All Problems

Constant Electric Field in a Vacuum

Problem 3.12

A thin nonconducting ring of radius RR has a linear charge density λ=λ0cos⁡φ,\lambda=\lambda_{0} \cos \varphi, where λ0\lambda_{0} is a constant, φ\varphi is the azimuthal angle. Find the magnitude of the electric field strength (a) at the centre of the ring; (b) on the axis of the ring as a function of the distance xx from its centre. Investigate the obtained function at x≫Rx \gg R.

Reveal Answer
3.12. (a) E=λ04ε0RE=\frac{\lambda_{0}}{4 \varepsilon_{0} R}; (b) E=λ0R24ε0(x2+R2)3/2E=\frac{\lambda_{0} R^{2}}{4 \varepsilon_{0}\left(x^{2}+R^{2}\right)^{3 / 2}}. For x≫Rx \gg R the strength E≈p4πεnx3,E \approx \frac{p}{4 \pi \varepsilon_{n} x^{3}}, where p=πR2λ0p=\pi R^{2} \lambda_{0}
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