Assertion (A): A changing magnetic flux induces an electric field.
Reason (R): An inductor always tends to keep the flux constant.
Faraday's Law states a changing magnetic flux induces an electric field. Inductors oppose change in flux, but don't keep it constant. Hence, Assertion is true and Reason is false.
Assertion (A): When a circuit having large inductance is switched off sparking occurs at the switch.
Reason (R): Emf induced in an inductor is given by \( |\text{E}| = \text{L} |\frac{\text{di}}{\text{dt}}| \).
When a circuit with large inductance is switched off, \( \text{di/dt} \) is very large. This induces a large \( \text{EMF} = \text{L di/dt} \) across the inductor, causing sparking.
Assertion (A): A metal ring is kept on a cardboard on top of a fixed current carrying solenoid. If current in the solenoid is switched off, the upward reaction of card board on the ring will increase.
Reason (R): Induced current in the ring will be in the same direction as in the solenoid.
When solenoid current is switched off, flux decreases. Induced current in the ring flows in the same direction as solenoid current (Lenz's Law), causing attraction. This increases the upward reaction.
Assertion (A): If a cylindrical bar magnet is dropped through a metallic pipe, it takes more time to come down a similar unmagnetised cylindrical iron bar dropped through the same metallic pipe.
Reason (R): For the magnet, eddy currents are produced in the metallic pipe.
As the magnet falls, changing flux induces eddy currents in the pipe. These currents oppose the magnet's motion (Lenz's Law), creating a retarding force that slows it down.
Assertion (A): The self inductance of a solenoid can be increased by decreasing length if number of turns are fixed.
Reason (R): Self inductance of a solenoid is directly proportional to current passing through it.
Self inductance of a solenoid is given by \(L = \frac{\mu_0 N^2 A}{l}\). So, Assertion (A) is true as \(L\) is inversely proportional to \(l\). Self inductance \(L\) is a property of the coil's geometry and material, not dependent on current. So, Reason (R) is false. Thus, (A) is true but (R) is false.
Assertion (A): If a coil carrying current in counter clockwise direction moves towards another stationary coil in the same plane, current induced in stationary coil will be counter clock wise.
Reason (R): Mutual induction between coils is independent of direction of current.
Assertion (A) is false: A counter-clockwise current in the moving coil creates a magnetic field pointing out of the page. As it moves towards the stationary coil, the outward flux through the stationary coil increases. By Lenz's law, the induced current will oppose this change by creating an inward magnetic field, which requires a clockwise current.
Reason (R) is true: Mutual inductance \(M\) is a geometric property of the coils and is independent of the direction of current. Since (A) is false and (R) is true, none of the provided options accurately describe the situation.
Assertion (A): If a bar magnet is moved towards a conducting coil in a direction perpendicular to the plane of coil, the work done in moving the magnet will be more if it is moved faster rather than slower.
Reason (R): If the magnet is moved at a faster rate towards the circular coil, then the induced current in the circular coil is more.
Assertion (A) is true: Faster movement causes a greater rate of change of magnetic flux (\(\frac{d\Phi}{dt}\)), leading to a larger induced emf (\(epsilon = -\frac{d\Phi}{dt}\)) and current (\(I = \frac{\epsilon}{R}\)). This results in a stronger opposing force (Lenz's Law), requiring more work. Reason (R) is true: Induced current is directly proportional to the rate of change of flux. Reason (R) correctly explains Assertion (A).
Assertion (A): The probability of burn out of a dc motor is maximum, when the motor is just switched on.
Reason (R): No back emf is developed in the armature of dc motor, when it is just switched on.
Assertion (A) is true: When a DC motor starts, its speed is zero, thus the back EMF (\(epsilon_b\)) is zero. This leads to the maximum current (\(I = \frac{V - \epsilon_b}{R_a}\)) drawn from the supply, which can cause burnout. Reason (R) is true: Back EMF is proportional to the motor's angular speed (\(epsilon_b = k\Phi\omega\)), so it is zero at startup (\(omega = 0\)). Reason (R) correctly explains Assertion (A).
Assertion (A): If a closed loop is kept in a space having time varying magnetic field, emf is always induced in the loop.
Reason (R): Induced emf in the loop is conservative in nature.
Assertion (A) is true: By Faraday's Law of electromagnetic induction, a time-varying magnetic flux (\(\Phi\)) through a closed loop will induce an electromotive force (\(epsilon = -\frac{d\Phi}{dt}\)). Reason (R) is false: The induced electric field and thus the induced emf, arising from a changing magnetic flux, are non-conservative in nature. If they were conservative, the line integral (emf) would be zero.
Assertion (A): If a magnet is allowed to fall co-axially through a long copper tube, its acceleration decreases with time.
Reason (R): The direction of force on magnet doesn’t change when it pass through a tube.
Assertion (A) is true: As the magnet falls, eddy currents are induced in the copper tube. By Lenz's law, these currents create a magnetic field that produces an upward braking force opposing the magnet's motion. This opposing force increases with the magnet's speed, thus reducing the net downward force and acceleration. Reason (R) is true: As the magnet falls downwards, the induced magnetic force always opposes the motion, meaning its direction is always upwards. However, (R) does not explain the *decrease* in acceleration, which is due to the *increasing magnitude* of the opposing force with speed. So, (R) is not the correct explanation of (A).