A thin metallic wire having cross sectional area of $10^{-4}\ \text{m}^2$ is used to make a ring of radius $30\ \text{cm}$. A positive charge of $2\pi\ \text{C}$ is uniformly distributed over the ring, while another positive charge of $30\ \text{pC}$ is kept at the centre of the ring. The tension in the ring is ______ N; provided that the ring does not get deformed (neglect the influence of gravity). (Given, $\dfrac{1}{4\pi\epsilon_0} = 9\times10^9$ SI units)
Numerical value type. Enter your answer.
Answer: 3
Take a small element of the ring subtending angle $d\theta$ at the centre. Its charge is $dQ = \dfrac{Q}{2\pi}d\theta$, and the outward electric force on it from the central charge $q$ is
$$dF = \frac{kq\,dQ}{R^2}$$
This is balanced by the inward component of the tension at its two ends, $2T\sin\frac{d\theta}{2} = T\,d\theta$:
$$T\,d\theta = \frac{kqQ}{2\pi R^2}d\theta \;\Rightarrow\; T = \frac{kqQ}{2\pi R^2}$$
$$T = \frac{9\times10^9\times30\times10^{-12}\times2\pi}{2\pi\times(0.3)^2} = \frac{0.27}{0.09} = 3\ \text{N}$$
Solution by Sreeraj P, M.Sc Physics