Question

A thin metallic wire having a cross-sectional area of $10^{-4} \, \text{m}^2$ is used to make a ring of radius 30 cm. A positive charge of $2\pi \, C$ is uniformly distributed over the ring, while another positive charge of 30 pC is kept at the center of the ring. The tension in the ring is ______ N; provided that the ring does not deform (neglect the influence of gravity).
(Given, $\frac{1}{4 \pi \varepsilon_0} = 9 \times 10^9 \, \text{SI units}$)

Answer 1

The linear charge density $\lambda$ of the ring is:

$\lambda = \frac{Q}{2 \pi R} = \frac{2\pi}{2\pi \times 0.3} = \frac{1}{0.3} \, \text{C/m}$

The force $F_e$ due to a small element of charge $dq$ at an angle $\theta$ on the ring is balanced by tension $T$ in the ring:

$2T \sin \frac{d\theta}{2} = \frac{kq_0 \lambda d\theta}{R^2}$

Expanding and simplifying for $T$:

$T = \frac{kq_0 \lambda}{2R}$

Substitute $k = 9 \times 10^9$, $q_0 = 30 \times 10^{-12} \, \text{C}$, $R = 0.3 \, \text{m}$:

$T = \frac{9 \times 10^9 \times 30 \times 10^{-12}}{2 \times 0.3}$

$T = 48 \, \text{N}$