Question 51:
easyAssertion (A): Change in magnetic flux w.r.t. time produces an induced emf.
Reason (R): Faraday established induced emf experimentally.
Assertion (A) is true based on Faraday's Law of electromagnetic induction, \( \epsilon = -\frac{d\Phi_B}{dt} \). Reason (R) is also true as Faraday's experiments confirmed this. (R) correctly explains (A).
Question 52:
easyAssertion (A): An emf is induced in a closed loop where magnetic flux is varied. The induced electric field is not a conservative field.
Reason (R): For induced electric field, the line integral \( \oint \vec{E} \cdot d\vec{l} \) around a closed path is non-zero.
Assertion (A) is true because varying magnetic flux induces a non-conservative electric field. Reason (R) is true; for a non-conservative field, \( \oint \vec{E} \cdot d\vec{l} \neq 0 \). (R) correctly explains why the induced electric field is non-conservative.
Question 53:
easyAssertion (A): It is more difficult to push a magnet into a coil with more loops.
Reason (R): This is because emf induced in each current loop resists the motion of the magnet.
Assertion (A) is true. More loops increase induced emf and current, strengthening the opposing magnetic field. Reason (R) is true due to Lenz's law, where induced effects oppose the cause. (R) provides the direct explanation for (A).
Question 54:
easyAssertion (A): When a coil is rotated in a uniform magnetic field about an axis perpendicular to the field, emf is induced in it which is maximum for the orientation of coil in which magnetic flux through the coil is zero.
Reason (R): In an electric generator, electrical energy is generated by rotating a coil in a magnetic field.
Assertion (A) is true because \( \epsilon = BA\omega sin(\omega t) \), which is max when \( \Phi_B = BA cos(\omega t) = 0 \). Reason (R) is also true, describing generators. However, (R) is an application and doesn't explain the condition for maximum emf in (A).
Question 55:
easyAssertion (A): If we use a battery across the primary of a step up transformer, no voltage is obtained across secondary.
Reason (R): Battery gives a steady current.
Assertion (A) is true. Transformers require a changing magnetic flux to induce voltage. Reason (R) is true; a battery provides DC current. This steady current causes constant flux, hence no induced emf in the secondary. (R) correctly explains (A).
Question 56:
easyAssertion (A): Only a change of magnetic flux with time, will maintain an induced current in the coil.
Reason (R): The presence of a large magnetic flux will maintain an induced current in the coil.
Assertion (A) is true based on Faraday's Law, which states that induced current is generated only by a change in magnetic flux. Reason (R) is false because a constant magnetic flux, regardless of its magnitude, does not induce a current.
Question 57:
easyAssertion (A): The mutual inductance of two coils is doubled if the self-inductance of the primary and secondary coil is doubled.
Reason (R): Mutual inductance \(M \propto \sqrt{L_1 L_2}\).
The mutual inductance is given by \(M = k \sqrt{L_1 L_2}\). If \(L_1\) and \(L_2\) are doubled, the new mutual inductance becomes \(M' = k \sqrt{(2L_1)(2L_2)} = 2k \sqrt{L_1 L_2} = 2M\). Thus, (A) is true and (R) correctly explains (A).
Question 58:
easyAssertion (A): If a charged particle is released from rest in a time varying magnetic field, it moves in a circle.
Reason (R): In a time varying magnetic field, conservative electric field is induced.
Assertion (A) is false; a charged particle released from rest experiences no magnetic force (\(F_B = q(v \times B)\)). While an induced electric field exists, it does not necessarily cause circular motion. Reason (R) is false; a time-varying magnetic field induces a non-conservative electric field.
Question 59:
easyAssertion (A): A system cannot have mutual inductance without having self inductance.
Reason (R): If mutual inductance of system is zero, its self-inductance must be zero.
Assertion (A) is true, as mutual inductance \(M\) depends on the self-inductances \(L_1\) and \(L_2\) (\(M \le \sqrt{L_1 L_2}\)). Reason (R) is false; if two coils are perfectly uncoupled (\(k=0\)), their mutual inductance is zero, but their self-inductances \(L_1\) and \(L_2\) can still be non-zero.
Question 60:
easyAssertion (A): At any instant, if the current through an inductor is zero, then the induced emf may not be zero.
Reason (R): An inductor tends to keep the flux constant.
Assertion (A) is true: The induced EMF is \(emf = -L \frac{dI}{dt}\). Even if \(I=0\) instantaneously, \(frac{dI}{dt}\) can be non-zero (e.g., during oscillation or switching). Reason (R) is true, describing Lenz's law. However, R is not the correct explanation for A, as A focuses on instantaneous values of \(I\) and \(frac{dI}{dt}\).