Careful measurement of the electric field at the surface of a black box indicates that the net outward flux through the surface of the box is 8.0 x 10³ Nm²/C.  (a) What is the net charge inside the box? (b) If the net outward flux through the surface of the box were zero, could you conclude that there were no charges inside the box? Why or why not?

Given,   $N\mathrm{et} \mathrm{outward} \mathrm{flux} \mathrm{through} \mathrm{the} \mathrm{surface} \mathrm{of} \mathrm{the} \mathrm{box}, \varphi = 8.0 \times {10}^3 {\mathrm{Nm}}^2/\mathrm{C}$

(a) According to gauss law as  $\boxed{\mathrm{\varphi }=\frac{\mathrm{q}}{{\mathrm{\epsilon }}_0}}$
Hence,             $\mathrm{q} = \mathrm{\varphi } . {\mathrm{\epsilon }}_0$
or                   $$\begin{gathered} \mathrm{q} = 8.0 \times {10}^3 \times 8.854 \times {10}^{-12} \\ = 0.07 \times {10}^{-6} \end{gathered}$$
i.e.,                $\mathrm{q}=0.07 \mathrm{\mu C}$

b) No, a conclusion cannot be made that there was no charge in the box. Perhaps, it can be concluded that the net charge inside the box is zero.