A magnetic field in a certain region is given by $\mathbf{B} = B_0 \cos(\omega t)\,\hat{k}$ and a coil of radius $a$ with resistance $R$ is placed in the $x$-$y$ plane with its centre at the origin in the magnetic field. Find the magnitude and the direction of the current at $(a, 0, 0)$ at $t = \dfrac{\pi}{2\omega}$, $t = \dfrac{\pi}{\omega}$ and $t = \dfrac{3\pi}{2\omega}$.

A magnetic field in a certain region is given by mathbfB = B₀ cos(ω t)hatk and a coil of radius a with resista

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\mathbf{B} = B_0 \cos(\omega t)\,\hat{k}

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Concept understanding — Electromagnetic Induction

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 $\vec{B}$ :

\Phi_B = \vec{B} \cdot \vec{A} = BA\cos\theta

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:

\mathcal{E} = -N\,\frac{d\Phi_B}{dt}

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

\mathcal{E} = Blv

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. …

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:

\Phi_B = \int \vec{B} \cdot d\vec{A} = BA\cos\theta

It can change three ways: by changing $B$ , by changing the area $A$ , or by rotating the loop (changing $\theta$ ).

Faraday's law

The induced EMF equals the rate of change of flux:

\mathcal{E} = -\frac{d\Phi_B}{dt}

For a coil of $N$ turns, $\mathcal{E} = -N\,\dfrac{d\Phi_B}{dt}$ . 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.

Lenz's law — the minus sign …

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The oscillating field gives $i(t)=\dfrac{B_0\pi a^2\omega}{R}\sin(\omega t)$ : magnitude $\dfrac{B_0\pi a^2\omega}{R}$ anticlockwise at $t=\pi/2\omega$ , zero at $t=\pi/\omega$ , and $\dfrac{B_0\pi a^2\omega}{R}$ clockwise at $t=3\pi/2\omega$ (as seen from $+z$ ).

Flux through the coil

The coil of radius $a$ lies in the $x$ – $y$ plane, so its area vector is $\vec A=\pi a^2\,\hat k$ , parallel to $\vec B=B_0\cos(\omega t)\,\hat k$ . Hence

\Phi = \vec B\cdot\vec A = \pi a^2 B_0\cos(\omega t).

Induced emf and current

By Faraday's law,

\varepsilon = -\frac{d\Phi}{dt} = -\pi a^2 B_0\big(-\omega\sin\omega t\big) = \pi a^2 B_0\,\omega\sin(\omega t),

and with the single resistance $R$ ,

i(t) = \frac{\varepsilon}{R} = \frac{\pi a^2 B_0\,\omega}{R}\,\sin(\omega t).

Magnitude and direction at each instant

Take the positive (anticlockwise, seen from $+z$ ) sense as the reference. The sign of $i$ already carries Lenz's law.

$t=\dfrac{\pi}{2\omega}$ : $\sin(\pi/2)=1$ , so $i=+\dfrac{\pi a^2 B_0\omega}{R}$ . Here the $+z$ flux is positive and decreasing, so the induced current reinforces it (produces $+z$ field) — anticlockwise.
$t=\dfrac{\pi}{\omega}$ : $\sin(\pi)=0$ , so $i=0$ . (The flux is at its extreme, momentarily not changing.) …

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Concept understanding — Electromagnetic Induction

Electromagnetic Induction

The Central Discovery

Magnetic Flux

Faraday's Law

Lenz's Law — the Minus Sign

Motional emf

Why this formula?

Electromagnetic Induction

Magnetic flux

Faraday's law

Lenz's law — the minus sign …

Flux through the coil

Induced emf and current

Magnitude and direction at each instant

Method: Finding Induced Current and Direction for a Sinusoidally Oscillating Field

Steps

Flux through the coil

Induced emf and current

Magnitude and direction at each instant

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