PhysicsCBSE Class 12 · Magnetism and Matter

Bar Magnet as an Equivalent Solenoid Class 12: Field Lines and Torque

Updated 15 October 2026 · Fact-checked

A bar magnet behaves like a current-carrying solenoid: their field lines have the same pattern outside and both act as magnetic dipoles of moment m. Its field points along m on the axis and against m on the equatorial line, and in a uniform field it feels a torque m × B that turns it along the field.

Understand Bar Magnet as an Equivalent Solenoid

Sprinkle iron filings around a bar magnet and around a current-carrying solenoid of the same size, and the patterns are almost identical. This is the key idea of the topic: a bar magnet can be thought of as a solenoid. The 2026-27 syllabus treats this topic qualitatively, so focus on the picture and the reasons, not on derivations.

The bar magnet. A bar magnet has two poles, north and south, near its ends. Like poles repel and unlike poles attract. A freely suspended magnet sets itself roughly north-south. Cutting a magnet never gives an isolated pole: each piece is a complete magnet with its own N and S.

Magnetic field lines. Field lines are a visual picture of the field:

Outside a magnet they go from the N pole to the S pole; inside, from S to N. So they form continuous closed loops, unlike electric field lines, which start on positive and end on negative charges.
The tangent at any point gives the direction of $B$ there (the way a small compass needle points).
Where lines are closer together, the field is stronger.
Two field lines never cross, because the field cannot have two directions at one point.

Bar magnet as an equivalent solenoid. Think of each turn of a solenoid as a small current loop, which is a tiny magnetic dipole with moment $I A$ (from the previous chapter). Placed side by side along the axis, these dipoles add: the solenoid has a total magnetic moment $m = N I A$ , pointing along the axis. Inside, their fields add to a strong, nearly uniform field; outside, the field spreads out from one end and returns to the other, exactly like a bar magnet. The end where the current looks anticlockwise behaves as a north pole. The field of a long solenoid at a far axial point, worked out from the field of each turn, has the same form as that of a bar magnet of the same moment. A bar magnet behaves this way because its atoms carry tiny current loops (electrons in motion) lined up in the same direction.

Field on the axis and on the equator (qualitative). At a far point on the axis, the field points along $m$ (away from the N end on that side). At a far point on the equatorial line, it points opposite to $m$ , parallel to the magnet, from the N end towards the S end. At the same distance, the axial field is about twice the equatorial field, and both fall off rapidly with distance, as $\frac{1}{r ^{3}}$ , just as for an electric dipole. (For reference, by analogy with the electric dipole, $B_{a x ia l} = \frac{μ _{0}}{4 π} \frac{2 m}{r ^{3}}$ and $B_{e q} = - \frac{μ _{0}}{4 π} \frac{m}{r ^{3}}$ ; no derivation is expected.)

Torque in a uniform field (qualitative). In a uniform field the forces on the two poles are equal and opposite, so there is no net force, but they form a couple. The torque $τ = m \times B$ tries to turn the magnet so that $m$ lines up with $B$ . It is largest when the magnet is at right angles to the field and zero when it is along or against the field. Along the field is stable (a nudge brings it back); against the field is unstable. A magnet released at an angle swings to and fro about the field direction. This is why a compass needle settles along the field.

Electrostatic analogy. A bar magnet corresponds to an electric dipole with $p \to m$ , $E \to B$ and $\frac{1}{ε _{0}} \to μ_{0}$ . The big difference is that magnetic field lines close on themselves, since isolated magnetic poles have never been found.

Bar Magnet as an Equivalent Solenoid: formulas and key facts

Magnetic moment of a solenoid (equivalent bar magnet): $m = N I A$; Along the axis; the end with anticlockwise current (seen from outside) is the N pole.
Direction of field at far axial and equatorial points: $B_{a x ia l} ∥ m, B_{e q} ∥ - m$; Axial field about twice the equatorial field at the same distance (qualitative).
Torque on a magnet in a uniform field: $τ = m \times B$; Largest at 90°, zero at 0° (stable) and 180° (unstable). Net force zero.
Electrostatic analogy: $p \to m, E \to B, \frac{1}{ε _{0}} \to μ_{0}$; Use it to write the bar magnet's field from the electric dipole's (no derivation needed).
Field lines: $closed loops: N \to S outside, S \to N inside$; Never cross; denser lines mean a stronger field.

How to solve Bar Magnet as an Equivalent Solenoid questions

Questions here are mainly explanations, diagrams and comparisons. Use this checklist.

1Draw the magnet (or solenoid) with N and S marked and the direction of $m$ (from S to N inside the magnet).
2For a solenoid, find the N end with the right-hand rule or the clockwise/anticlockwise rule, and its moment from m = NIA.
3For field direction at a point, decide whether the point is on the axis (field along m) or on the equator (field opposite to m).
4For torque questions, find the angle between m and B: 90° gives the largest torque, 0° and 180° give none; say which is stable.
5For field-line questions, state the rule (closed loops, no crossing, tangent gives direction, density gives strength) and the reason behind it.
6Use the electric dipole analogy to support your answer, but remember the syllabus treats this topic qualitatively.

Bar Magnet as an Equivalent Solenoid in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

A solenoid acting as a bar magnet

Suppose you wind 400 turns of insulated wire on a straw of cross-section 2 cm² and pass 0.5 A through it. You then hang it horizontally from a thread so that it can turn freely in a uniform field of 0.05 T.

(a) Find its magnetic moment. (b) How do you decide which end is its north pole? (c) In which direction does it settle, and what is the largest torque it can feel?

Try it first, then show the solutionSolution

$m = N I A = 400 \times 0.5 \times 2 \times 1 0^{- 4} = 0.04$ A m².
Look at one end: if the current there circulates anticlockwise, that end is the N pole (right-hand thumb rule gives m pointing out of it).
It settles with $m$ along $B$ , like a compass needle. The largest torque, when it is at right angles to B, is $m B = 0.04 \times 0.05 = 2 \times 1 0^{- 3}$ N m.

Answer: (a) 0.04 A m²; (b) the end where the current looks anticlockwise is N; (c) it settles along B; largest torque $2 \times 1 0^{- 3}$ N m.

Memory hook: A coil with current is a magnet you can switch on and off.

Compass needles around a bar magnet

Imagine placing two small compass needles near a bar magnet lying on a table: one on the line of the magnet's axis, beyond its N pole, and one on the perpendicular bisector of the magnet, some distance to the side.

(a) Which way does each needle point? (b) If both needles are the same distance from the magnet's centre (far compared with its length), which feels the stronger field? (c) What happens to both readings if the magnet is turned round end to end?

Try it first, then show the solutionSolution

Axial needle: the field there is along $m$ , so its north end points away from the magnet's N pole, along the axis.
Equatorial needle: the field is opposite to $m$ , so the needle lies parallel to the magnet with its north end pointing towards the magnet's S-pole end.
At the same large distance, the axial field is about twice the equatorial field, so the axial needle feels the stronger field.
Reversing the magnet reverses $m$ , so both needles swing round through 180°.

Answer: (a) Axial: pointing away from the magnet along its axis; equatorial: parallel to the magnet, north end towards its S end. (b) The axial one, about twice as strong. (c) Both reverse.

Memory hook: On the axis the field follows m; beside the magnet it runs back against m.

Tricks to remember Bar Magnet as an Equivalent Solenoid

Short memory aids. Say each one aloud once, then use it on a question the same day.

NIA makes a magnet
Turns × current × area gives a solenoid its magnetic moment, just like a bar magnet's m.
Use it when: Linking solenoid and bar magnet.
Clock rule for poles
Look at an end: anticlockwise current = N (draw an N with arrows anticlockwise), clockwise = S.
Use it when: Polarity of a solenoid or loop.
Loops, not lines
Magnetic lines close on themselves; electric lines start and stop on charges.
Use it when: Comparing field lines.
Axis follows, equator opposes
Axial field along m, equatorial field against m, axial about double.
Use it when: Field direction questions.

Bar Magnet as an Equivalent Solenoid: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Field lines of a bar magnet and a solenoid · 2 marks · Very short answerBoard level

(a) Draw (describe) the field lines of a bar magnet and of a current-carrying solenoid of the same size. (b) Give two similarities between them.

Try it first, then show the solutionSolution

(a) Both: lines leave one end (N), curve round outside and enter the other end (S), continuing inside from S to N to form closed loops; inside the solenoid the lines are nearly parallel and evenly spaced. (1)
(b) Similarities: the outside patterns are the same; both have a magnetic moment along the axis from S to N and behave as magnetic dipoles (a freely suspended one sets along the field). (1)

Answer: Identical outside patterns forming closed loops; both behave as magnetic dipoles with moment along the axis.

Example 2: Why a bar magnet is an equivalent solenoid · 3 marks · Short answerBoard level

Explain qualitatively why a bar magnet can be regarded as an equivalent solenoid. Find the magnetic moment of a solenoid of 500 turns, cross-section 2 cm², carrying 1.5 A.

Try it first, then show the solutionSolution

Each turn of a solenoid is a current loop, which is a magnetic dipole of moment IA along its axis. The turns are lined up, so the moments add: $m = N I A$ , along the axis. (1)
The fields of all the turns add up to a field outside that has the same pattern as a bar magnet's, and the far-field on the axis has the same form as that of a bar magnet of moment m. In a magnet, aligned atomic current loops play the role of the turns. (1)
$m = 500 \times 1.5 \times 2 \times 1 0^{- 4} = 0.15$ A m². (1)

Answer: Each turn is a small dipole; together they form a dipole of moment NIA with a bar-magnet field pattern; m = 0.15 A m².

Example 3: Rules for magnetic field lines · 3 marks · Short answerHarder board level

State three properties of magnetic field lines and give the reason for each. How do they differ from electric field lines of a charge?

Try it first, then show the solutionSolution

They form closed loops (outside N to S, inside S to N), because there are no isolated magnetic poles for them to start or end on. (1)
Two lines never cross, because the field at a point has only one direction. The tangent at a point gives the direction of B. (1)
The density of lines shows the field's strength. Electric lines of a static charge start on positive and end on negative charges, so they are not closed loops. (1)

Answer: Closed loops, never cross, tangent gives direction (density gives strength); electric lines begin and end on charges.

Example 4: A magnet in a uniform field · 3 marks · Short answerHarder board level

A small bar magnet is placed in a uniform magnetic field and released with its axis at right angles to the field.
(a) Is there a net force on it? (b) Describe the torque and the motion that follows. (c) Name the positions of stable and unstable equilibrium.

Try it first, then show the solutionSolution

(a) No: the forces on the two poles are equal and opposite in a uniform field. (1)
(b) They form a couple: the torque $m \times B$ is largest at 90° and turns the magnet towards the field; the magnet overshoots and swings to and fro about the field direction (settling there if there is friction). (1)
(c) Stable: $m$ along $B$ . Unstable: $m$ opposite to $B$ ; there the torque is zero but a small nudge makes it turn away. (1)

Answer: (a) No; (b) a turning couple, largest at 90°, causing oscillation about B; (c) stable along B, unstable against B.

Example 5: Cutting a magnet and a solenoid · 3 marks · Short answerHOTS / competency

A bar magnet of moment m is cut into two equal pieces (a) perpendicular to its length and (b) along its length. Find the moment of each piece in each case and explain, using the solenoid picture, why no piece is a single pole.

Try it first, then show the solutionSolution

(a) Cut across: each piece is half as long with the same pole strength, so each has moment $\frac{m}{2}$ . (1)
(b) Cut along: each piece has the full length but half the cross-section (half the pole strength), so again $\frac{m}{2}$ . (1)
A magnet is like a solenoid made of many turns (atomic current loops). Cutting a solenoid gives two shorter solenoids, each with two ends, N and S. In the same way each magnet piece has both poles. (1)

Answer: m/2 in both cases; every piece is a complete magnet because cutting a 'solenoid' just gives shorter solenoids.

Example 6: Comparing points around a magnet · 4 marks · Case-basedHarder board level

Read the passage and answer.

Point P is on the axis of a short bar magnet and point Q on its equatorial line, both at the same large distance r from its centre.
(a) Give the direction of the field at P and at Q relative to the magnetic moment. (1)
(b) Which point has the stronger field, and roughly by what factor? (1)
(c) How would the fields change if the distance were doubled? (1)
(d) Name the electric arrangement whose field has the same pattern. (1)

Try it first, then show the solutionSolution

(a) At P, along $m$ ; at Q, opposite to $m$ . (1)
(b) P, about twice as strong as Q. (1)
(c) Both fall to about one-eighth, since the dipole field falls as $\frac{1}{r ^{3}}$ . (1)
(d) An electric dipole: its axial and equatorial fields have the same directions relative to $p$ and the same 2 : 1 ratio. (1)

Answer: (a) P along m, Q opposite to m; (b) P, about twice; (c) each falls to about 1/8; (d) an electric dipole.

Where students lose marks in Bar Magnet as an Equivalent Solenoid

The slips examiners see most often, and what to write instead.

Mistake: Drawing field lines that end at the poles.
Copying the electric-field picture.
Do this: Continue every line through the magnet from S to N so it closes on itself.
Usually costs: 1 mark in a diagram.
Mistake: Saying a magnet cut in half gives an N piece and an S piece.
Thinking poles are separate objects.
Do this: Each piece is a complete magnet with both poles; use the solenoid picture to explain.
Usually costs: 1 mark.
Mistake: Writing a full derivation of the axial field of a bar magnet.
Older books derive it.
Do this: For 2026-27 the treatment is qualitative; describe and compare, and use the electric-dipole analogy.
Usually costs: Time, which may cost marks elsewhere.
Mistake: Saying a uniform field pulls a magnet along.
Confusing torque with force.
Do this: In a uniform field the net force is zero; only a torque acts.
Usually costs: 1 mark.
Mistake: Getting the solenoid's polarity backwards.
Mixing up clockwise and anticlockwise.
Do this: Anticlockwise current seen from an end means that end is N; check with the right-hand thumb rule.
Usually costs: ½ to 1 mark.

Exam tips

Neat field-line diagrams of a bar magnet and a solenoid, side by side, answer most 2-mark questions on this topic.
When asked to 'explain' the equivalence, mention current loops as dipoles, m = NIA and the matching field pattern.
Earth's magnetism, Gauss's law for magnetism and the mathematical treatment of the bar magnet are not in the 2026-27 syllabus.
The 2026 board paper 55/1/1 tested this topic with a 1-mark item (Q13); prepare the reasons behind each fact, not just the facts.
Use the words 'stable' and 'unstable' correctly for the two zero-torque positions.

Practice questions on Bar Magnet as an Equivalent Solenoid

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
Inside a bar magnet, the magnetic field lines go
(a) from N to S (b) from S to N (c) nowhere, the field is zero (d) from the middle outwards
Hint
Field lines are closed loops.
Answer
(b) from S to N.
Q2 · 1 mark · Very short answerBoard level
Why can two magnetic field lines never intersect?
Hint
What would the tangent mean at a crossing?
Answer
At an intersection there would be two tangents, i.e. two directions of the field at one point, which is impossible.
Q3 · 2 marks · Very short answerHarder board level
A solenoid of 200 turns and radius 1 cm carries 2 A. Find its magnetic moment and state which end is its north pole if the current appears clockwise when viewed from end X.
Hint
m = NIπr².
Answer
$m = 200 \times 2 \times π \times 1 0^{- 4} \approx 0.126$ A m². Clockwise from X means X is the S pole, so the other end is north.
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): Magnetic field lines always form closed loops.
Reason (R): Isolated magnetic poles do not exist.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Where could a line start or end?
Answer
(a).
Q5 · 2 marks · Very short answerHarder board level
In which orientations does a bar magnet in a uniform field feel (a) the largest torque and (b) no torque? Which of the zero-torque positions is stable?
Hint
τ depends on the sine of the angle between m and B.
Answer
(a) When its axis is at right angles to B. (b) When its axis is along B or opposite to B. Along B (m parallel to B) is stable; opposite is unstable.
Q6 · 2 marks · Very short answerHOTS / competency
The current in a small solenoid hanging freely in a uniform field is reversed. What happens to its magnetic moment and to the torque on it, and where does it finally settle?
Hint
m = NIA changes sign with I.
Answer
Its moment reverses, so it is now opposite to B (unstable). The torque reverses in sense; any small disturbance makes it swing round through 180° and it settles again with the new m along B.
Q7 · 2 marks · Very short answerHOTS / competency
A compass is moved along the perpendicular bisector of a bar magnet, far from it. Explain why the needle stays parallel to the magnet but points the 'wrong' way compared with a compass on the axis beyond the N pole.
Hint
Compare the field direction on the axis and on the equator.
Answer
On the axis the field is along m, so the needle points away from the N pole. On the equatorial line the field is opposite to m, parallel to the magnet, so the needle lies parallel to the magnet with its north end towards the magnet's S-pole end: opposite to the axial needle.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Magnetic moment of a solenoid?Show answer
m = NIA, along its axis.
2.Direction of field lines inside a bar magnet?Show answer
From S to N.
3.Field direction at a far equatorial point of a bar magnet?Show answer
Opposite to m.
4.Net force on a magnet in a uniform field?Show answer
Zero; only a torque acts.
5.When is the torque on a bar magnet largest?Show answer
When its axis is perpendicular to the field.
6.Which end of a solenoid is north?Show answer
The end where the current appears anticlockwise.

Bar Magnet as an Equivalent Solenoid: frequently asked questions

Why is a bar magnet called an equivalent solenoid?

Its field pattern and its behaviour in a field are the same as those of a current-carrying solenoid with the same magnetic moment. In the magnet, aligned atomic current loops play the role of the solenoid's turns.

Do I need to derive the field of a bar magnet for CBSE 2026-27?

No. The syllabus treats the bar magnet as an equivalent solenoid, its axial and equatorial field and the torque on it qualitatively only. Know the directions, the comparisons and the reasons.

Why do magnetic field lines form closed loops?

Because isolated magnetic poles have never been found. A field line has nowhere to start or end, so it continues through the magnet from S to N and closes on itself.

What happens when a bar magnet is placed in a non-uniform field?

The forces on its two poles are unequal, so it feels a net force as well as a torque. It is pulled towards the stronger-field region when lined up with the field.

Is Earth's magnetism part of the 2026-27 syllabus?

No. Earth's magnetism, declination and dip are listed as not in the 2026-27 CBSE content for this chapter.