Electromagnetic induction is the phenomenon in which a changing magnetic flux through a circuit produces an electromotive force (emf) — and hence a current, if the circuit is closed. It is the single idea behind generators, transformers, inductors, and the entire AC power grid.
The Central Discovery
Michael Faraday found (1831) that a current is induced in a coil not when a magnet sits still near it, but only while the magnet moves — that is, only while the magnetic flux linked with the coil is changing. A steady magnet, however strong, induces nothing.
Magnetic Flux
The key quantity is magnetic flux ΦB through a surface of area A in a field B:
ΦB=B⋅A=BAcosθ
where θ is the angle between B and the area's normal. Its SI unit is the weber (Wb), where 1Wb=1T⋅m2.
Flux can change in three distinct ways, and any of them induces an emf:
the field strength B changes,
the area A of the loop changes,
the orientation θ changes (a coil rotating in a field — the basis of the generator).
Faraday's Law
The induced emf equals the negative rate of change of flux. For a coil of N turns:
E=−NdtdΦB
The faster the flux changes, the larger the emf. This is why a magnet dropped quickly through a coil gives a bigger deflection than one moved slowly.
Lenz's Law — the Minus Sign
The negative sign expresses Lenz's law: the induced current flows in the direction that opposes the change producing it. Push a magnet's north pole toward a coil, and the coil's near face becomes a north pole to repel it; pull it away, and the face becomes a south pole to attract it. This is simply energy conservation — you must do work against this opposition, and that work becomes the electrical energy of the induced current.
Motional emf
A special, very useful case: a conducting rod of length l moving with speed v perpendicular to a field B sweeps out area and develops an emf
E=Blv
Here the emf arises because the free charges in the rod experience a magnetic force qv×B, which drives them along the rod. …
Why this formula?
Electromagnetic Induction
Electromagnetic induction is the effect discovered by Faraday: a changing magnetic flux through a circuit drives an induced EMF (and hence a current). The key word is changing — a steady field, however strong, induces nothing.
Magnetic flux
Flux measures how many field lines thread a surface bounded by the loop:
ΦB=∫B⋅dA=BAcosθ
It can change three ways: by changing B, by changing the area A, or by rotating the loop (changing θ).
Faraday's law
The induced EMF equals the rate of change of flux:
E=−dtdΦB
For a coil of N turns, E=−NdtdΦB. The EMF depends on how fast the flux changes, not on the flux itself — a slow change gives a small EMF, a rapid change a large one.
A plate heats through Joule dissipation I2R, so any conduction current — direct (a) or direct/alternating (d) — heats it. A time-varying magnetic field induces eddy currents (ε=−dΦB/dt=0) that also heat it (b). A field that varies only in space, not in time, gives no changing f …
A plate heats whenever a current dissipates energy in it (I2R) — from a direct or alternating conduction current, or from eddy currents induced by a time-varying magnetic field. A steady (only space-varying) field on a stationary plate induces nothing. Correct options: (a), (b), (d).
Concept understanding. Heating in a conductor is Joule heating: power P=I2R. So the plate heats whenever a current flows in it, whatever the source of that current.
Why (a) and (d): A direct current through the plate dissipates I2R and heats it (a). More generally, any conduction current — direct or alternating — does the same, so (d) is also true. …
Method: Reasoning Pattern for "What Causes This Heating/Effect?" Multi-Select Questions
Use this systematic elimination approach for any conceptual multi-select MCQ built around a single governing physical requirement — here, Joule heating requires an actual current, so the question reduces to "does each scenario produce a current?"
Steps
Step 1: Identify the one physical mechanism the question is really testing
Heating in a conductor is always P=I2R — so the real question hiding behind every option is simply: does this scenario produce a current in the plate?
Step 2: Check each option against the relevant law
For an option describing a conduction current directly (direct or alternating), current obviously flows — Ohm's law applies trivially. For an option describing a magnetic field instead of a direct current, you must instead check Faraday's law: does the flux through (or linking) the plate actually change with time? Only a genuinely time-varying field induces eddy currents.
Step 3: Eliminate options with no time variation …