Why electric field outside the sphere that carries polarization is 0?

There is a sphere with radius $R$ that carries a polarization $$\vec P(\vec{r})=k\vec r$$

$\vec r$ is the vector from the center.

I found electric field the outside generated by the sphere is $\vec E=0$

Can you, please, explain it in physical terms?

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1 Answer

Your polarized sphere is spherically symmetric. Therefore outside the sphere, in the vaccuum, the solution must be spherically symmetric. However, the only spherically symmetric solution of Maxwell's Equations in vacuum is the electric field of a point charge. Since your sphere has zero net charge, it must be the electric field of a point with zero charge. Thus $\vec{E} = 0$.

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