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(C) The induced electromotive force $(emf)$ across a rotating conducting rod in a uniform magnetic field is given by the formula $e = \frac{1}{2} B \o

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$\frac{1}{2}B\omega(L^2 + l^2)$

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(C) The induced electromotive force $(emf)$ across a rotating conducting rod in a uniform magnetic field is given by the formula $e = \frac{1}{2} B \omega r^2$,where $r$ is the perpendicular distance from the axis of rotation to the end of the rod.In this case,the rod is $J$-shaped,but the induced $emf$ depends only on the displacement vector between the two endpoints of the rod relative to the axis of rotation.The effective distance $r$ from the pivot point $P$ to the other end $Q$ is the straight-line distance between them. Based on the geometry of the rod,this distance is $r = \sqrt{l^2 + L^2}$.Substituting this into the formula,we get:$e = \frac{1}{2} B \omega (\sqrt{l^2 + L^2})^2$Simplifying the expression:$e = \frac{1}{2} B \omega (l^2 + L^2)$

Column-$I$	Column-$II$
$A$. Halophiles	$i$. Gram-negative bacteria
$B$. Thermoacidophiles	$ii$. Inhabit extremely salty areas
$C$. Spirochetes	$iii$. Responsible for methane production
$D$. Firmicutes	$iv$. Survive in acidic and high-temperature conditions
	$v$. Gram-positive bacteria

When a $J$-shaped conducting rod is rotating in its own plane with constant angular velocity $\omega$,about one of its ends $P$,in a uniform magnetic field $\vec B$ (directed normally into the plane of the paper),then the magnitude of the emf induced across it will be:

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