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Constant Electric Field in a Vacuum

Problem 3.27

A space is filled up with a charge with volume density ρ=ρ0e−αr3,\rho=\rho_{0} \mathrm{e}^{-\alpha r^{3}}, where ρ0\rho_{0} and α\alpha are positive constants, rr is the distance from the centre of this system. Find the magnitude of the electric field strength vector as a function of rr. Investigate the obtained expression for the small and large values of r,r, i.e. at αr3≪1\alpha r^{3} \ll 1 and αr3≫1\alpha r^{3} \gg 1

Reveal Answer
 3.27. E=ρ03ε0αr2(1−e−αr3). Accordingly, E≈ρ0r3ε0 and E≈≈ρ03εnαr2\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}
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