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

Correct Answer: 48

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

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

\[ T = 48 \, \text{N} \]