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Constant Electric Field in a Vacuum

Problem 3.43

Two coaxial rings, each of radius R,R, made of thin wire are separated by a small distance l(l≪R)l(l \ll R) and carry the charges qq and −q.-q . Find the electric field potential and strength at the axis of the system as a function of the xx coordinate (Fig. 3.6). Show in the same drawing the approximate plots of the functions obtained. Investigate these functions at ∣x∣≫R|x| \gg R

Reveal Answer
\begin{aligned} &\varphi=\frac{q l}{4 \pi \varepsilon_{0}} \frac{x}{\left(R^{2}+x^{2}\right)^{3 / 2}}, E_{x}=-\frac{q l}{4 \pi \varepsilon_{0}} \frac{R^{2}-2 x^{2}}{\left(R^{2}+x^{2}\right)^{5 / 2}}\\ &\text { where } \end{aligned} ExE_{x} is the projection of the vector E\mathbf{E} on the xx axis. The functions are plotted in Fig. 17. If ∣x∣≫R,|x| \gg R, then φ≈ql4πε0x2\varphi \approx \frac{q l}{4 \pi \varepsilon_{0} x^{2}} and Ex≈ql2πε0x3E_{x} \approx \frac{q l}{2 \pi \varepsilon_{0} x^{3}}
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