All Problems Constant Electric Field in a Vacuum
Problem 3.25
A ball of radius \(R\) carries a positive charge whose volume density depends only on a separation \(r\) from the ball's centre as \(\rho=\rho_{0}(1-r / R),\) where \(\rho_{0}\) is a constant. Assuming the permittivities of the ball and the environment to be equal to unity, find: (a) the magnitude of the electric field strength as a function of the distance \(r\) both inside and outside the ball; (b) the maximum intensity \(E_{\max }\) and the correspond ing distance \(r_{m}\).
3.25. (a) \(E=\frac{\rho_{0} r}{3 \varepsilon_{0}}\left(1-\frac{3 r}{4 R}\right)\) for \(r \leqslant R, E=\frac{\rho_{0} R^{3}}{12 \varepsilon_{0} r^{2}}\) for \(r \geqslant R\); (b) \(E_{\max }=1 / 9 \rho_{0} R / \varepsilon_{0}\) for \(r_{m}=2 / 3 R\).