Find the Charge!

At any point $P(x,y,z)$ on the sphere, a unit vector perpendicular to the sphere radially outwards is $\hat{n}=\dfrac{x}{R} \hat{i} + \dfrac{y}{R} \hat{j} + \dfrac{z}{R} \hat{k}$ The electric flux passing through an elemental area $dS$ at point $P$ on the sphere can be given as $d\Phi=\vec{E} \cdot \hat{n} dS= \left(\dfrac{2x^2}{5(x^2+y^2)}+\dfrac{2y^2}{5(x^2+y^2)}\right) dS$

Which implies $d\Phi=\dfrac{2}{5} dS$ . Since the flux is independent of coordinates, thus total flux can be integrated for whole sphere which is given as $\int d\Phi= \Phi=\dfrac{2}{5} \int dS=\dfrac{2}{5} \times 4\pi(5^2)=40\pi$

Hence from Gauss's law , charge enclosed can be given as $40\pi\epsilon_{0} \approx 1.112 \times 10^{-9}$ .

Hence $\lfloor 10^{12}k \rfloor = \boxed{1112}$ .

Problem treated in spherical coordinates:

$x = R \sin{\theta}\cos{\phi}$ $y = R \sin{\theta}\sin{\phi}$ $z = R\cos{\theta}$

$\vec{r} = x \ \hat{i} + y \ \hat{j} +z \ \hat{k}$ $\hat{r} = \frac{\vec{r}}{\lvert \vec{r} \rvert}$

The electric field vector can be written in terms of the $R$ , $\theta$ and $\phi$ . The surface area element on the sphere is:

$d\vec{S} = R^2\sin{\theta} \ d\theta \ d\phi \ \hat{r}$

The elementary flux through the element is:

$d\Phi = \vec{E} \cdot d\vec{S}$

Plugging in expressions and simplifying gives: $d\Phi = 10 \sin{\theta} \ d\theta \ d\phi$

The total flux through the sphere is:

$\Phi = \int_{0}^{2 \pi} \int_{0}^{\pi} 10 \sin{\theta} \ d\theta \ d\phi$ $\Phi = 40 \pi$

According to Gauss' law:

$\Phi = \frac{Q_{enc}}{\epsilon_o}$

From here, the required answer can be computed.

$\boxed{\lfloor 10^{12}Q_{enc} \rfloor= 1112}$

What I find strange about this problem is when I try to solve it using Maxwell's equation:

$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_o}$

Where $\rho$ is the charge per unit volume in space. The divergence of the electric field evaluates to zero which implies that the charge per unit volume is zero, which implies that the total charge should be zero. I get a contradictory result using Maxwell's equation. Can someone clarify this for me? Thanks in advance