Proving DC Current Loop Does Not Radiate w/ Lienard-Wiechert Potential

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SUMMARY

The discussion centers on proving that a direct current (DC) current loop does not radiate electromagnetic waves, utilizing the Lienard-Wiechert potentials as outlined in Jackson's "Classical Electrodynamics." The problem requires transitioning to a continuous medium by considering the limits of charge (q) approaching zero, distance (\Delta) approaching zero, and the number of charges (N) approaching infinity. The expected outcome is to derive the known Biot-Savart law, indicating that contributions from the charges in a closed path cancel out, resulting in no radiation.

PREREQUISITES
  • Understanding of Lienard-Wiechert potentials
  • Familiarity with Biot-Savart law
  • Knowledge of classical electrodynamics principles
  • Ability to manipulate limits in mathematical expressions
NEXT STEPS
  • Study the derivation of Lienard-Wiechert potentials in detail
  • Explore the implications of the Biot-Savart law in electromagnetic theory
  • Investigate the concept of linear charge density and its applications
  • Review classical electrodynamics problems related to radiation and charge motion
USEFUL FOR

This discussion is beneficial for physics students, particularly those studying classical electrodynamics, as well as educators and researchers interested in electromagnetic radiation and charge dynamics.

Rafa_Tapia
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Homework Statement


Hi guys,
on chapter 14 in Jackson's classical electrodynamics, problems 24 asks to prove that a dc current loop does not radiate, starting from Lienard-Wiechert potentials for the individual charges q. The distance between them is \Delta.
We should move to the continuous media by considering q->0, \Delta->0 and N->Infty. Then, the fields should go to the known Biot Savart law. The charge's speed is constant but there exists an acceleration.

Homework Equations



We should use just the Lienard-Wiechert potentials.

The Attempt at a Solution



I've being trying this problem for a long time with no success. I've moved from q/Delta to a linear charge density and try to see the properties of the potential along the total path. I suppose the contributions should cancel out, but I don't get it for an arbitrary closed path.

Has any of you have the solution or at least some hint, it would be of great help!
Thanks a lot!
 
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Rafa_Tapia said:

Homework Statement


Hi guys,
on chapter 14 in Jackson's classical electrodynamics, problems 24 asks to prove that a dc current loop does not radiate, starting from Lienard-Wiechert potentials for the individual charges q. The distance between them is \Delta.
We should move to the continuous media by considering q->0, \Delta->0 and N->Infty. Then, the fields should go to the known Biot Savart law. The charge's speed is constant but there exists an acceleration.

Homework Equations



We should use just the Lienard-Wiechert potentials.

The Attempt at a Solution



I've being trying this problem for a long time with no success. I've moved from q/Delta to a linear charge density and try to see the properties of the potential along the total path. I suppose the contributions should cancel out, but I don't get it for an arbitrary closed path.

Has any of you have the solution or at least some hint, it would be of great help!
Thanks a lot!
What is the (classical) expression for the energy density in terms of electric and magnetic fields?
 

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