Consider the two idealized systems: (i) a parallel plate capacitor with large plates and small separation and

NCERT Exemplar · Q3

Q. Consider the two idealized systems:

(i) a parallel plate capacitor with large plates and small separation and

(ii) a long solenoid of length

L \gg R

, radius of cross-section. In

(i)

E

is ideally treated as a constant between plates and zero outside. In

(ii) magnetic field is constant inside the solenoid and zero outside. These idealised assumptions, however, contradict fundamental laws as below:

(a) case

(i) contradicts Gauss’s law for electrostatic fields.

(b) case

(ii) contradicts Gauss’s law for magnetic fields.

(i) agrees with ∮E.dl = 0.

(d) case

(ii) contradicts ∮H.dl = Ien

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Concept understanding — Magnetic Field Lines

Magnetic Field Lines

A magnet or a current-carrying wire fills the space around it with a magnetic field. We cannot see this field, so we picture it using magnetic field lines — continuous curves that map both the direction and the strength of the field at every point.

What a field line represents

The tangent to a field line at any point gives the direction of the magnetic field $\vec{B}$ there. If you place a tiny compass needle at that point, it aligns along the tangent, its north pole pointing the way the line runs. The density of the lines (how closely packed they are) represents the magnitude of $\vec{B}$ : crowded lines mean a strong field, widely spaced lines mean a weak field.

Key properties (exam essentials)

Important

Outside a magnet the lines run from the north pole to the south pole, but they are continuous closed loops — inside the magnet they run south to north, so every line closes on itself.
Two field lines never intersect. If they did, a compass at the crossing point would have to point in two directions at once, which is impossible.
Lines are crowded where the field is strong (near the poles) and spread out where it is weak.
They form smooth, continuous curves with no free ends.

The fact that magnetic field lines always close on themselves is deep: it means there are no isolated magnetic poles (monopoles). This is Gauss's law for magnetism:

\oint \vec{B} \cdot d\vec{A} = 0

The net magnetic flux through any closed surface is zero — every line that enters the surface also leaves it.

Contrast with electric field lines

Note

Electric field lines start on positive charges and end on negative charges — they are open curves. Magnetic field lines have no such start or end; they are always closed loops. This single difference reflects that isolated electric charges exist, but isolated magnetic poles do not.

Uniform field

When field lines are parallel, equally spaced, and straight, the field is uniform — the same magnitude and direction everywhere. The region deep inside a long solenoid, or between the flat poles of a large magnet, is very nearly uniform.

Why this matters

Field-line diagrams let you read a field at a glance: where it is strong, which way it points, and whether it is uniform. They underpin magnetic flux, $\Phi_B = \vec{B}\cdot\vec{A}$ , Gauss's law for magnetism, and the study of electromagnetic induction. Sketching the correct pattern for a bar magnet, a straight wire (concentric circles around it), and a solenoid (uniform inside, bar-magnet-like outside) is a standard exam skill.

The properties of magnetic field lines, including why they always form closed loops, are introduced in the NCERT Class 12 Physics chapter on magnetism and matter, and "properties of magnetic field lines class 12 physics important questions" is a common CBSE board short-answer topic. This closed-loop explanation, linked to the absence of magnetic monopoles, matches the NCERT-prescribed reasoning.

Why this formula?

Magnetic Field Lines

A magnetic field line is an imaginary curve we draw to picture an invisible field. Its purpose is to encode two things at once: the direction of the field $\vec{B}$ (the tangent to the line at any point) and the strength of the field (how densely the lines are packed). They are a map, not physical objects.

The four defining rules

1. The tangent gives the field direction. At every point, $\vec{B}$ points along the tangent to the field line through that point. A compass needle placed on the line aligns with it.

2. Field lines form closed loops. Unlike electric field lines, which begin and end on charges, magnetic field lines never start or stop. This is Gauss's law for magnetism:

\oint \vec{B} \cdot d\vec{A} = 0

The net flux through any closed surface is zero because isolated magnetic poles (monopoles) do not exist — every north pole is paired with a south pole. So for a bar magnet the lines emerge from the north pole outside, curve around to the south pole, and continue through the interior of the magnet back to the north, closing the loop.

3. Field lines never cross. If two lines crossed, the tangent — and hence $\vec{B}$ — would have two directions at that point. Since the field has a single, unique direction everywhere, crossings are impossible.

4. Density represents strength. Where lines are crowded, the field is strong; where they spread out, it is weak. The flux through a small perpendicular area $dA$ is $d\Phi_B = B\,dA$ , so packing more lines through the same area means a larger $B$ (as near the poles of a magnet).

A quick picture

For a straight wire carrying current $I$ , the field lines are concentric circles around the wire, tightly spaced close to the wire and spreading out farther away. Grip the wire with your right hand, thumb along the current — your curled fingers trace the direction of the loops. This is exactly the closed-loop, non-crossing, density-encodes-strength behaviour the rules above describe.

Sharp cut-off fields are unphysical. The solenoid idealisation ( $B$ uniform inside, exactly zero outside) violates Gauss's law for magnetism and Ampère's law. Correct options: (b) and (d).

Concept understanding.

Case (i) — capacitor: Taking $E$ uniform between the plates and zero outside is consistent with Gauss's law (a pill-box's flux still equals $q_{enc}/\varepsilon_0$ ), so (a) is wrong. But a loop running from inside (where $E\neq0$ ) to outside (where $E=0$ ) would give $\oint \vec{E}\cdot d\vec{l}\neq 0$ , contradicting the electrostatic condition $\oint\vec{E}\cdot d\vec{l}=0$ . Hence (c) is wrong — case (i) does not agree with that law, it violates it (which is exactly why fringing fields must exist).

Case (ii) — solenoid: If $B$ were exactly zero outside, a closed Gaussian surface straddling the end face would have flux entering (inside, $B\neq0$ ) with none leaving, giving net $\oint\vec{B}\cdot d\vec{A}\neq0$ and violating Gauss's law for magnetism $\nabla\cdot\vec B=0$ (field lines must close). So (b) is correct. Similarly, an Amperian loop encircling the solenoid from outside encloses the net winding current, yet $B=0$ there would give $\oint\vec{H}\cdot d\vec{l}=0$ , contradicting $\oint\vec{H}\cdot d\vec{l}=I_{en}$ . So (d) is correct.

✓Final answer

\oint\vec H\cdot d\vec l=I_{en}

Method: Testing an Idealized Field Against Gauss's and Ampere's Laws

Use this method whenever a problem presents an idealized field (sharp cutoff at a boundary: uniform inside a bounded region, exactly zero immediately outside) and asks which fundamental law that idealization actually violates.

Steps

Step 1: State the idealization precisely

Write down exactly what is assumed uniform and where it is assumed to drop to zero — e.g. "field is constant inside a bounded region and exactly zero immediately outside it, with no transition region."

Step 2: Test against Gauss's law for the relevant field

For $\vec{E}$ : a Gaussian surface drawn entirely inside, or entirely outside, the region gives flux equal to enclosed charge over $\varepsilon_0$ — a sharp uniform-then-zero field is often still consistent with this, since it correctly reflects the enclosed charge. For $\vec{B}$ : a Gaussian surface straddling the boundary (e.g. a pillbox capping the end of a solenoid) has flux entering through one face with none leaving anywhere else if $B=0$ just outside — that gives a nonzero net flux, which violates $\nabla\cdot\vec{B}=0$ , since real $\vec{B}$ field lines must always close on themselves.

Step 3: Test against the relevant circulation law

For electrostatics: draw a closed loop running from a region of nonzero $\vec{E}$ into a region of zero $\vec{E}$ . The electrostatic condition $\oint\vec{E}\cdot d\vec{l}=0$ must hold for ANY closed loop — if the sharp idealization makes this loop integral come out nonzero, it violates that condition (this is exactly why real fringing fields must exist at the edges). For magnetism: an Amperian loop that encloses the true winding current but passes through the idealized "zero field" region outside gives $\oint\vec{H}\cdot d\vec{l}=0$ from that region's contribution, contradicting $\oint\vec{H}\cdot d\vec{l}=I_{en}$ when the enclosed current is genuinely nonzero.

Step 4 (Applying to this problem): Separate "is consistent with" from "agrees with"

Distinguish a law the idealization happens to satisfy on a bulk Gaussian surface from a circulation-type law it silently violates through the artificial sharp edge. A field can pass one test and fail the other — always check flux-type and circulation-type laws separately rather than assuming one violation implies the other.

Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

KCET 2020Set A-11 markMCQ
$Q. In a permanent magnet at room temperature (A) Magnetic moment of each molecule is zero. (B) The individual molecules have non-zero magnetic moment which are all perfectly aligned. (C) Domains are partially aligned. (D) Domains are all perfectly aligned.$
›Reveal solutionSolution
A permanent magnet retains its magnetism because its microscopic magnetic domains are partially aligned, not perfectly aligned — perfect alignment would require absolute zero or an infinitely strong field.

The key to this question lies in understanding magnetic domains — tiny regions within a ferromagnetic material where the magnetic moments of atoms are already aligned. In an unmagnetised piece of iron, these domains point in random directions, so the net magnetic field cancels out. When you magnetise it (by placing it in an external field or stroking it with another magnet), the domains that happen to be aligned with the field grow at the expense of others, and some domains rotate to align partially. The result is a net magnetisation that persists even after the external field is removed — that’s what makes it a permanent magnet.

Now, at room temperature, thermal energy constantly jostles the atoms. This thermal agitation prevents perfect alignment of all domains. If every single domain were perfectly aligned, the material would be magnetically saturated — a state that requires either an extremely strong external field or cooling to near absolute zero. A permanent magnet sitting on your desk is far from that condition.

Let’s examine each option carefully.

Option (A): “Magnetic moment of each molecule is zero.”

This is false. In a ferromagnetic material like iron, each atom (or molecule) has a permanent magnetic moment due to unpaired electrons in its d- or f-orbitals. These moments are never zero — they’re the very reason the material can be magnetic at all. What changes is the alignment of these moments, not their existence.

Option (B): “The individual molecules have non-zero magnetic moment which are all perfectly aligned.”

This describes a perfectly saturated magnet. But at room temperature, thermal vibrations prevent perfect alignment. Even the strongest permanent magnets (like neodymium) have only about 70–90% of their domains aligned. Perfect alignment would mean every atomic moment points exactly the same way — that’s impossible at room temperature because thermal energy randomises some fraction of them.

Option (C): “Domains are partially aligned.”

This is correct. In a permanent magnet, a majority of domains are aligned in one direction, but some remain misaligned or only partially rotated. The net magnetisation is the vector sum of all domain magnetisations, which is less than the saturation value. This partial alignment is stable at room temperature due to the material’s high coercivity (resistance to demagnetisation).

Option (D): “Domains are all perfectly aligned.”

This is the same as option (B) but at the domain level. Perfect domain alignment would mean the material is a single domain with all atomic moments parallel — a state that requires extreme conditions. At room temperature, domain walls exist, and domains are never perfectly aligned.

Watch out

A common mistake is to think that a permanent magnet has all its domains aligned. In reality, only a majority are aligned — the rest are still random or partially aligned. If all domains were perfectly aligned, the magnet would be at saturation, which is not the case for ordinary permanent magnets at room temperature.

Tip

Think of domains like a crowd of people in a stadium. In an unmagnetised material, everyone faces random directions. When you magnetise it, you ask everyone to face north — but some people are stubborn or jostled by neighbours, so only about 70% face north. That’s a permanent magnet: partial alignment that persists.

✓Final answer
$The correct option is (C) — domains are partially aligned.$
KCET 2019Set A-11 markMCQ
$Q. In a permanent magnet at room temperature (A) magnetic moment of each molecule is zero. (B) the individual molecules have non zero magnetic moment which are all perfectly aligned. (C) domains are partially aligned. (D) domains are all perfectly aligned.$
›Reveal solutionSolution
In a permanent magnet at room temperature, the material is ferromagnetic and its magnetic domains are partially aligned — not perfectly aligned — because thermal agitation prevents perfect ordering. The correct option is (C).

The key to this question lies in understanding how permanent magnets work at the microscopic level. A permanent magnet is made of a ferromagnetic material (like iron, nickel, or cobalt). In such materials, individual atoms have permanent magnetic moments due to unpaired electrons. However, these moments do not act independently — they strongly interact with neighbours, causing large groups of atoms (about $10^{12}$ to $10^{15}$ atoms) to align their moments spontaneously. These groups are called domains.

Each domain is like a tiny magnet with all its atomic moments aligned. But in an unmagnetised piece, the domains themselves point in random directions, so the net magnetic field cancels out. To make a permanent magnet, we apply an external magnetic field, which causes domains that are aligned with the field to grow at the expense of others, and also rotates domain magnetisation. After the field is removed, some of this alignment remains — the material is now a permanent magnet.

Now, at room temperature, thermal energy constantly jostles the atoms. This thermal agitation prevents the domains from being perfectly aligned. Some domains will still point in directions that are not exactly along the net magnetisation. So the alignment is partial, not perfect.

Let's examine each option:

Option (A): "Magnetic moment of each molecule is zero."

This is false. In a ferromagnetic material, each atom (or molecule) has a non-zero magnetic moment due to unpaired electrons. If the moments were zero, there would be no magnetism at all.

Option (B): "The individual molecules have non zero magnetic moment which are all perfectly aligned."

This describes a perfectly ordered state. But at room temperature, thermal energy disrupts perfect alignment. Perfect alignment occurs only at absolute zero ( $0\ \text{K}$ ). At room temperature, the alignment is not perfect — it's partial.

Option (C): "Domains are partially aligned."

This is correct. In a permanent magnet, the domains are not all pointing in the same direction. Some domains are aligned with the net magnetisation, others are partially misaligned due to thermal effects and material imperfections. The net magnetisation is the vector sum of all domain magnetisations, which is less than the saturation value.

Option (D): "Domains are all perfectly aligned."

This would mean the material is magnetically saturated — all domains point exactly the same way. That happens only under a very strong external field, and even then, at room temperature, thermal fluctuations prevent perfect alignment. Once the field is removed, the alignment relaxes to a partially aligned state.

Watch out

A common mistake is to think that a permanent magnet has all its domains perfectly aligned. In reality, perfect alignment is an idealisation that occurs only at absolute zero or under an infinitely strong field. At room temperature, thermal energy always causes some misalignment.

Tip

Think of domains like a crowd of people all facing roughly the same direction, but each person is slightly turned by jostling. The net direction is clear, but no one is perfectly aligned. That's a permanent magnet at room temperature.

✓Final answer
$The correct option is (C) — domains are partially aligned.$

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Q.Consider the two idealized systems:

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