Q.A rectangular loop of wire ABCD, of resistance R, lies in the plane of and close to an infinitely long straight wire carrying a current I(t) = I₀(1 − t/T) for 0 ≤ t ≤ T, with I = 0 for t > T (so the current falls linearly from I₀ at t = 0 to zero at t = T). The loop's sides AB and DC, each of length L₁, are parallel to the long wire; the nearer side DC is at perpendicular distance x from the wire and the farther side AB at perpendicular distance x + L₂ (loop breadth L₂). Find the total charge that passes through any given point of the loop during the time interval from t = 0 to t = T.

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. …