Past exam of the physics B course of the University of Cambridge / 2011 / Paper 1 / Section A / A1 / Solution

For an outward radial electric field, Gauss's law gives the enclosed chargeTaking the enclosing sphere to infinity gives $Q_{\mathrm{total}}=0$. A nonzero local field is compatible with zero net charge. Indeed, for $r>0$,Positive inner charge is balanced by negative outer charge. There is no point charge at the origin, since ${\lim }_{r\to 0}4\pi {ϵ}_0{r}^2{E}_r=0$. The field's magnitude does not specify its inward/outward sign, but reversing a globally radial direction leaves the zero total unchanged. This is net charge from the asymptotic radial electric field.