All Problems Constant Electric Field in a Vacuum
Problem 3.45
An electric capacitor consists of thin round parallel plates, each of radius \(R,\) separated by a distance \(l(l \ll R)\) and uniformly charged with surface densities \(\sigma\) and \(-\sigma .\) Find the potential of the electric field and the magnitude of its strength vector at the axes of the capacitor as functions of a distance \(x\) from the plates if \(x\) \(\gg l\) Investigate the obtained expressions at \(x \gg R\).
3.45. \(\varphi \approx \pm \frac{\sigma l}{2 \varepsilon_{0}}\left(1-\frac{x}{\sqrt{x^{2}+R^{2}}}\right), E \approx \frac{\sigma l R^{2}}{2 \varepsilon_{0}\left(x^{2}+R^{2}\right)^{3 / 2}} .\) If \(x \gg R\) then \(\varphi \approx \pm \frac{p}{4 \sigma \varepsilon_{0} x^{2}}\) and \(E \approx \frac{p}{2 \pi \varepsilon_{0} x^{3}},\) where \(p=\pi R^{2} \sigma l .\) In the formulas for the potential \(\varphi\) the plus sign corresponds to the space adjoining the positively charged plate and the minus sign to the space adjoining the negatively charged pläte.