Impulse of electromagnetic field

[Petar Mali]

We have

$$\vec{F}=\int_V\vec{f}dV=-\frac{d}{dt}\int_V(\vec{D}\times \vec{B})dV$$

$$\vec{g}=\vec{D}\times \vec{B}$$

$$\vec{F}=-\frac{d}{dt}\int_V\vec{g}dV$$

$$\vec{F}=\frac{d\vec{p}_{mech}}{dt}$$

$$\frac{d}{dt}(\vec{p}_{mech}+\int_V\vec{g}dV)=0$$

$$\vec{p}_{mech}+\int_V\vec{g}dV=\vec{const}$$

In total field law of conservation of impulse

$$\vec{p}_{mech}$$ - mechanical impulse of particles in field

In one book I found that $$\int_V\vec{g}dV=\vec{const}$$ is necessary but not always sufficient condition for law of action and reaction in electrodynamics.

My question is when is $$\int_V\vec{g}dV=\vec{const}$$ necessary and sufficient condition for this law? Thanks for your answer!

 

[Petar Mali]

Does anyone know this?

 

[Meir Achuz]

$$\vec{p}_{mech}+\int_V\vec{g}dV=\vec{const}$$

In total field law of conservation of impulse

$$\vec{p}_{mech}$$ - mechanical impulse of particles in field

In one book I found that $$\int_V\vec{g}dV=\vec{const}$$ is necessary but not always sufficient condition for law of action and reaction in electrodynamics.

My question is when is $$\int_V\vec{g}dV=\vec{const}$$ necessary and sufficient condition for this law? Thanks for your answer!

If you believe
$$\vec{p}_{mech}+\int_V\vec{g}dV=\vec{const}$$,
then isn't the answer obviously, yes?
I assume you mean NIII for the forces on objects.

 

[Petar Mali]

NIII?

No! It's not obviously, I think. What is your idea?

 

[sardar]

I would like to first clarify that the equations and concepts mentioned in the content are related to the dynamics of electromagnetic fields, specifically the relationship between force, impulse, and the law of conservation of impulse. This is an important topic in the field of electromagnetism and has been extensively studied and verified through experiments.

To answer your question, the condition \int_V\vec{g}dV=\vec{const} is necessary and sufficient for the law of action and reaction in electrodynamics when the electromagnetic fields are in a steady state. In other words, when the fields are not changing with time and there are no external forces acting on the system, the law of conservation of impulse holds true and the total impulse of the system remains constant.

However, when the fields are not in a steady state and are changing with time, the condition \int_V\vec{g}dV=\vec{const} may not be sufficient for the law of action and reaction to hold. This is because in such cases, the total impulse of the system may not remain constant due to the presence of external forces or the emission of electromagnetic waves. In these scenarios, additional terms may need to be considered in the equation to fully account for the dynamics of the system.

In conclusion, the condition \int_V\vec{g}dV=\vec{const} is necessary and sufficient for the law of action and reaction in electrodynamics only in the case of steady state fields. In other cases, further considerations and calculations may be needed to accurately describe the behavior of the system.