Determine the net charge in a region in which the electric field is...

According to Gauss's law, the net electric flux through any closed surface is given by $\oint \vec{E} \cdot d\vec{A} = \dfrac{q_{net}}{\varepsilon_0}$.

Since the electric field $\vec{E}$ is uniform at all points in the region, it is a constant vector. The net flux through any closed surface in this region is $\vec{E} \cdot \oint d\vec{A}$.

For any closed surface, the vector sum of the area elements is zero, i.e., $\oint d\vec{A} = 0$.

Thus, the net flux is zero, which implies $q_{net} = 0$.

Alternatively, using the differential form of Gauss's law, $\nabla \cdot \vec{E} = \dfrac{\rho}{\varepsilon_0}$. Since $\vec{E}$ is constant, its divergence is zero, giving $\rho = 0$. Therefore, the net charge in the region is zero.