S is the surface of a lump of magnetic material. (a) Lines of B are necessarily continuous across S. (b) Some

NCERT Exemplar · Q5

Q. S is the surface of a lump of magnetic material.

(a) Lines of B are necessarily continuous across S.

(b) Some lines of B must be discontinuous across S.

(d) Lines of H cannot all be continuous across S.

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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 …

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. …

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Because $\nabla\cdot\vec{B}=0$ always, $\vec{B}$ -lines are continuous across the surface of a magnetic lump. But $\vec{H}$ is sourced by magnetic poles at that surface, so $\vec{H}$ -lines cannot all be continuous. Correct options: (a) and (d).

Concept understanding. The two magnetic fields obey

\nabla\cdot\vec{B}=0,\qquad \vec{B}=\mu_0(\vec{H}+\vec{M}),\qquad \nabla\cdot\vec{H}=-\nabla\cdot\vec{M}.

The first says $\vec{B}$ has no sources or sinks: its field lines are always closed loops and its normal component is continuous across any boundary,

B_{n,\text{in}} = B_{n,\text{out}}.

Inside the material $\vec{M}\neq 0$ ; just outside $\vec{M}=0$ . So $\vec{M}$ drops abruptly across $S$ , and this jump acts as an effective surface pole density $\sigma_m = \vec{M}\cdot\hat{n}$ . From $\vec{H}=\vec{B}/\mu_0-\vec{M}$ , the normal component of $\vec{H}$ jumps:

H_{n,\text{out}} - H_{n,\text{in}} = M_n = \vec{M}\cdot\hat{n}.

Testing each option.

(a) Lines of $\vec{B}$ are necessarily continuous across $S$ (normal component continuous; $\vec{B}$ -lines are closed). Correct. …

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Q.S is the surface of a lump of magnetic material.

Concept understanding — Magnetic Field Lines

Magnetic Field Lines

What a field line represents

Key properties (exam essentials)

Contrast with electric field lines

Uniform field …

Why this formula?

Magnetic Field Lines

The four defining rules

Method: Boundary Conditions for B and H at a Material Surface

Steps

Q.S is the surface of a lump of magnetic material.

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