The self inductance $L$ of a solenoid of length $l$ and area of cross-section $A$, with a fixed number of turns $N$, increases as (a) l and A increase. (b) l decreases and A increases. (c) l increases and A decreases. (d) both l and A decrease.

The self inductance L of a solenoid of length l and area of cross-section A, with a fixed number of turns N, i

NCERT Exemplar · Q6

L

(a) l and A increase.

(b) l decreases and A increases.

(d) both l and A decrease.

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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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Since $L=\mu_0 N^2 A/l$ for a solenoid, at fixed $N$ the inductance grows when the length $l$ decreases and the area $A$ increases — that combined condition is the correct choice.

Deriving the formula

For a long solenoid of length $l$ with $N$ turns carrying current $I$ , the interior field is

B = \mu_0\,\frac{N}{l}\,I .

The flux through one turn of area $A$ is $\Phi_1 = BA = \mu_0\dfrac{N}{l}IA$ , and the total flux linkage over $N$ turns is

N\Phi_1 = \mu_0\,\frac{N^2}{l}\,I A .

By definition $L = N\Phi_1/I$ , so

\boxed{\,L = \frac{\mu_0 N^2 A}{l}\,}.

Applying the fixed- $N$ condition

The question fixes $N$ , so treat $\mu_0$ and $N$ as constants and read off the two remaining dependences:

Area: $L \propto A$ . Widening the solenoid lets each turn enclose more flux, so $L$ rises when $A$ increases. …

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Q.The self inductance $L$ of a solenoid of length $l$ and area of cross-section $A$ , with a fixed number of turns $N$ , increases as

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 …

Deriving the formula

Applying the fixed- $N$ condition

Method: Reading Off How a Quantity Changes From Its Own Formula (Proportionality Reasoning)

Steps

Q.The self inductance LLL of a solenoid of length lll and area of cross-section AAA, with a fixed number of turns NNN, increases as

Deriving the formula

Applying the fixed-NNN condition

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Q.The self inductance $L$ of a solenoid of length $l$ and area of cross-section $A$ , with a fixed number of turns $N$ , increases as

Applying the fixed- $N$ condition

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