Q.The magnetic flux passing through a coil perpendicular to its plane is a function of time and is given by $\Phi_B = (2t^3 + 4t^2 + 8t + 8)$ Wb. If the resistance of the coil is 5 $\Omega$, determine the induced current through the coil at a time t = 3 second.

Electromagnetic Induction

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 $\Phi_B$ through a surface of area $A$ in a field $\vec{B}$:

where $\theta$ is the angle between $\vec{B}$ and the area's normal. Its SI unit is the weber (Wb), where $1\ \text{Wb} = 1\ \text{T}\cdot\text{m}^2$.

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 $\theta$ 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:

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

Here the emf arises because the free charges in the rod experience a magnetic force $q\vec{v}\times\vec{B}$, which drives them along the rod.

Where It Leads

Once a coil's own changing current induces an emf in itself, we call it self-inductance ($L$); when one coil's changing current induces emf in a neighbour, that is mutual inductance ($M$). Both are direct consequences of Faraday's law. Rotate a coil steadily in a magnetic field and the sinusoidal emf it produces is exactly the alternating voltage that runs the AC circuits studied in this chapter.

Faraday's and Lenz's laws of electromagnetic induction form one of the highest-weightage chapters in NCERT Class 12 Physics, tested extensively in CBSE boards, JEE Main and NEET. Anyone searching "Faraday's law of electromagnetic induction formula and examples class 12 physics" will find this changing-flux explanation, including the motional emf case, is exactly how NCERT presents the chapter.