Jackson Electrodynamics 14.24: Prove that a DC current loop does not radiate, starting from Lienard-Wiechert potentials

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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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Answers and Replies

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nrqed
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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!!
What is the (classical) expression for the energy density in terms of electric and magnetic fields?
 

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