Problem 3.9

A thin wire ring of radius \(r\) carries a charge \(q .\) Find the magnitude of the electric field strength on the axis of the ring as a function of distance \(l\) from its centre. Investigate the obtained function at \(l \gg r\). Find the maximum strength magnitude and the corresponding distance \(l\). Draw the approximate plot of the function \(E(l)\).

Reveal Answer

\begin{aligned} &\text { 3.9. } E=\frac{q l}{4 \pi \varepsilon_{0}\left(r^{2}+l^{2}\right)^{3 / 2}} . \text { For } l \gg r \text { the strength } E \approx \frac{q}{4 \pi \varepsilon_{0} l^{2}}, \text { as }\\ &\text { in the case of a point charge. } E_{\max }=\frac{q}{6 \sqrt{\overline{3} \pi \varepsilon_{0} r^{2}}} \text { for } l=r / \sqrt{2} \text { . } \end{aligned}

Constant Electric Field in a Vacuum