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A thin metallic wire having cross sectional area of $10^{-4} \mathrm{~m}^2$ is used to make a ring of radius $30 \mathrm{~cm}$. A positive charge of $2 \pi \mathrm{C}$ is uniformly distributed over the ring, while another positive charge of $30$ $\mathrm{pC}$ is kept at the centre of the ring. The tension in the ring is__________ $\mathrm{N}$; provided that the ring does not get deformed (neglect the influence of gravity). (given, $\frac{1}{4 \pi \epsilon_0}=9 \times 10^9 \mathrm{SI}$ units)
$7$
$3$
$5$
$6$
Solution
$ 2 \mathrm{~T} \sin \frac{\mathrm{d} \theta}{2}=\frac{\mathrm{kq}_0}{\mathrm{R}^2} \cdot \lambda \mathrm{Rd} \theta $
$ {\left[\lambda=\frac{\mathrm{Q}}{2 \pi \mathrm{R}}\right]}$
$ \Rightarrow \mathrm{T}=\frac{\mathrm{Kq} \mathrm{q}_0 \mathrm{Q}}{\left(\mathrm{R}^2\right) \times 2 \pi} $
$ =\frac{\left(9 \times 10^9\right)\left(2 \pi \times 30 \times 10^{-12}\right)}{(0.30)^2 \times 2 \pi} $
$ =\frac{9 \times 10^{-3} \times 30}{9 \times 10^{-2}}=3 \mathrm{~N}$
