Q.An infinitely long straight wire lies in the plane of the paper and carries a current I(t) whose rate of change dI/dt = λ is constant. A rectangular loop ABCD of wire, of resistance R, lies in the same plane with its two sides AB and DC (each of length l) parallel to the long wire. The nearer side DC is at a perpendicular distance x₀ from the wire and the farther side AB at a perpendicular distance x₀ + r (so the loop's breadth measured away from the wire is r). Find the magnitude of the current induced in the loop.

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. …