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I have seen Gauss Law being used for a uniformly charged hollow sphere rotating with $\omega$. How is that valid to use Gauss law since it is an electrostatic law and if it is valid, why do we get a net constant electric field outside the sphere inspite of the accelerating charges - does the net time dependence cancel out ? How do I see this cancellation intuitively ? Also, can I see intuitively why the net radiation emitted is zero instead of integrating the Poynting flux ?

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  • $\begingroup$ What are the assumptions on the sphere? Is the charge uniformly and continuously distributed across the surface? $\endgroup$ Aug 26, 2014 at 7:34
  • $\begingroup$ yes, it is a uniform distribution; it could be a hollow sphere in which case, it is a surface distribution and as a special case, it is also true then for the solid sphere with a uniform volume charge distribution $\endgroup$
    – user56199
    Aug 26, 2014 at 8:33

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Take a look at the conventional form of Maxwell equations. They tell us that Gauss's law actually applies every time. However, to get the field $\vec{E}$ from the charge distribution by the usual methods, we also need to know that $$\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} = 0$$ Because otherwise the field could not be generated by the electrostatic potential. We know that the magnetic field is always divergenceless and $$\nabla \times \vec{B} = \mu_0 (\vec{j} + \varepsilon_0\frac{\partial \vec{E}}{\partial t})$$ So if the charge distribution is fixed (not to change $\vec{E}$) and the current also, then we have no changes in the magnetic field $\vec{B}$, $\partial_t \vec{B}$ is zero and we can use our usual electrostatic methods to solve the problem.

That is, accelerating charges are not enough to spoil the usual methods of electrostatics, it is the non-stationarity of current which does. But yes, once you take a look at the actually atomic structure of the charges on the sphere, you will see electromagnetic radiation being created by the non-stationarity of the charge-no-charge movement. This will induce a noise on the background calculated by neglecting the fine-grained structure.

Why? The electromagnetic field is linear and the radiation introduced by charges in the sphere will have a different phase. Interference of these waves will cause that most of the effects of these waves cancel or randomize and the result can be averaged out into the macroscopic picture, with some of the energy of the rotation transferred into radiated electromagnetic noise.

To conclude, the fact is that the electric field is only influenced by the immediate charge distribution and changes in the magnetic field. The magnetic field on the other hand is influenced only by immediate currents and changes in the electric field. If the immediate distributions of charge and current are static, we will get only static fields.


EDIT: Before someone smacks me, the last statement has obviously some exceptions if we consider different initial and boundary conditions. A planar electromagnetic wave could be passing by and scattering off the sphere etc. I implicitly assume there are no $\partial_t \vec{B},\partial_t \vec{E}$ to begin with and that the boundary conditions (the boundary could be at finite or infinite distance) do not change in time.

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