A conducing loop in the shape of a right angled isosceles triangle of height 10 cm is kept such that the $90^{0}$ vertex is very close to an infinitely long conducting wire (see the fig.) The wire is electrically insulated from the loop. The hypotenuse of the triangle is parallel to the wire. They current in the triangular loop is in counterclockwise direction and increased at a constant rate of $10 A s^{- 1}$ . Which of the following statement(s) is/are correct?

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If the loop is rotated at a constant angular speed about the wire, an additional emf of $(\frac{μ_{0}}{π})$ volt is induced in the wire

The induced current in the wire is in opposite direction to the current along the hypotenuse

The magnitude of induced emf in the wire is $(\frac{μ_{0}}{π})$ volt

There is a repulsive force between the wire and the loop

answer is A, C.

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Detailed Solution

By reciprocity theorem of mutual induction, it can be assumed that current in infinite wire is varying at 10 A/s and EMF is induced in triangular loop.

Flux of magnetic field through triangle loop, if current in infinite wire is $ϕ,$ can be calculated as follows.

$d ϕ = \frac{μ_{0} i}{2 π y} 2 y d y$

$d ϕ = \frac{μ_{0} i}{π} d y$

$ϕ = \frac{μ_{0} i}{π} (\frac{1}{10})$

$\Rightarrow E M F = | \frac{d ϕ}{d t} | = \frac{μ_{0}}{π} (\frac{1}{10}) \frac{d i}{d t} = \frac{μ_{0}}{π} \frac{1}{10} (10 \frac{A}{s}) = \frac{μ_{0}}{π} v o l t$