A uniformly charged rotating sphere does not radiate, why not?

[wykk]

The problem says I have a spherically symmetric spinning constant charge distribution of charge Q and angular momentum w; I saw two possible explanations but none of them has made me realize why it is zero, one mentions thata constant w somehow implies a constant E which would mean there is no B and poynting vector would be zero.
Another mentions that the charge distribution rho is constant therefore J the current density is too and B becomes zero but I don't know how to derive an expression that relates B and J

 

[kuruman]

Hi @wykk and welcome to PF.

If this is a homework problem, it should be posted under Introductory Physics Homework with the template provided. Please read the forum homework help guidelines before posting there. I would also strongly recommend posting the exact description of the question as given to you. Providing links to the explanations that you saw would also be helpful as it is possible that you may have misconstrued what you read.

 

[vanhees71]

It's because the charge and current distributions are time-independent. Thus you also have static fields. I suppose it's meant that the angular velocity $\vec{\omega}=\text{const}$.

 

[sophiecentaur]

It's because the charge and current distributions are time-independent. Thus you also have static fields. I suppose it's meant that the angular velocity $\vec{\omega}=\text{const}$.

It's the equivalent of DC passing round a loop of wire; no radiation except at switch on.

 

[HomogenousCow]

Look at the form of the Jefimenko solutions, which are the solutions to electromagnetism in free space. Only the terms that go as $1/|r-r'|$ contribute to the radiation field, all three of these terms are proportional to $\dot \rho$ or $\dot{\mathbf J}$ which are both zero in the case of a spherically symmetric spinning body (or even an axially symmetric one).