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I've been thinking about a distribution of charge that does not follow Maxwell's euations, and I can't understand what's wrong with my reasoning. If we have a constant distribution of charge $\rho(x,y,z)=\rho_0$ in all space then we have $\vec\nabla\cdot \vec{E}=\rho_0/\varepsilon_0$. But the charge' symmetry implies that $\vec{E}=(0,0,0)$ in all of space (it as to be the same after a translation or a rotation), so $\vec\nabla\cdot \vec{E}=0$. Can someone help me?

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