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 infinity 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. 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 $\left(\frac{{\mu }_0}{\pi }\right)$ volt is induced in the wire
• C. The magnitude of induced emf in the wire is $\left(\frac{{\mu }_0}{\pi }\right)$ volt
• D. The induced current in the wire is in opposite direction to the current along the hypotenuse

Answer 1

Answer: (A), (C)

Here $\frac{di}{dt}=10A/S$
for mutual inductance
$d\varphi =\frac{{\mu }_0}{2\pi }\frac{i}{r}(2rdr)$
$\varphi =\frac{{\mu }_0}{\pi }i\underset{0}{\overset{4}{\int }}dr$
$\varphi =\left(\frac{{\mu }_0}{\pi }d\right)i=Mi$
$M=\frac{{\mu }_0}{\pi }d=0.1\frac{{\mu }_0}{\pi }$
When current through loop increases then magnetic flux
$\varphi =Mi$
$\epsilon =\frac{d\varphi }{dt}$
$\epsilon =M\frac{di}{dt}=\frac{{\mu }_0}{\pi }\times 10\times 0.1$
$\epsilon =\frac{{\mu }_0}{\pi }$
Force between loop and conductor is repulsive.