Question

Assertion (A): When a circular coil, placed in a region with its plane parallel to a magnetic field, expands radially outwards, no emf is induced in it. Reason (R): There is a constant magnet field in the perpendicular (to the plane of the coil) direction.

• A. Both (A) and (R) are true. (R) is correct explanation of (A).
• B. Both (A) and (R) are true. (R) is not correct explanation of (A).
• C. (A) is true, (R) is false
• D. (A) is false, (R) is true

Hint:

Faraday's Law of Induction: $\mathcal{E} = -d\Phi_B/dt$.
Magnetic Flux: $\Phi_B = BA\cos\theta$. $\theta$ is angle between $\vec{B}$ and normal to area $\vec{A}$.
If plane of coil is parallel to $\vec{B}$, then $\theta=90^\circ$, $\cos(90^\circ)=0$, so $\Phi_B=0$.
If $\Phi_B$ remains zero during expansion, $d\Phi_B/dt=0$, so $\mathcal{E}=0$.
If there is a constant non-zero magnetic field component perpendicular to the plane of the coil ($B_{\perp} \neq 0$), and the area $A$ changes, then flux $\Phi_B = B_{\perp}A$ changes, and an EMF is induced. So (R) is false if it's meant as a reason for no EMF under expansion.

Answer 1

The Correct Option is C

Assertion (A): Induced emf ($\mathcal{E}$) in a coil is given by Faraday's law: $\mathcal{E} = -\frac{d\Phi_B}{dt}$, where $\Phi_B$ is the magnetic flux through the coil. Magnetic flux $\Phi_B = \vec{B} \cdot \vec{A} = BA \cos\theta$, where $B$ is the magnetic field strength, $A$ is the area of the coil, and $\theta$ is the angle between the magnetic field vector $\vec{B}$ and the area vector $\vec{A}$ (which is perpendicular to the plane of the coil). The assertion states the coil's plane is *parallel* to the magnetic field. If the plane of the coil is parallel to $\vec{B}$, then the area vector $\vec{A}$ (normal to the plane) is perpendicular to $\vec{B}$. So, the angle $\theta$ between $\vec{A}$ and $\vec{B}$ is $90^\circ$. Therefore, $\cos\theta = \cos(90^\circ) = 0$. This means the initial magnetic flux $\Phi_B = BA \cos(90^\circ) = 0$. When the coil expands radially outwards, its area $A$ changes. However, if the orientation remains such that its plane is still parallel to $\vec{B}$ (and $\vec{B}$ itself is uniform and constant in direction), then $\theta$ remains $90^\circ$. The flux will remain $\Phi_B = B A(t) \cos(90^\circ) = 0$ at all times during expansion. Since the flux is constantly zero, the rate of change of flux $\frac{d\Phi_B}{dt} = \frac{d(0)}{dt} = 0$. Therefore, no emf is induced in the coil ($\mathcal{E} = 0$). Assertion (A) is true. Reason (R): "There is a constant magnet field in the perpendicular (to the plane of the coil) direction." This statement contradicts the condition given in Assertion (A). Assertion (A) specifies that the coil's plane is *parallel* to the magnetic field, meaning the field is IN the plane of the coil, not perpendicular to it. If the field were perpendicular to the plane of the coil ($\theta=0^\circ$), then flux would be $\Phi_B = BA$. If $A$ changes, $d\Phi_B/dt = B (dA/dt) \neq 0$, and an emf would be induced. The Reason (R) describes a scenario different from that in Assertion (A). If the magnetic field mentioned in (R) is the *actual* field condition for the scenario in (A), then (R) is stating that the component of the magnetic field perpendicular to the plane of the coil is constant (specifically, it could be a constant zero if the field is entirely parallel to the plane). However, the wording "a constant magnet field in the perpendicular... direction" usually implies a non-zero field component $B_{\perp}$ that is constant. If $B_{\perp}$ (field component perpendicular to coil's plane) is constant (even if zero), and area $A$ changes, then flux $\Phi_B = B_{\perp} A$. Then $d\Phi_B/dt = B_{\perp} (dA/dt)$. If $B_{\perp}$ is non-zero and constant, an emf would be induced. The crucial part for (A) to be true is that $B_{\perp}=0$. Reason (R) says there is a constant magnetic field in the perpendicular direction. If this means $B_{\perp} = \text{constant value } C$. If $C \neq 0$, then flux is $CA(t)$, and $d\Phi_B/dt = C (dA/dt) \neq 0$. This would induce an EMF, contradicting (A). So for (A) to be true, $B_{\perp}$ must be zero. If (R) means "$B_{\perp}$ is a constant, which could be zero", it's technically true for (A)'s scenario. But "a constant magnetic field in the perpendicular direction" usually means $B_{\perp} \neq 0$. If $B_{\perp} \neq 0$ and constant, then as $A$ changes, flux changes, and emf is induced. This scenario contradicts (A)'s conclusion. Thus, if (A) is true (no emf), it must be that $B_{\perp}=0$. If (R) is claiming $B_{\perp} = \text{constant (non-zero)}$, then (R) is false for the outcome of (A) to hold. The condition for (A) to be true is that the component of $\vec{B}$ perpendicular to the coil's area is zero. (R) states there is a constant magnetic field in the perpendicular direction. If this means a non-zero constant field, then (R) is false as a condition that would make (A) true. Given the options, and that (A) is true, let's analyze (R). If (R) is interpreted as "The component of the magnetic field perpendicular to the plane of the coil is a constant value $B_0$." If $B_0 \neq 0$, then $\Phi_B = B_0 A$. As $A$ changes, $\Phi_B$ changes, $d\Phi_B/dt \neq 0$, so $\mathcal{E} \neq 0$. This contradicts (A). Thus, for (A) to be true, the field component perpendicular to the plane must be zero. Reason (R) states "There is a constant magnet field in the perpendicular ... direction". This phrase itself doesn't imply it's zero. It implies some $B_{\perp}$ exists and is constant. If such a $B_{\perp}$ (non-zero) existed, (A) would be false. Since we've shown (A) is true (because the problem stated field is parallel to plane, so $B_{\perp}=0$), (R) must be describing a condition not met, or a condition that would lead to a different outcome. Therefore, Reason (R) as a general statement (implying $B_{\perp}$ could be non-zero) would lead to an induced EMF if $A$ changes. The premise for (A) is that field is *parallel* to the plane, meaning $B_{\perp}=0$. "No EMF is induced" is true for this. Reason (R) says "There is a constant magnetic field in the perpendicular ... direction". If this field were non-zero, EMF would be induced. Thus (R) cannot be the explanation for (A), and indeed if (R) implies a non-zero $B_{\perp}$, then (R) is false for the specific context of (A) where $B_{\perp}$ must be zero. The most direct interpretation: % Option (A) Field is parallel to plane ($B_{\perp}=0$). Coil expands. Flux is $0 \times A = 0$. $d\Phi_B/dt = 0$. No EMF. (A) is true. % Option (R) Claims there is a constant field perpendicular to plane. This can be $B_{\perp}=C$. If $C \neq 0$, then expanding coil leads to EMF. If $C=0$, then (R) just says "the perpendicular component is constant zero". If we take (R) as "The reason for no EMF is that $B_{\perp}$ is constant (and in the scenario of A, this constant is zero)". This is a poor reason. A better way to interpret "Reason (R)" is as a general statement about physics. Is it true that if there's a constant $B_{\perp}$, no EMF is induced when coil expands? No, that's false. If $B_{\perp}$ is constant (and non-zero) and $A$ changes, EMF is induced. So (R) is false as a general statement of physics that would explain (A). Thus, (A) is true, (R) is false. This matches option (c). \[ \boxed{\text{(A) is true, (R) is false}} \]