A conducting loop in the shape of a right angled isosceles triangle of height 10 cm is kept such that the vertex is very close to an infinitely long conducting wire (see the figure). The wire is electrically insulated from the loop. The hypotenuse of the

Question

A conducting loop in the shape of a right angled isosceles triangle of height 10 cm is kept such that the $90 °$ vertex is very close to an infinitely long conducting wire (see the figure). The wire is electrically insulated from the loop. The hypotenuse of the triangle is parallel to the wire. The 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) true [2016]

Smart Achievers · Accepted Answer

(1, 4)

The flux passing through the triangular wire if $i$ current flows through the infinitely long conducting wire

$d ϕ = \int_{0}^{0.1} \frac{μ_{0} i}{2 π x} \times 2 π d x$

$ϕ = \frac{μ_{0} i}{10 π} = M i$

$∴$ $M = \frac{μ_{0}}{10 π}$ $(\frac{d i}{d t} = 10 {As}^{- 1} given)$

Induced emf in the wire, $e = M \frac{d i}{d t} = \frac{μ_{0}}{10 π} \times 10 = \frac{μ_{0}}{π} V$

There will be no extra induced emf in the wire because there is no change in the magnetic flux due to rotation of the loop.

As the current in the triangular wire is decreasing, the induced current in AB is in the same direction as the current in the hypotenuse of the triangular wire. Therefore, force will be repulsive.

Solution

1 The magnitude of induced $e m f$ in the wire is $(\frac{μ_{0}}{π})$ volt

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

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

4 There is a repulsive force between the wire and the loop.

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