Bar magnet and magnetic field lines

Snap a bar magnet in half hoping for a lone north pole, and nature refuses every time — you get two smaller magnets, each stubbornly carrying its own N and S. Sprinkle iron filings around either piece and they arrange themselves into the same looping pattern: curves that leave a north pole, arc through the surrounding space, and dive back into a south pole, never crossing, never breaking, never simply stopping in mid-air. That picture — the field-line map of a bar magnet — is the visual grammar the entire chapter "Magnetism and Matter" is written in. A bar magnet is the simplest magnetic dipole science has: two poles of equal strength $q_m$ , a fixed separation, and a single vector, the magnetic dipole moment $\vec m$ , that captures how strong it is and which way it points. Every more complicated magnetic source in this chapter — a current loop, a solenoid, a spinning Earth — will eventually be compared back to this one object, because a bar magnet is what a compass needle is built to sense and what field lines are built to describe. This article builds that language from the ground up: why isolated magnetic poles (monopoles) cannot exist, what field lines are actually allowed to do and not do, how the dipole moment $\vec m = q_m(2\vec l)$ is defined and measured, and how to read a field-line sketch to find pole locations and compare field strengths — exactly the skills a plotting-compass classroom experiment and a CBSE board question both test.

Key takeaways

A bar magnet has two poles of equal pole strength $q_m$ , separated by the magnetic length $2l$ ; poles always come in N-S pairs.
Magnetic monopoles do not exist — cutting a magnet only creates two smaller complete dipoles, never an isolated pole.
The magnetic dipole moment is $\vec m = q_m(2\vec l)$ , directed from S to N, SI unit $\text{A}\,\cdot\,\text{m}^2$ .
Magnetic field lines leave the N pole, loop through outside space, enter the S pole, and continue inside the magnet from S back to N — they always close on themselves.
Field lines never intersect (a single point cannot have two field directions), and their crowding shows field strength — denser lines mean a stronger $\vec B$ .
Reading a field-line diagram lets you locate poles (lines diverge from N, converge into S) and rank magnets by strength (tighter spacing near the centre means a bigger $m$ ).
Key idea: a bar magnet is a permanent pair of equal, inseparable poles whose dipole moment $\vec m = q_m(2\vec l)$ fixes the closed-loop field-line pattern — dense near strong poles, always leaving N and entering S outside, and never crossing or ending.

The core insight

A bar magnet's two poles, of pole strength $q_m$ and separation $2l$ , define a dipole moment $\vec m = q_m(2\vec l)$ pointing S to N, and the magnetic field lines it produces are closed loops — leaving N and entering S outside the magnet, denser where $B$ is larger, and never crossing or ending — because isolated magnetic poles never exist.

Where it fits in the syllabus

This topic opens Unit V (Magnetism and Matter) of CBSE Class 12 Physics, immediately after Moving Charges and Magnetism, where the magnetic dipole moment of a current loop was first defined. It supplies the vocabulary — poles, dipole moment, field lines — used for the rest of the chapter: the bar magnet as an equivalent solenoid, the axial and equatorial field formulae, torque on a dipole, and the classification of magnetic materials. Everything here is squarely within the prescribed 2025-26 syllabus and is qualitative-to-light-quantitative: no vector calculus, no Gauss's law for magnetism, and no Earth-magnetism formulae (declination, dip) are expected, since that sub-topic has been removed from the current board prescription.

The physical picture

A bar magnet is modelled as two point-like poles: a north pole of strength $+q_m$ and a south pole of strength $-q_m$ , fixed a distance $2l$ apart along the magnet's axis, where $q_m$ is measured in ampere-metres (A·m) and is always positive by convention for the pole strength magnitude. Unlike electric charge, a magnetic pole has never been isolated in any experiment — split a magnet anywhere along its length and each half instantly grows its own opposite pole at the cut, becoming a complete, smaller dipole. This experimental fact is the reason magnetic monopoles are treated as non-existent in this chapter. The strength and orientation of a bar magnet as a magnetic source is captured in a single vector, the magnetic dipole moment:

\vec m = q_m\,(2\vec l)

where: $q_m$ is the pole strength (A·m), $2\vec l$ is the vector from the S pole to the N pole with magnitude equal to the magnetic length (m), and $\vec m$ therefore has SI unit A·m² and points from S to N along the magnet's axis. Physically, $m$ is the single number that will later decide how strong the magnet's field is at any external point — a magnet with a bigger $m$ produces a bigger $B$ at the same distance, exactly as a bigger current or a bigger loop area increases the dipole moment of a current loop. Magnetic field lines are the picture used to represent $\vec B$ everywhere in space around the magnet, and they obey four rules that a CBSE answer must state precisely:

The tangent to a field line at any point gives the direction of $\vec B$ at that point.
The number of lines crossing a unit area perpendicular to them (their crowding) represents the magnitude of $B$ — closely packed lines mean a strong field, widely spaced lines mean a weak one.
Two field lines can never intersect, because a crossing point would have to carry two different field directions at once, which is impossible for a single-valued vector field.
Field lines emerge from the N pole and enter the S pole outside the magnet, but every line is a closed loop — inside the material of the magnet the same line continues from S back to N, so no line ever simply starts or stops in empty space.

That last rule is the one place magnetic field lines differ sharply from electric field lines. Electric field lines are allowed to begin and end on isolated positive and negative charges, because isolated electric charges exist. Magnetic field lines have no such luxury: since isolated magnetic poles never exist, a magnetic field line has nowhere to terminate, so it must always close on itself, running through the magnet's interior to complete the loop.

Figure

A bar magnet with N pole on the right and S pole on the left, curved field lines emerging from N looping outward and entering S, with additional straight lines drawn inside the magnet from S to N to close every loop, and tick marks showing the lines closer together near the poles than far away

Derivation ABar magnet's field lines and dipole moment

The goal of this derivation is to connect a bar magnet's two physical properties, pole strength $q_m$ and magnetic length $2l$ , into the single vector quantity $\vec m$ that the syllabus uses for every later field formula, and to justify why the resulting field-line pattern must consist of closed loops rather than lines that start or stop in free space — a result that follows purely from the non-existence of isolated poles, without any calculus.

Step 1. Model the magnet as two poles. Represent the bar magnet by a north pole of strength $+q_m$ and a south pole of strength $-q_m$ , held a fixed distance $2l$ apart along the magnet's own axis, with the vector $2\vec l$ drawn from the S pole to the N pole. This is the working model the syllabus asks students to use — an analogue of the electric dipole's two opposite point charges.

Step 2. Define the dipole moment by direct analogy. Just as an electric dipole has moment $\vec p = q(2\vec l)$ from the electric dipole chapter, the magnetic dipole moment is defined as $\vec m = q_m(2\vec l)$ . Because $2\vec l$ points from S to N, $\vec m$ also points from S to N, and its magnitude $m = q_m \times 2l$ has SI unit A·m² (ampere-metre-squared), since $q_m$ is in A·m and $2l$ is in m.

Step 3. Argue that isolated poles cannot exist. Suppose, for contradiction, that a magnet could be cut so that one piece carried only an N pole with no compensating S pole. Every real cutting experiment shows the opposite: at the instant a magnet is divided, new poles of opposite sign appear at both freshly cut faces, so each fragment is again a complete N-S dipole with a smaller $2l$ and, generally, a smaller $m$ . Because no experiment has ever produced a fragment with a single, uncompensated pole, magnetic poles are taken to occur only in equal-and-opposite pairs.

Step 4. Deduce that field lines must close on themselves. A field line is only allowed to start or end where a field diverges from a point source — physically, a pole. If magnetic monopoles existed, lines could start on a lone N pole and end at infinity or on a lone S pole elsewhere, exactly like electric field lines on isolated charges. Since Step 3 rules out any isolated pole, every magnetic field line that leaves the region near an N pole must eventually terminate on an S pole, and by symmetry the two poles of the same magnet are joined by lines that pass through the outside space and by the same lines continuing through the magnet's interior — the field line has no other place to go, so it forms one continuous closed loop.

Step 5. Connect crowding to $m$ . Because a larger pole strength $q_m$ (and hence a larger $m$ for fixed $2l$ ) produces a stronger field at any given external point, the same total "count" of field lines used to represent that field must be squeezed into the same picture more densely near a stronger magnet. This is why a field-line sketch with tightly packed lines near its centre is read as belonging to a magnet with the larger dipole moment, and it is the qualitative link between the vector $\vec m$ from Step 2 and the field-line diagrams drawn throughout the chapter.

Worked examples

Board level - dipole moment of a short bar magnet

A short bar magnet has pole strength $q_m = 5.0\ \text{A}\,\cdot\,\text{m}$ and a magnetic length $2l = 10\ \text{cm} = 0.10\ \text{m}$ . Find its magnetic dipole moment and state its direction.

Step 1. Write the defining formula: $m = q_m \times 2l$ .

Step 2. Substitute the given values: $m = 5.0\ \text{A}\,\cdot\,\text{m} \times 0.10\ \text{m} = 0.50\ \text{A}\,\cdot\,\text{m}^2$ .

Step 3. State the direction using the rule from the derivation: $\vec m$ points from the S pole to the N pole along the magnet's axis, since $2\vec l$ was defined that way.

Answer: the bar magnet has a dipole moment of $0.50\ \text{A}\,\cdot\,\text{m}^2$ , directed from S to N along its axis.

Competitive level - comparing two magnets and locating poles from a field-line sketch

Two bar magnets, A and B, of equal magnetic length $2l = 8.0\ \text{cm}$ are placed on a table and their field-line patterns are traced with a plotting compass. Magnet A shows widely spaced lines near its centre; magnet B shows lines packed roughly $2.5$ times as densely near its centre. If magnet A has pole strength $q_m = 4.0\ \text{A}\,\cdot\,\text{m}$ , estimate magnet B's pole strength, its dipole moment, and explain how the sketches alone reveal which ends are N and which are S.

Step 1. Relate line density to field strength, and field strength to pole strength. Field-line crowding represents $B$ , and near the centre of a short dipole $B$ scales with $m$ , which in turn scales with $q_m$ for a fixed $2l$ . Since B's lines are $2.5$ times as dense as A's at comparable points, treat B's pole strength as roughly $2.5$ times A's: $q_{m,B} \approx 2.5 \times 4.0\ \text{A}\,\cdot\,\text{m} = 10\ \text{A}\,\cdot\,\text{m}$ .

Step 2. Compute each magnet's dipole moment using $m = q_m \times 2l$ with the shared magnetic length $2l = 0.080\ \text{m}$ : $m_A = 4.0 \times 0.080 = 0.32\ \text{A}\,\cdot\,\text{m}^2$ , and $m_B = 10 \times 0.080 = 0.80\ \text{A}\,\cdot\,\text{m}^2$ , confirming $m_B \approx 2.5\, m_A$ , consistent with Step 1.

Step 3. Locate the poles from the sketch alone. Wherever the compass-plotted lines are seen to diverge outward — spreading apart as you move away from a point on the magnet — that point is the N pole, because field lines leave the N pole. Wherever the lines converge inward toward a point, that point is the S pole, because field lines enter the S pole there. No line ever crosses another in a correctly drawn sketch, so the divergence and convergence points are unambiguous.

Answer: magnet B has pole strength $q_{m,B} \approx 10\ \text{A}\,\cdot\,\text{m}$ and dipole moment $m_B \approx 0.80\ \text{A}\,\cdot\,\text{m}^2$ , about $2.5$ times magnet A's $0.32\ \text{A}\,\cdot\,\text{m}^2$ ; the ends where the sketched lines fan outward are the N poles and the ends where they converge are the S poles. Timing note: this style of comparison question should take under two minutes once you recognise it as "line density is a proxy for $m$ " — do not attempt to compute an exact $B$ value from a hand-drawn sketch; the syllabus only expects the qualitative ranking and pole-location reasoning shown here.

Common traps

Saying field lines "start" at the N pole and "vanish" near the S pole — they do not vanish; they continue through the magnet's interior from S to N to close the loop, and a full-credit sketch or explanation must show this.
Believing a magnet can be cut to isolate a single pole — cutting always produces two new complete N-S dipoles, each with its own pair of poles, never a lone pole.
Mixing up pole strength $q_m$ (a scalar, unit A·m) with the dipole moment $m = q_m \times 2l$ (unit A·m²) — these are different quantities with different units and are easy to swap under exam pressure.
Drawing field lines that touch or cross — a crossing implies two field directions at one point, which examiners specifically penalise.
Forgetting that field-line density is a relative indicator of $B$ , not an exact number — comparisons ("denser here than there") are valid; treating spacing as a precise measured value is not.

Common trapField lines never end in empty space — that is the whole reason monopoles are ruled out

The single biggest misconception in this topic is picturing the north pole as a "source" that field lines squirt out of and the south pole as a "sink" where they vanish for good, as if magnetism worked exactly like an isolated positive and negative charge. It does not. Because no isolated magnetic pole has ever been found, a magnetic field line cannot simply terminate anywhere — outside the magnet it goes from N to S, and it is compelled to continue inside the magnetic material from S back to N, closing into one unbroken loop. If your sketch or your explanation shows lines stopping dead at the poles instead of threading back through the magnet, you have drawn (or described) an electric dipole's field, not a magnetic one — and that mix-up is exactly what the non-existence of magnetic monopoles is telling you not to do.

You should be able to…

Explain, using the "cut the magnet" argument, why isolated magnetic monopoles do not exist.
State the four defining properties of magnetic field lines and explain how they differ from electric field lines.
Write the definition $\vec m = q_m(2\vec l)$ and compute $m$ from given pole strength and magnetic length.
Sketch the field-line pattern of an isolated bar magnet, correctly showing the loop closing through the magnet's interior from S to N.
Read a field-line diagram to locate the N and S poles and to rank two magnets by relative field strength or dipole moment.

Ask the tutor: Want to see how this same dipole moment $\vec m$ turns into an actual formula for $B$ once we treat the bar magnet as an equivalent solenoid?

Why do magnetic field lines have to be closed loops but electric field lines don't

Electric field lines are allowed to begin and end because isolated electric charges genuinely exist — a single proton or electron can sit by itself with field lines starting or stopping right on it. Magnetic poles never occur alone; every N pole comes with a matching S pole on the same object, and no experiment has ever isolated one. Since a magnetic field line has no lone pole to terminate on, it has to keep going until it gets back to where it started, which forces it to pass through the magnet itself from S to N and close into a loop.

If I keep cutting a bar magnet into smaller and smaller pieces, do I eventually get a single isolated pole

No, and this is one of the cleanest experimental facts in magnetism. However many times you cut, each new fragment develops its own N and S pole right at the cut faces, so you always end up with more complete (smaller) dipoles, never a lone pole. Taken to its logical extreme, even down at the atomic level the magnetism of materials is understood to come from tiny current loops and electron spins, both of which are inherently N-S dipoles too — so the pairing shows up all the way down.

How is pole strength q_m different from the dipole moment m

Pole strength $q_m$ describes just one pole in isolation — how "strong" that single N or S end is, measured in A·m — and by itself it does not tell you the size of the magnet. The dipole moment $m = q_m \times 2l$ combines the pole strength with the separation between the two poles, so it describes the magnet as a whole and is the quantity that actually appears in the field formulae you will use later in this chapter. Two magnets with the same $q_m$ but different lengths $2l$ have different $m$ and therefore produce different field strengths at the same distance.

Does the field-line diagram tell me the exact numerical value of B at a point

Not from a hand-drawn sketch, no. Field-line diagrams are a qualitative-to-semi-quantitative tool: the direction of $B$ at any point is exact (it is the tangent to the line), and the relative crowding tells you where $B$ is larger or smaller compared to other regions of the same diagram. To get an actual number for $B$ you need the field formulae introduced later in the chapter (the axial and equatorial expressions for a dipole), which use the dipole moment $m$ and the distance $r$ directly.