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The Larmor Formula suggests that the power radiated by an accelerating charge is non zero. But we know that a uniformly charged non-conducting ring rotating about its central axis does not radiate. Why is this so?

This is somehow very strange to me. For example, suppose I charge a very small portion of the non conducting ring with charge q. Now when I rotate this ring with a constant angular velocity, there should be radiation. Now, all I have to do is spray some charge uniformly on the rotating ring, and the radiation just ceases! Isn't this strange? We can negate radiation simply by spraying charge carefully?

Please explain why there is no radiation in this case.

sammy gerbil
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Lelouch
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5 Answers5

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To radiate, you must have a fluctuation with time

If you take a small portion of the ring, charge it, and then make it rotate, the distribution of charges does change with time.

If the ring is evenly charged, with no additional currents, and you make the ring rotate, then in this (ideal) situation the distribution of charges between two times is always the same. There would be no way to differentiate the situation at $t_1$ and $t_2$, so no radiation. I assume that if you compute the EM fields radiated by each charge, and sum them over the ring, they cancel out.

Albert Beton
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Photons are their own anti-particle and can cancel out. Or put another way, light is a wave and waves can destructively interfere. With the rotating ring of charge, there are many radiating fields, but the symmetry of the problem lets them cancel destructively so there is no net wave and no power lost to radiation. Radiation here is just an EM wave carrying power.

Think of a somewhat similar situation from electrostatics: If you add electrical charge to a perfect conductor, the charges arrange themselves on the surface and the electric field is 0 within the conductor. There are many charges with many fields, but they all cancel out. This is easiest to calculate for a sphere, in case you want to try it.

EL_DON
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I address this question in my answer Why don't loop currents produce light?. As you divide the single charged particle into more and more pieces while keeping the total charge and average current fixed, the charge distribution varies less and less over time and the radiated power smoothly decreases to zero.

Community
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tparker
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If we were to describe emission from radiation from Maxwell equations, then what couples to electromagnetic radiation is really current, not the charge itself (although there may be some complicated situations, depending on the gauge chosen.) A rotating homogeneous ring or a standing wave do not correspond to current flows changing in time, even though particles constituting them do move.

Another way to see it is by disentangling Lagrangian and Eulerian descriptions of the current and charges: the particles do move (Lagrangian view), but the charge distribution and the current distribution remain unchanged in every point of space.

Roger V.
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When a single electron turns in a circle it emits radiation to help conserve momentum along the axis it has just turned from . But a ring of many electrons replaces the momentum of the turning electron by another electron which takes its former position.