Question

Shown in the figure is 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 semicylinder. If the current flowing through the straight wire be $i_{0}$ then the force on the semicylinder is

• A. $\frac{\mu_{0}ii_{0}}{\pi R^{2}}$
• B. $\frac{\mu_{0}i_{0}i}{\pi^{2}R}$
• C. $\frac{\mu_{0}i_{0}^{2}i}{\pi^{2}R}$
• D. None of these

Answer 1

Answer: (B)

$\frac{\mu_{0}i_{0}i}{\pi^{2}R}$

Answer 2

The net magnetic force on the conducting wire

$= \text{F } = \ \int_{}^{}{\text{2dFcos}\theta}$

⇒ F = $\int_{}^{}{2\left\lbrack \frac{\mu_{0}(di)i_{0}}{2\pi R} \right\rbrack\cos\theta}$

⇒ F = $\frac{\mu_{0}i_{0}}{\pi R}\int_{}^{}{di\cos\theta}$When $di = \frac{i}{\pi R}\text{ x Rd}\theta = \frac{\text{id}\theta}{\pi}$

⇒$F = \frac{\mu_{0}i_{0}}{\pi R}\int_{}^{}\frac{(id\theta)\cos\theta}{\pi}$

⇒$F = \frac{\mu_{0}i_{o}i}{\pi^{2}R}\int_{0}^{\pi/2}{\cos\theta d\theta = \frac{\mu_{0}i_{0}i}{\pi^{2}R}}$