i want to calculate the electric field due to some charge distribution with radial symmetry.
i know the charge in the region so i calculated the flux $\Phi=\dfrac{2e_\text{charge}-5\mathrm{e}^{-2}e_\text{charge}}{\varepsilon_0}$ and i want to use gauss's law to find the electric filed on the face of a sphere through which the flux is $\Phi$
$$\iint_S \vec{E}\cdot d\vec{S}=\Phi$$
or in spherical coordinates
$$\int_0^\pi \int_0^{2\pi} \vec{E} \cdot \left( \matrix{ 1 \\ 0 \\ 0} \right) r^2 \sin(\phi) \, d\theta \, d\phi=\Phi$$
how do i calculate $\vec{E}$, i know that it has only a component in the $\hat{r}$ direction
thank you internet people.