Induced EMF and Force on Loop Near Infinitely Long Wire

To determine which statements are correct, we analyze the system using Faraday's Law of Electromagnetic Induction, the concept of Mutual Inductance, Lenz's Law, and magnetic force interactions.

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1. Calculation of Mutual Inductance and Induced EMF (Option A)

By the Reciprocity Theorem of Mutual Inductance, the mutual inductance $M$ between two circuits is symmetric ($M_{12} = M_{21}$).

To find $M$, let us assume a current $I$ flows through the infinitely long straight wire. The magnetic field at a distance $y$ from the wire is given by Ampere's Law: $B(y) = \frac{\mu_0 I}{2\pi y}$

The triangular loop is a right-angled isosceles triangle with its $90^\circ$ vertex located at $y \to 0$ (extremely close to the wire) and its hypotenuse parallel to the wire at a distance $h = 10\text{ cm} = 0.1\text{ m}$.

Since the angle at the vertex is $90^\circ$, the altitude of length $h$ bisects this vertex into two $45^\circ$ angles. Therefore, at a distance $y$ from the vertex, the width of the triangular strip parallel to the wire is: $w(y) = 2 \cdot (y \tan 45^\circ) = 2y$

The magnetic flux $d\Phi$ through an infinitesimal horizontal strip of thickness $dy$ at a distance $y$ from the wire is: $d\Phi = B(y) \cdot w(y) \, dy = \left(\frac{\mu_0 I}{2\pi y}\right) (2y \, dy) = \frac{\mu_0 I}{\pi} \, dy$

Integrating $y$ from $0$ to $h$: $\Phi = \int_{0}^{h} \frac{\mu_0 I}{\pi} \, dy = \frac{\mu_0 I h}{\pi}$

Thus, the mutual inductance $M$ between the wire and the loop is: $M = \frac{\Phi}{I} = \frac{\mu_0 h}{\pi}$

Given that the current in the triangular loop increases at a constant rate $\frac{dI}{dt} = 10\text{ A s}^{-1}$ and $h = 0.1\text{ m}$, the magnitude of the induced electromotive force ($emf$) in the wire is: $\mathcal{E} = M \frac{dI}{dt} = \left(\frac{\mu_0 \times 0.1}{\pi}\right) \times 10 = \frac{\mu_0}{\pi}\text{ V}$

Statement (A) is TRUE.

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2. Effect of Rotation on Induced EMF (Option B)

The magnetic field produced by an infinitely long straight wire possesses cylindrical symmetry around the wire. Consequently, rotating the triangular loop with a uniform angular velocity around the wire as an axis does not change the magnetic flux linking the loop and the wire.

Since $\frac{d\Phi_{\text{rot}}}{dt} = 0$, no additional rotational (motional) $emf$ is induced in the wire.

Statement (B) is FALSE.

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3. Direction of Induced Current (Option C)

1. The current in the triangular loop flows in a counterclockwise direction.

   • Along the horizontal hypotenuse at the bottom, the current flows from left to right.
   • Along the legs near the top vertex, the current flows towards the top vertex along the right leg and away along the left leg, giving a net leftward horizontal component near the vertex.

2. The counterclockwise current creates a magnetic flux directed out of the page ($\odot$) inside the loop. As the current increases, this out-of-page flux increases.

3. According to Lenz's Law, the induced current in the straight wire must create an into-the-page ($\otimes$) magnetic field below the wire to oppose the increasing out-of-page flux.

4. By the right-hand thumb rule, a current in the straight wire flowing to the right produces an into-the-page magnetic field in the region below it.

5. Since both the current along the hypotenuse and the induced current in the wire flow to the right, the induced current in the wire is in the same direction as the current along the hypotenuse, not opposite.

Statement (C) is FALSE.

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4. Force Between the Wire and the Loop (Option D)

• Near the top vertex (closest to the straight wire), the horizontal component of the loop's current flows to the left.
• The induced current in the straight wire flows to the right.
• Since antiparallel currents repel each other, and the top vertex is closest to the wire, this repulsive force dominates over any forces on the more distant parts of the loop.
• Alternatively, by Lenz's Law, any mechanical action resulting from induced currents must oppose the cause (the increase in flux coupling), resulting in a net repulsive force between the wire and the loop.

Statement (D) is TRUE.

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Conclusion

The correct options are A and D.