All Problems Conductors and Dielectrics in an Electric Field
Problem 3.62
A thin wire ring of radius \(R\) carries a charge \(q\). The ring is oriented parallel to an infinite conducting plane and is separated by a distance \(l\) from it. Find: (a) the surface charge density at the point of the plane symmetrical with respect to the ring; (b) the strength and the potential of the electric field at the centre of the ring.
3.62. (a) \(\sigma=\frac{l q}{2 \pi\left(l^{2}+R^{2}\right)^{3 / 2}}\) (b) \(E=\frac{1}{4 \pi \varepsilon_{0}} \frac{q}{4 l^{2}\left[1+1 / 4(R / l)^{2}\right]^{3 / 2}}\), \(\varphi=\frac{1}{4 \pi \varepsilon_{0}} \frac{q}{R}\left(1-\frac{1}{\sqrt{1+4(l / R)^{2}}}\right)\)