Problem 3.27

A space is filled up with a charge with volume density \(\rho=\rho_{0} \mathrm{e}^{-\alpha r^{3}},\) where \(\rho_{0}\) and \(\alpha\) are positive constants, \(r\) is the distance from the centre of this system. Find the magnitude of the electric field strength vector as a function of \(r\). Investigate the obtained expression for the small and large values of \(r,\) i.e. at \(\alpha r^{3} \ll 1\) and \(\alpha r^{3} \gg 1\)

\begin{aligned} &\text { 3.27. } E=\frac{\rho_{0}}{3 \varepsilon_{0} \alpha r^{2}}\left(1-\mathrm{e}^{-\alpha r^{3}}\right) . \quad \text { Accordingly, } \quad E \approx \frac{\rho_{0^{r}}}{3 \varepsilon_{0}} \text { and } E \approx\\ &\approx \frac{\rho_{0}}{3 \varepsilon_{n} \alpha r^{2}} \end{aligned}

Constant Electric Field in a Vacuum