Electromagnetic Induction and Alternating Current 1 Question 10
10. A conducting loop in the shape of a right angled isosceles triangle of height $10 cm$ 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 $10 As^{-1}$. Which of the following statement(s) is (are) true?
(2016 Adv.)
(a) There is a repulsive force between the wire and the loop.
(b) If the loop is rotated at a constant angular speed about the wire, an additional emf of $\frac{\mu _0}{\pi}$ volt is induced in the wire.
(c) The magnitude of induced emf in the wire is $\frac{\mu _0}{\pi}$ volt.
(d) The induced current in the wire is in opposite direction to the current along the hypotenuse.
Show Answer
Answer:
Correct Answer: 10. $(a, c)$
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 $\varphi$, can be calculated as follows
$$ \begin{aligned} & d \varphi=\frac{\mu _0 i}{2 \pi y} \cdot 2 y d y \\ & d \varphi=\frac{\mu _0 i}{\pi} d y \end{aligned} $$
$$ \begin{aligned} \varphi & =\frac{\mu _0 i}{\pi} \frac{l}{\sqrt{2}} \\ \Rightarrow \quad EMF & =\left|\frac{d \varphi}{d t}\right|=\frac{\mu _0}{\pi} \frac{l}{\sqrt{2}} \cdot \frac{d i}{d t} \\ & =\frac{\mu _0}{\pi}(10 cm) 10 \frac{A}{s}=\frac{\mu _0}{\pi} \text { volt } \end{aligned} $$
If we assume the current in the wire towards right then as the flux in the loop increases we know that the induced current in the wire is counter clockwise. Hence, the current in the wire is towards right.
Field due to triangular loop at the location of infinite wire is into the paper. Hence, force on infinite wire is away from the loop. By cylindrical symmetry about infinite wire, rotation of triangular loop will not cause any additional EMF.
