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Motional EMF

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Question

A conducting square loop is placed in a magnetic field $B$ with its plane perpendicular to the field. Some how the sides of the loop start shrinking at a constant rate $α$ . The induced emf in the loop at an instant when its side is $a$ , is :

A

2 a α B

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B

a^{2} α B

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C

2 a^{2} α B

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D

a α B

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Solution

The correct option is A $2 a α B$
At time $t$
Side of square $l = l_{i} - d t$
$S = l^{2} = {(l_{i} - α t)}^{2}$
At given time
$l = l_{i} - α t = a$
$ϕ = B S = B {(l_{i} - α t)}^{2}$
$e = ∣ ∣ ∣ \frac{d ϕ}{d t} ∣ ∣ ∣ = 2 B α (l_{i} - α t)$
But $(l_{i} - α t) = a$
$∴$ $e = 2 a α B$

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