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Electric field produced by a uniformly polarized sphere
You don't need to specify the $z$ direction.
To get a uniform polarization in a linear dielectric, you need to immerse it in a uniform external field, $E$, where $E$ is the value far from the field.
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What is $E$ with respect to the relation between an electric field and its polarization density?
The $\mathbf{E}$ in the formula is the electric field at equilibrium $\mathbf{E}_\mathrm{total}$ (In the following answer I shall use $\mathbf{E}_\mathrm{total}$ in place of $\mathbf{E}_\mathrm{n}$).
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