is the unit of measurement for electromagnetic linear momentum?

The unit of measurement for electromagnetic linear momentum is kilogram-meter per second (kg-m/s).

4. How does the relative motion of the two charges affect the electromagnetic linear momentum of the system?

The relative motion of the two charges affects the direction and magnitude of the electromagnetic linear momentum. If the charges are moving in the same direction, the momentum will be greater than if they are moving in opposite directions.

5. Can electromagnetic linear momentum be transferred between two systems of moving charges?

Yes, electromagnetic linear momentum can be transferred between two systems of moving charges through the exchange of electromagnetic fields. This transfer can result in changes in the motion of the charges and can be observed as a force acting on the charges.