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In Spherical coordinates if we have a source at the origin generating $E$ in the $\hat{r}$ direction and $H$ in the $\hat{\phi}$ direction then our Poynting Vector will be in the $\hat{\theta}$ direction. When considering the power of such a source we know that the total power is the flux of the Poynting vector through a surface that encloses the sphere.

If we take this surface to be a sphere of radius $R$, then when taking the flux at every point it is zero since $\hat{\theta} \times \hat{n}=0$ where $\hat{n}$ is the unit vector normal to the surface given by $\hat{r}$, so is the power of the source $0$?

This doesn't seem entirely correct to me since there is obviously fields being generated, and the magnetic and electric field phases can be switched to alter the power factor, but here it seems as though regardless of what the fields are their directionality ensures $0$ power.

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  • $\begingroup$ Not all field configurations must radiate, not all surfaces will experience a net power flux .... $\endgroup$
    – ohneVal
    Aug 14, 2021 at 14:10

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It's possible to pose questions about field configurations that can't be made! An everywhere radial $\mathbf E$ field suggests a net charge with a spherically-symmetrical distribution, which is fine. But an everywhere azimuthal $\mathbf H$ field suggests a current in the $z$-direction, which is inconsistent with a spherical charge distribution (it leads to a time-dependent dipole moment). So I think you should first specify a physically possible field configuration.

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