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Standard XII

Physics

Motional EMF

A wire bent a...

Question

A wire bent as a parabola $y = a x^{2}$ is located in a uniform magnetic field of induction $\to B$ , the vector $\to B$ being perpendicular to the $x y$ plane. At the moment $t = 0$ a connector starts sliding in $y$ direction from the parabola apex with a constant acceleration $ω$ (Figure). Find the emf of electromagnetic induction in the loop thus formed as a function of $y$ .

y B \sqrt{\frac{8 ω}{a}}

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y B \sqrt{\frac{4 ω}{a}}

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y B \sqrt{\frac{2 ω}{a}}

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None of these

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Solution

The correct option is A $y B \sqrt{\frac{8 ω}{a}}$
The emf in the loop is the motional emf induced in length $A C$ of the sliding connector.
$P C = x = \sqrt{\frac{y}{a}}$
Length $A C = 2 x = 2 \sqrt{\frac{y}{a}}$
Velocity of the connector after it gets displaced by $y$ is given by
$V^{2} = 2 ω y$
or, $V = \sqrt{2 ω y}$
$∴$ Emf induced in $A C$ is $ε = B l V$
$= B (2 \sqrt{\frac{y}{a}}) (\sqrt{2 ω y})$
$= y B \sqrt{\frac{8 ω}{a}}$

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