Consider a metallic pipe with an inner radius of $1\ \text{cm}$. If a cylindrical bar magnet of radius $0.8\ \text{cm}$ is dropped through the pipe, it takes more time to come down than it takes for a similar unmagnetised cylindrical iron bar dropped through the metallic pipe. Explain.

Consider a metallic pipe with an inner radius of 1cm. If a cylindrical bar magnet of radius 0.8cm is dropped t

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1\ \text{cm}

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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 key idea is Lenz’s law: the falling magnet induces eddy currents in the metallic pipe, which create an opposing magnetic field that slows the magnet down. An unmagnetised iron bar does not produce a changing magnetic flux, so no eddy currents are induced and it falls freely. Hence the magnet takes longer.

Why does the magnet slow down?

The phenomenon is a beautiful demonstration of electromagnetic induction and Lenz’s law. When a magnet moves through a conductor, the changing magnetic field induces circulating currents (called eddy currents) in the conductor. These currents themselves produce a magnetic field that opposes the motion that caused them — that’s Lenz’s law in action. The result is a braking force on the magnet.

An unmagnetised iron bar, on the other hand, produces no magnetic field of its own. As it falls through the pipe, the magnetic flux through any loop in the pipe does not change — so no induced currents, no opposing force, and it falls under gravity alone.

Let’s walk through the details step by step.

What happens when the magnet falls? The cylindrical bar magnet has its own magnetic field. As it moves downward through the metallic pipe, the magnetic flux through any imaginary loop in the pipe wall changes with time. By Faraday’s law, the induced emf in a loop is

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

where $\Phi_B$ is the magnetic flux through that loop. The negative sign is Lenz’s law — the induced current will oppose the change in flux.

Eddy currents are set up in the pipe wall.

The pipe is a conductor (say copper or aluminium). The induced emf drives eddy currents — closed loops of current swirling within the metal. These currents are not negligible; the pipe’s low resistance allows large currents to flow.
These eddy currents produce their own magnetic field.

According to Lenz’s law, the magnetic field of the eddy currents opposes the motion of the magnet. If the magnet is falling downward, the induced field above it repels it (like poles repel) and the induced field below it attracts it (opposite poles attract). Both effects act upward — a braking force.

Watch out

A common mistake is to think the pipe is attracted to the magnet. Actually, the eddy currents create a force that opposes the motion, not the position. So the force is always upward while the magnet falls downward, regardless of which pole is leading.
The braking force reduces acceleration.

The net downward force on the magnet is

F_{\text{net}} = mg - F_{\text{braking}}

where $F_{\text{braking}}$ depends on the speed of the magnet (faster motion → larger $d\Phi/dt$ → stronger eddy currents → larger opposing force). Eventually, the magnet may even reach a terminal velocity where $F_{\text{braking}} = mg$ , and it falls at constant speed — much slower than free fall.

Now consider the unmagnetised iron bar. …

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Q.Consider a metallic pipe with an inner radius of $1\ \text{cm}$ . If a cylindrical bar magnet of radius $0.8\ \text{cm}$ is dropped through the pipe, it takes more time to come down than it takes for a similar unmagnetised cylindrical iron bar dropped through the metallic pipe. Explain.

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 …

Why does the magnet slow down?

Method: Recognizing and Analyzing Eddy-Current (Magnetic) Braking

Steps

Why does the magnet slow down?

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