Question

A long thin walled hollow cylinder of radius r is carrying a current I. A very long current carrying straight conductor is passing through the axis of the hollow cylinder, carrying a current $i_0$. Find the tension per unit length developed in the hollow cylinder due to the interaction of the straight current carrying conductor.

Answer 1

Considering the magnetic forces between $i_0$ and the differential current $di$ we obtain
$dF=\frac{{\mu }_0{i}_0}{2\pi r}\left(di\right)(di=\frac{id\theta }{2\pi })$
Where dF is the force per unit length.
Since the elementary portion is in equilibrium
$2Tsin\left(d\theta /2\right)=dF$ (T is tension per unit length)
$\Rightarrow Td\theta =dF$ [as $d\theta$is very small, $sin\frac{d\theta }{2}=\frac{d\theta }{2}]$
$\Rightarrow Td\theta =\frac{{\mu }_0{i}_0}{2\pi r}\left(\frac{id\theta }{2\pi }\right)$
$\Rightarrow T=\frac{{\mu }_0{i}_{0}i}{4{\pi }^{2}r}.$