Question

A thin wire ring of radius r has a electric charge q. What will be the increment in the force stretching the wire if a charge Q is placed at the center of the ring.

Answer 1

When the wire is given charge $q=+Q$.it comes intention, called hoop tension. When another charge$q_0=+Q$ is placed at the centre of the ring, the tension in the wire increases to keep this segment into equilibrium again.

Consider an elementary strip subtend in g angled $\theta$ at its centre

The string is in equillibrium in X-direction.For equillibrium in y-direction,$F_e=2Tsin\frac{d\theta }{2}$

charge on elementary strip$=\frac{q}{2\pi }\ast d\theta$

$F=\frac{1}{4\pi E_o}\ast \frac{q}{2\pi }\ast \frac{d\theta \ast q_0}{r^2}$

Therefore $2T\frac{d\theta }{2}=\frac{1}{4\pi E_o}\ast \frac{q}{2\pi }\ast \frac{d\theta \ast q_0}{r^2}$

$T=\frac{q{q}_0}{8{\pi }^2{E}_o{r}^2}$