All Problems

Constant Electric Field in a Vacuum

Problem 3.9

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

Reveal Answer
 3.9. E=ql4πε0(r2+l2)3/2. For l≫r the strength E≈q4πε0l2, as  in the case of a point charge. Emax⁡=q63‾πε0r2 for l=r/2 . \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}
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