Problem 3.44

Two infinite planes separated by a distance $l$ carry a uniform surface charge of densities $\sigma$ and $-\sigma$ (Fig. 3.7). The planes have round coaxial holes of radius $R,$ with $l \ll R .$ Taking the origin $O$ and the $x$ coordinate axis as shown in the figure, find the potential of the electric field and the projection of its strength vector $E_{x}$ on the axes of the system as functions of the $x$ coordinate. Draw the approximate plot $\varphi(x)$ .

\varphi=\frac{\sigma l}{2 \varepsilon_{0} \sqrt{x^{2}+R^{2}}}, E_{x}=-\frac{\sigma l R^{2}}{2 \varepsilon_{0}\left(x^{2}+R^{2}\right)^{3 / 2}} . \text { See Fig. }

Constant Electric Field in a Vacuum