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Motional EMF

There is a un...

Question

There is a uniform magnetic field $B$ directed in negative $z$ direction. A conductor $A B C$ has length $A B = l_{1}$ , parallel to the $x -$ axis, and length $B C = l_{2}$ parallel to the $y -$ axis. $A B C$ moves in the $x y$ plane with velocity $v_{x}^i + v_{y}^j$ . The potential difference between $A$ and $C$ is proportional to

v_{x} l_{1} + v_{y} l_{2}

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v_{x} l_{2} + v_{y} l_{1}

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v_{y} l_{1} - v_{x} l_{2}

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v_{x} l_{1} - v_{y} l_{2}

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Solution

The correct option is C $v_{y} l_{1} - v_{x} l_{2}$
Emf induced is due to velocity component perpendicular to the length of the conductor.
Emf induced in $A B$ is $ε_{1} = B l_{1} v_{y}$ [ $A$ positive]
Emf induced in $B C$ is $ε_{2} = B l_{2} v_{x}$ [ $C$ positive]
$∴ ε_{A C} = l_{1} v_{y} - l_{2} v_{x}$

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