All Problems Constant Electric Field in a Vacuum
Problem 3.12
A thin nonconducting ring of radius \(R\) has a linear charge density \(\lambda=\lambda_{0} \cos \varphi,\) where \(\lambda_{0}\) is a constant, \(\varphi\) is the azimuthal angle. Find the magnitude of the electric field strength (a) at the centre of the ring; (b) on the axis of the ring as a function of the distance \(x\) from its centre. Investigate the obtained function at \(x \gg R\).
3.12. (a) \(E=\frac{\lambda_{0}}{4 \varepsilon_{0} R}\); (b) \(E=\frac{\lambda_{0} R^{2}}{4 \varepsilon_{0}\left(x^{2}+R^{2}\right)^{3 / 2}}\). For \(x \gg R\) the strength \(E \approx \frac{p}{4 \pi \varepsilon_{n} x^{3}},\) where \(p=\pi R^{2} \lambda_{0}\)