Undergrad Interaction Term in EM Lagrangian - Explained

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The discussion centers on the role of the interaction term -Aμjμ in the Lagrangian for electrodynamics, which includes the field energy density -FμνFμν/4. It highlights that while field energy can explain forces between charged particles, the interaction term is essential for generating a non-zero electromagnetic field from charges. Without this term, the equations of motion derived from the Euler-Lagrange equation would not accurately describe the dynamics of charged particles. The conversation also notes that calculations often consider the static limit, which can lead to misunderstandings about the necessity of the interaction term. Overall, the interaction term is crucial for a complete understanding of electromagnetic interactions.
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The (classical, relativistic) Lagrangian for electrodynamics contains the field energy density -FμνFμν/4 and the interaction term -Aμjμ. I understand the maths of that - for one thing, the equations of motion turn out right if you plug this into the Euler Lagrange equantion.
Now I recall having learned that you can explain the forces between charged particles solely with the field energy: pushing 2 electrons together increases field energy because it goes with the square of the field strength, and pushing an electron and a positron together decreases field energy. If this is true, why do we need the interaction term at all? What am I missing?
 
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If you do not have an interaction term, the charges do not generate a non-zero electromagnetic field.

Furthermore, it is worth noting that you are considering the static limit when you are doing that type of calculation.
 
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Perfect answer, thanks! Makes me feel a bit stupid, though.
 

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