Interaction Term in EM Lagrangian - Explained

In summary, the Lagrangian for electrodynamics includes a field energy density term and an interaction term. The equations of motion are satisfied when this is plugged into the Euler Lagrange equation. It is possible to explain the forces between charged particles solely with the field energy, but without an interaction term, the charges would not generate a non-zero electromagnetic field. This is especially relevant in the static limit.
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Anypodetos
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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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  • #2
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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Likes Anypodetos
  • #3
Perfect answer, thanks! Makes me feel a bit stupid, though.
 

1. What is an interaction term in EM Lagrangian?

An interaction term in EM Lagrangian is a mathematical term that represents the interaction between two or more fields in the electromagnetic (EM) theory. It is used to describe the behavior and dynamics of particles in an electromagnetic field.

2. How is an interaction term calculated in EM Lagrangian?

The interaction term in EM Lagrangian is calculated by taking the product of the two fields that are interacting with each other. This product is then integrated over the entire space to calculate the total interaction energy.

3. What is the significance of an interaction term in EM Lagrangian?

The interaction term in EM Lagrangian is significant because it allows us to understand and predict the behavior of particles in an electromagnetic field. It helps us to describe the forces and interactions between particles and fields, and is a crucial component in understanding the dynamics of electromagnetic systems.

4. Can an interaction term be negative in EM Lagrangian?

Yes, an interaction term in EM Lagrangian can be negative. This indicates that the two fields are interacting in an attractive manner, while a positive interaction term indicates a repulsive interaction. The sign of the interaction term depends on the type and strength of the interaction between the fields.

5. How does an interaction term affect the equations of motion in EM Lagrangian?

The presence of an interaction term in EM Lagrangian affects the equations of motion by introducing additional terms that account for the interactions between fields. These additional terms can change the behavior of particles in an electromagnetic field, resulting in more complex and non-linear equations of motion.

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