Question

Shown in the figure is a very long semi-cylindrical conducting shell of radius R and carrying a current i. An infinitely long straight current carrying conductor lies 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 :

• A. $\frac{{\mu }_{0}i\times i_0}{{\pi }^{2}R}$
• B. $\frac{{\mu }_{0}i\times i_0}{\pi R^2}$
• C. $\frac{{\mu }_0{i}_o^2\times i}{{\pi }^{2}R}$
• D. None of these

Answer 1

The correct option is A $\frac{{\mu }_{0}i\times i_0}{{\pi }^{2}R}$

$df=B\left(di\right)\left(1\right)$ $di=\frac{i}{\pi R}.Rd\theta =\frac{id\theta }{\pi }$

$\Rightarrow df=B.\frac{id\theta }{\pi }$

$\therefore d{f}_{net}=df\sin \theta$

$=B\frac{i}{\pi }d\theta \sin \theta$

$\Rightarrow F_{net}=\frac{Bi}{\pi }{\int }_0^{\pi }\sin \theta d\theta$

$=\frac{2Bi}{\pi }=\frac{2{\mu }_{\circ }{i}_{\circ }}{2\pi R}.\frac{I}{\pi }$

$=\frac{{\mu }_{\circ }i{i}_{\circ }}{{\pi }^{2}R}$