In which of the following cases emf would be induced

The correct answer is Option (3) → Whenever the magnetic flux linked with a circuit changes even if the circuit is open.

By Faraday's law, an emf is induced whenever the magnetic flux linked with a circuit changes, \( \varepsilon = -d\Phi_B/dt \). A closed path is needed only for a current to flow, not for the emf to appear.

Option (1) When a coil and a magnet move with the same velocity in the same direction. — Incorrect. There is no relative motion between the coil and the magnet, so the flux through the coil does not change.

Option (2) When a magnet is rotated about its own axis. — Incorrect. The field of a magnet is symmetric about its own axis, so rotating it about that axis does not change the flux through a nearby circuit.

Option (4) When a closed loop moves totally inside a uniform magnetic field. — Incorrect. Inside a uniform field the flux through the loop stays \( BA \) as it moves, so it does not change and no emf is induced.

NCERT (2026-27):
Faraday’s Law of Induction – “… The magnitude of the induced emf in a circuit is equal to the time rate of change of magnetic flux through the circuit.”

Lenz’s Law and Conservation of Energy – “… What will happen if an open circuit is used in place of the closed loop in the above example? In this case too, an emf is induced across the open ends of the circuit.”