Question

Shown in the figure is a very long semi-cylindrical conducting shell of radius R and carrying 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_0i0​, then the force per unit length on the conducting wire is:

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Answer 1

Explanation:

The net magnetic force on the conducting wire per unit length $=\mathrm{F}=\int 2 \mathrm{dF} \cos \theta$

$\Rightarrow \mathrm{F}=\int 2\left[\frac{\mu_{0}(\mathrm{di}) \mathrm{i}_{0}}{2 \pi \mathrm{R}}\right] \cos \theta$

$\Rightarrow \mathrm{F}=\frac{\mu_{0} \mathrm{i}_{0}}{\pi \mathrm{R}} \int \{dicos} \theta$

When $\mathrm{di}=\frac{\mathrm{i}}{\pi \mathrm{R}} \times \mathrm{Rd} \theta=\frac{\mathrm{id} \theta}{\pi}$

$\Rightarrow \mathrm{F}=\frac{\mu_{0} \mathrm{i}_{0}}{\pi \mathrm{R}} \int \frac{(\mathrm{id} \theta) \cos \theta}{\pi}$

$\Rightarrow \mathrm{F}=\frac{\mu_{0} \mathrm{i}_{0} \mathrm{i}}{\pi^{2} \mathrm{R}} \int_{0}^{\pi / 2} \cos \theta \mathrm{d} \theta=\frac{\mu_{0} \mathrm{i}_{0} \mathrm{i}}{\pi^{2} \mathrm{R}}$