Question

A conducting loop in the shape of a right angled isosceles triangle of height $10cm$ is kept such that the ${90}^{\circ }$ 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 $10A{s}^{-1}$. Which of the following statement(s) is (are) true?

• A. The induced current in the wire is in opposite direction to the current along the hypotenuse
• B. There is a repulsive force between the wire and the loop
• C. The magnitude of induced emf in the wire is $\left(\frac{{\mu }_0}{\pi }\right)volt$
• D. If the loop is rotated at a constant angular speed about the wire, an additional emf of $\left(\frac{{\mu }_0}{\pi }\right)volt$ is induced in the wire.

Answer 1

Answer: (B), (C)

The correct options are
B There is a repulsive force between the wire and the loop
C The magnitude of induced emf in the wire is $\left(\frac{{\mu }_0}{\pi }\right)volt$

By reciprocity theorem of mutual induction, it can be assumed that current in infinite wire is varying with $10A/s$ and EMF is produced in the loop.
From above diagram using similarity of triangles
$\frac{x}{b}=\frac{y}{h}$
$\Rightarrow x=\left(\frac{by}{h}\right)$
as given triangle is rightangled isosceles $tan{45}^{\circ }=\frac{h}{b/2}\Rightarrow b=2h$
a) Flux through the element of length $x$ and width $dy$ due to infinitely long conducting wire is $d\varphi =\overrightarrow{B}.\overrightarrow{A}=\left(\frac{{\mu }_{0}i}{2\pi y}\right)xdy=\frac{{\mu }_{0}i}{2\pi }\left(\frac{b}{h}\right)dy=\frac{{\mu }_{0}i}{\pi }dy$
Flux through whole loop
$\varphi =\frac{{\mu }_{0}i}{\pi }{\int }_0^{h}dy=\frac{{\mu }_{0}ih}{\pi }$
b) Emf generated$E=\frac{d\varphi }{dt}=\frac{{\mu }_{0}h}{\pi }\frac{di}{dt}=\frac{{\mu }_0}{\pi }[10\times 10\times {10}^{-2}]$
$E=\frac{{\mu }_0}{\pi }$ [ $c$ is correct]
c) Rotation of loop does not change $\varphi$ . Hence $D$ is incorrect
d) Given that current in the loop is counter clockwise, it produces $\overrightarrow{B}$ along outward normal.
Because of mutual Induction the wire produces $\overrightarrow{B}$ into the loop (perpendicular to it).
From Lenz law (induced current opposes generated magnetic field) induced current in the wire should be in the same direction as that of hypotenuse.[$A$ is incorrect]

$F_1=BIa$
the net downward Force due to $F_1$ > due to $F_3$ as the value of $\overrightarrow{B}$ on $BC$ is smaller (as it is at a greater distance). Hence repulsive. [$B$ is correct]

or
Field due to triangular loop at the location of infinite wire is into the paper.Hence force on the infinite wire is away from the loop(repulsive).[$B$ is correct]