Adjoining figure shows a very long semicylindrical conducting shell of radius $R$ and carrying a current $i$. An infinitely long straight current carrying conductor is lying along the axis of the semi-cylinder. If the current flowing through the straight wire be $i_0$, then the force per unit length on the conducting wire is
Diffcult
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The net magnetic force on the conducing wire per unit length
$\mathrm{F}=\int 2 \mathrm{dF} \cos \theta$
$=\int 2\left[\frac{\mu_{0}(\mathrm{d} i) i_{0}}{2 \pi \mathrm{R}}\right] \cos \theta$
$=\frac{\mu_{0} i_{0}}{\pi R} \int d i \cos \theta$
where,
$\mathrm{d} i=\frac{i}{\pi \mathrm{R}} \times \mathrm{R} \mathrm{d} \theta=\frac{i \mathrm{d} \theta}{\pi}$
$\therefore \,\,\, \mathrm{F}=\frac{\mu_{0} i_{0}}{\pi \mathrm{R}} \int \frac{(i \mathrm{d} \theta) \cos \theta}{\pi}$
$=\frac{\mu_{0} i_{0} i}{\pi^{2} \mathrm{R}} \int_{0}^{\pi / 2} \cos \theta \mathrm{d} \theta=\frac{\mu_{0} i_{0} i}{\pi^{2} \mathrm{R}}$
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