A circular conducting loop of radius $R$ carries a current $I.$ Another straight infinite conductor carrying current $I$ passes through the diameter of this loop as shown in the figure. The magnitude of force exerted by the straight conductor on the loop is
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$\mathrm{d} \mathrm{F}_{\mathrm{res}}=2(\mathrm{dF}) \sin \theta$
$=2 \mathrm{I}(\mathrm{d} \ell) \mathrm{B} \sin \theta$
$=2 \mathrm{I}(\mathrm{R} \mathrm{d} \theta)\left(\frac{\mu_{0} \mathrm{I}}{2 \pi r}\right) \sin \theta$
$=\frac{\mu_{0} I^{2} R}{\pi} \frac{\sin \theta d \theta}{(r)}$
$=\frac{\mu_{0} I^{2} R}{\pi} \frac{\sin \theta}{(R \sin \theta)} d \theta=\frac{\mu_{0} I^{2}}{\pi}(d \theta)$
$\mathrm{F}_{\mathrm{res}}=\int \mathrm{dF}_{\mathrm{res}}=\frac{\mu_{0} \mathrm{I}^{2}}{\pi} \int_{0}^{\pi} \mathrm{d} \theta=\mu_{0} \mathrm{I}^{2}$
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