For paramagnets, χ = C/T (susceptibility inversely proportional to absolute temperature). Higher T means more thermal randomization, weaker response.

Magnetism and Matter

Q: What is a magnetic dipole moment and its SI unit?

M = md for a bar magnet (pole strength × length) or M = NIA for a current loop. SI unit: A·m² (or J/T).

Q: How is a bar magnet equivalent to a solenoid?

Atomic current loops inside a magnetized bar produce a surface current pattern like a solenoid winding. Both have the same net dipole moment and produce identical external fields.

Q: What is the magnetic field on the axis and equatorial line of a short bar magnet?

Axial: (μ₀/4π)(2M/r³) along M. Equatorial: (μ₀/4π)(M/r³) opposite to M. Axial is twice equatorial at equal distances.

Q: What are diamagnetic, paramagnetic, and ferromagnetic materials?

Diamagnetic: small negative susceptibility, weakly repelled (Cu, water, Bi). Paramagnetic: small positive, weakly attracted (Al, O₂). Ferromagnetic: very large positive, strongly attracted, retains magnetization (Fe, Ni, Co).

Q: What is the Curie temperature?

Temperature above which a ferromagnet loses its ferromagnetism and becomes paramagnetic. Iron: 770°C; Ni: 358°C; Co: 1131°C. Above T_c: χ = C/(T - T_c) (Curie-Weiss law).

Q: Why do diamagnetic materials get repelled by magnets?

An applied field induces atomic currents that oppose the change (Lenz's law), giving an antiparallel moment. In a non-uniform field, this is pushed toward weaker regions, away from the poles.

Q: What is the difference between magnetic field B and magnetic intensity H?

H is the field from free currents alone (A/m). B is the total field including magnetization: B = μ₀(H + M_m). In free space B = μ₀H; in material B = μH with μ = μ₀(1+χ).

Q: Can you isolate a north or south magnetic pole?

No. Cutting a magnet always produces two smaller magnets, each with N and S. This is captured by Gauss's law for magnetism: ∮B·dA = 0.

Q: What are magnetic domains?

Small regions in a ferromagnet where atomic moments are spontaneously aligned. In an unmagnetized sample, domains point randomly and cancel. External field grows favourable domains and rotates others, producing large net magnetization.

PhysicsMagnetic Effects of Current and MagnetismFor JEE aspirants

Magnetism and Matter connects the current-loop picture of Concept 261 to the properties of permanent magnets and materials. A bar magnet is equivalent to a solenoid: both have the same dipole field. A magnetic dipole of moment $M$ produces field $B_{axial} = \frac{μ _{0}}{4 π} \frac{2 M}{r ^{3}}$ on its axis and $B_{eq} = \frac{μ _{0}}{4 π} \frac{M}{r ^{3}}$ on its equatorial line. In a uniform field it feels torque $τ = M \times B$ and has energy $U = - M \cdot B$ . All materials respond weakly or strongly to a magnetic field, classified as diamagnetic, paramagnetic, or ferromagnetic. Paramagnets follow Curie's law ( $χ \propto 1/ T$ ); ferromagnets lose their permanent magnetization above the Curie temperature. This concept is explicitly listed in both JEE Main 2026 and NEET 2026 syllabi.

Key Formulas - Quick Reference

Magnetic dipole moment: $M = m d$ (pole strength $\times$ magnetic length), or $M = N I A \overset{n}{^}$ for a coil
Bar magnet as equivalent solenoid: same $M$ , same far-field pattern
Field on axis of a short bar magnet (distance $r ≫ d$ ): $B_{axial} = \frac{μ _{0}}{4 π} \frac{2 M}{r ^{3}}$
Field on equatorial line: $B_{eq} = \frac{μ _{0}}{4 π} \frac{M}{r ^{3}}$
General point ( $θ$ from axis): $B = \frac{μ _{0}}{4 π} \frac{M}{r ^{3}} 1 + 3 cos^{2} θ$
Torque in uniform field: $τ = M \times B$ , magnitude $M B sin θ$
Potential energy: $U = - M \cdot B = - M B cos θ$
Magnetization: $M_{m}$ (magnetic moment per unit volume) - not to be confused with dipole moment $M$
Magnetic intensity $H$ and induction $B$ : $B = μ_{0} (H + M_{m}) = μ_{0} μ_{r} H$
Susceptibility $χ = M_{m} / H$ ; permeability $μ = μ_{0} (1 + χ) = μ_{0} μ_{r}$
Curie's law (paramagnets): $χ = C / T$
Ferromagnets above Curie temperature $T_{c}$ become paramagnetic: $χ = C / (T - T_{c})$ (Curie-Weiss law)

1. Bar Magnet - Pole Strength and Dipole Moment

A bar magnet has a north pole (+m) and south pole (−m) separated by the magnetic length $d$ (from S to N). The pole strength $m$ measures the "magnetic charge" of a pole in ampere-metres (A·m). The magnetic dipole moment is:

M = m d

Direction: from south to north pole inside the magnet. Unit: $A\cdotm^{2}$ .

Poles always come in pairs: cutting a magnet in half gives two smaller dipoles, never an isolated pole.
Magnetic length $d_{m}$ is slightly less than the geometric length $d_{g}$ : $d_{m} \approx 0.84 d_{g}$ typically.
Two similar poles repel; two unlike poles attract - inverse-square law for pole-pole force.

Figure 1: Magnetic poles are not separable objects the way electric charges are. Each fragment immediately develops its own N and S. This is the microscopic reason behind

\oint B \cdot d A = 0

: there is no magnetic charge for the flux to come out of.

2. Bar Magnet as an Equivalent Solenoid

Imagine a solenoid of $N$ turns, length $L$ , area $A$ , carrying current $I$ . Its dipole moment $M_{s} = N I A$ . Now, atomic currents inside a magnetized bar produce a net "surface current" that circulates like a solenoid winding. The magnetic field outside a bar magnet is identical to that outside a solenoid of the same overall dipole moment $M$ .

Figure 2: Outside the material the two field maps are indistinguishable. A magnetised bar is really a stack of atomic current loops whose interior currents cancel, leaving a net surface current that circulates exactly like a solenoid winding. Both have

M = N I A

and both obey

τ = M \times B

3. Magnetic Field due to a Bar Magnet (Short Dipole)

For a bar magnet treated as a point dipole ( $r ≫ d$ ):

3.1 Axial (end-on) field

B_{axial} = \frac{μ _{0}}{4 π} \frac{2 M}{r ^{3}}

Direction: along $M$ (i.e., from S to N through the magnet, extended along the axis).

3.2 Equatorial (broadside-on) field

B_{eq} = - \frac{μ _{0}}{4 π} \frac{M}{r ^{3}}

Direction: opposite to $M$ (i.e., from N to S in the plane perpendicular to the magnet).

Note: $B_{axial} = 2 B_{eq}$ at the same distance - the axial field is exactly twice the equatorial field of a short dipole.

3.3 General point at angle $θ$ from the axis

$B_{r} = \frac{μ _{0}}{4 π} \frac{2 M cos θ}{r ^{3}}$ (radial), $B_{θ} = \frac{μ _{0}}{4 π} \frac{M sin θ}{r ^{3}}$ (tangential)

Resultant: $B = \frac{μ _{0}}{4 π} \frac{M}{r ^{3}} 1 + 3 cos^{2} θ$ ; makes angle $α$ with radial with $tan α = tan θ /2$ .

Figure 3: At equal distances the axial field is twice the equatorial field and points the opposite way:

B_{axial} = \frac{μ _{0}}{4 π} \frac{2 M}{r ^{3}}

along

M

, while

B_{eq} = \frac{μ _{0}}{4 π} \frac{M}{r ^{3}}

points against

M

. Both distances are measured from the centre of the magnet.

4. Torque, Energy, and Oscillation in Uniform Field

All results from Concept 261 apply directly to bar magnets:

Torque: $τ = M \times B$ , magnitude $M B sin θ$ .
Potential energy: $U = - M \cdot B = - M B cos θ$ . Stable at $θ = 0°$ ; unstable at $θ = 180°$ .
Work done rotating from $θ_{1}$ to $θ_{2}$ : $W = M B (cos θ_{1} - cos θ_{2})$ .
Small oscillation period (vibration magnetometer): $T = 2 π I_{inertia} / (M B)$ .

Solved Example 1

A short bar magnet has moment

M = 0.5 A\cdotm^{2}

. Find the magnetic field on its axis at

20 cm

from the centre.

Solution:

$B_{axial} = \frac{μ _{0}}{4 π} \frac{2 M}{r ^{3}} = 1 0^{- 7} \times \frac{2 \times 0.5}{( 0.2 ) ^{3}} = \frac{1 0 ^{- 7}}{0.008} = 1.25 \times 1 0^{- 5} T = 12.5 μ T$ .

Solved Example 2

A bar magnet of moment

1 0^{- 4} J/T

sits in a horizontal field

B = 4 \times 1 0^{- 5} T

. Find (a) the maximum torque, (b) the potential energy when

M

makes

60°

with

B

Solution:

(a) $τ_{m a x} = M B = 1 0^{- 4} \times 4 \times 1 0^{- 5} = 4 \times 1 0^{- 9} N\cdotm$ .

(b) $U = - M B cos 60° = - 4 \times 1 0^{- 9} \times 0.5 = - 2 \times 1 0^{- 9} J$ .

5. Magnetization, Magnetic Intensity, and Susceptibility

5.1 Magnetization $M_{m}$

The magnetization $M_{m}$ (denoted here $M_{m}$ to distinguish from dipole moment $M$ ) is the magnetic moment per unit volume:

$M_{m} = \frac{net magnetic moment}{volume}$ (SI unit: A/m)

5.2 Magnetic intensity $H$ and induction $B$

Inside a magnetized material, the total field $B$ has contributions from both the applied field and the magnetization:

B = μ_{0} (H + M_{m})

Here $H$ (magnetic intensity) is what the applied current alone would produce; SI unit A/m. Both $H$ and $M_{m}$ have the same units, since $B$ (T) $= μ_{0} \times$ (A/m).

5.3 Susceptibility and permeability

For linear materials, magnetization is proportional to applied intensity:

M_{m} = χ H

where $χ$ is the magnetic susceptibility (dimensionless). Then:

$B = μ_{0} (1 + χ) H = μ H = μ_{0} μ_{r} H$

where $μ = μ_{0} (1 + χ)$ is the permeability and $μ_{r} = 1 + χ$ is the relative permeability.

6. Classification of Magnetic Materials

Property	Diamagnetic	Paramagnetic	Ferromagnetic
Susceptibility $χ$	small, negative	small, positive	very large, positive
Relative permeability $μ_{r}$	slightly $< 1$	slightly $> 1$	$≫ 1$ (thousands)
Direction of induced moment	opposite to applied field	along applied field	along applied field (strong)
Behaviour in non-uniform $B$	repelled from strong region	attracted to strong region	strongly attracted
Effect of temperature	independent (mostly)	$χ \propto 1/ T$ (Curie's law)	ferromagnetic below $T_{c}$ , paramagnetic above
Examples	Bi, Cu, water, gold, silver, N₂, most organic	Al, Pt, Na, Ca, O₂, Mn	Fe, Ni, Co, Gd; ferrites

Figure 4: The line density inside the sample is the sign of

χ

. A diamagnet thins the field inside itself and so is pushed towards weaker field; a paramagnet concentrates it slightly and a ferromagnet enormously, so both are pulled towards stronger field. Nothing here depends on the field being non-uniform - that only decides whether there is a net force.

6.1 Diamagnetism

Every atom, even one with no intrinsic magnetic moment, develops a small moment opposite to any applied field (Lenz's law at atomic scale). All materials have this contribution, but it is masked in para- and ferromagnetic materials by their stronger intrinsic moments.

$χ_{dia}$ is small and negative (typically $- 1 0^{- 6}$ to $- 1 0^{- 5}$ ).
Independent of temperature.
A diamagnetic rod suspended in a non-uniform field aligns perpendicular to the field and drifts toward weaker regions.
Superconductors are perfect diamagnets ( $χ = - 1$ , $μ_{r} = 0$ ) - they expel all field (Meissner effect).

6.2 Paramagnetism

Atoms have permanent magnetic moments (from unpaired electron spins), but thermal motion keeps them randomly oriented in the absence of a field. An applied field partially aligns them; the net magnetization is small.

$χ_{para}$ is small and positive (typically $1 0^{- 5}$ to $1 0^{- 3}$ ).
Aligns along the field; drifts toward stronger regions.
Obeys Curie's law: $χ = C / T$ , where $C$ is Curie's constant. Higher $T$ increases thermal randomization, weakening the response.

6.3 Ferromagnetism

Atomic moments spontaneously align in domains even without external field, due to strong quantum-mechanical exchange interaction. An external field grows and orients favourable domains, producing very large magnetization.

$χ_{ferro}$ is very large and positive ( $1 0^{2}$ to $1 0^{5}$ ); depends on field strength (non-linear).
Exhibits hysteresis: magnetization lags behind the field, producing a hysteresis loop (JEE only, off NEET syllabus).
Retains remanent magnetization even after field is removed (permanent magnets).
Above the Curie temperature $T_{c}$ , thermal energy overcomes exchange coupling and ferromagnetism disappears; the material becomes paramagnetic. Above $T_{c}$ , Curie-Weiss law: $χ = \frac{C}{T - T _{c}}$ .

Figure 5: A ferromagnet is already magnetised inside each domain even with no external field; what an unmagnetised sample lacks is agreement between domains. The applied field grows the favourably aligned domains, which is why the response is so large and why it partly survives when the field is removed.

JEE Advanced / JEE Main only - off the NEET syllabus

Hysteresis. Because domain walls do not move back freely, the magnetisation of a ferromagnet depends on how the field was applied, not just on its present value. Taking the material once round a full cycle of $H$ traces a closed hysteresis loop.

Retentivity (remanence) $B_{r}$ : the induction left in the material when $H$ is brought back to zero.
Coercivity $H_{c}$ : the reverse field needed to drive $B$ back to zero.
Energy loss per cycle $\propto$ the area of the loop, dissipated as heat.
Hard magnetic materials (steel, alnico): large $B_{r}$ and large $H_{c}$ , fat loop - permanent magnets.
Soft magnetic materials (soft iron, ferrites): large $B_{r}$ but small $H_{c}$ , thin loop - transformer cores, electromagnets.

Figure 6:

B

lags behind

H

, so the material remembers its history. Retentivity

B_{r}

is what is left when

H

returns to zero; coercivity

H_{c}

is the reverse field needed to erase it. A fat loop (steel) makes a good permanent magnet; a thin loop (soft iron) wastes little energy and suits transformer cores and electromagnets.

Curie temperatures: Iron

\approx 770° C

(

1043

K), Nickel

\approx 358° C

(

631

K), Cobalt

\approx 1131° C

(

1404

K). Heat a magnet above its

T_{c}

and it loses its magnetism permanently unless re-magnetized.

7. Effect of Temperature

Figure 7: Plotted separately because the vertical scales differ by several orders of magnitude. A paramagnet's

χ

falls as

C / T

and stays small. A ferromagnet's

χ

is enormous below

T_{c}

, collapses at the Curie point, and above it behaves like a paramagnet with

χ = \frac{C}{T - T _{c}}

Solved Example 3

A paramagnetic salt has susceptibility

χ = 6 \times 1 0^{- 3}

300

K. Find its susceptibility at

150

K, assuming Curie's law.

Solution:

$χ \propto 1/ T ⟹ χ_{2} / χ_{1} = T_{1} / T_{2} = 300/150 = 2$ .

$χ_{2} = 2 \times 6 \times 1 0^{- 3} = 1.2 \times 1 0^{- 2}$ .

Solved Example 4

Classify: (i) copper coin, (ii) aluminium foil, (iii) iron nail, (iv) water. Which will be attracted, which repelled, by a strong magnet?

Solution:

Copper: diamagnetic - weakly repelled.

Aluminium: paramagnetic - weakly attracted.

Iron: ferromagnetic - strongly attracted.

Water: diamagnetic - weakly repelled (a strong enough magnet can levitate a droplet of water).

Common Mistakes to Avoid

Watch out

Using axial-field formula on equatorial points (or vice versa): axial is $\frac{μ _{0}}{4 π} \frac{2 M}{r ^{3}}$ ; equatorial is $\frac{μ _{0}}{4 π} \frac{M}{r ^{3}}$ - factor of 2 difference, and opposite direction.
Confusing magnetization $M_{m}$ (per unit volume) with dipole moment $M$ : different quantities, different units. Both often written as $M$ .
Mixing up $B$ and $H$ : $H$ depends only on free currents; $B$ includes contributions from magnetization. Related by $B = μ_{0} (H + M_{m})$ .
Ignoring sign of $χ$ : negative for diamagnets, positive for para/ferro. This determines whether a substance is attracted or repelled.
Applying Curie's law to ferromagnets below $T_{c}$ : only valid for paramagnets (and for ferromagnets in their paramagnetic phase above $T_{c}$ , with modification to Curie-Weiss).
Assuming a ferromagnet always retains magnetism: above the Curie temperature ( $770° C$ for iron), it loses ferromagnetism entirely and becomes paramagnetic.
Confusing pole strength $m$ with dipole moment $M$ : $M = m d$ ; pole strength is analogous to charge, dipole moment analogous to $q \cdot d$ .

Frequently Asked Questions

Q1. What is a magnetic dipole moment and its SI unit?

The magnetic dipole moment $M$ of a bar magnet is $M = m d$ , where $m$ is pole strength (A·m) and $d$ is magnetic length. For a current loop, $M = N I A$ . SI unit is $A\cdotm^{2}$ (or equivalently, joule/tesla).

Q2. How is a bar magnet equivalent to a solenoid?

The atomic current loops inside a magnetized bar produce a net surface current pattern identical to that of a solenoid's winding. Both have the same net dipole moment and produce identical fields outside. This equivalence lets us apply solenoid results to bar magnets and vice versa.

Q3. What is the magnetic field on the axis and equatorial line of a short bar magnet?

Axial: $B_{axial} = \frac{μ _{0}}{4 π} \frac{2 M}{r ^{3}}$ , along $M$ . Equatorial: $B_{eq} = \frac{μ _{0}}{4 π} \frac{M}{r ^{3}}$ , opposite to $M$ . At equal distances, axial field is exactly twice equatorial field.

Q4. What are diamagnetic, paramagnetic, and ferromagnetic materials?

Diamagnetic materials have small negative susceptibility and are weakly repelled by magnets (e.g., copper, water, bismuth). Paramagnetic have small positive susceptibility and are weakly attracted (e.g., aluminium, oxygen). Ferromagnetic have very large positive susceptibility, are strongly attracted, and retain magnetization (e.g., iron, nickel, cobalt).

Q5. What is Curie's law?

For a paramagnetic material, the magnetic susceptibility varies inversely with absolute temperature: $χ = C / T$ , where $C$ is Curie's constant. Higher temperature means more thermal randomization of atomic moments, so weaker net magnetization at fixed applied field.

Q6. What is the Curie temperature?

The Curie temperature $T_{c}$ is the temperature above which a ferromagnetic material loses its ferromagnetism and becomes paramagnetic. For iron $T_{c} \approx 770° C$ ( $1043$ K); for nickel $\approx 358° C$ ; for cobalt $\approx 1131° C$ . Above $T_{c}$ , susceptibility follows the Curie-Weiss law $χ = C / (T - T_{c})$ .

Q7. Why do diamagnetic materials get repelled by magnets?

Applying an external field induces atomic currents that (by Lenz's law) oppose the change. This creates a small magnetic moment antiparallel to the applied field. In a non-uniform field, this antiparallel moment is pushed toward weaker regions, i.e., away from magnetic poles - hence repulsion. All materials have this effect; it dominates only in true diamagnets.

Q8. What is the difference between magnetic field $B$ and magnetic intensity $H$ ?

$H$ is the field due only to free (conduction) currents, in A/m. $B$ is the total field, including contributions from magnetization: $B = μ_{0} (H + M_{m})$ , in tesla. In free space (no magnetic material), $B = μ_{0} H$ . In a material, $B = μ H$ where $μ = μ_{0} (1 + χ)$ .

Q9. Can you isolate a north or south magnetic pole?

No. Every attempt to cut a magnet in half produces two smaller bar magnets, each with a complete N-S pair. This experimental fact is captured by Gauss's law for magnetism ( $\oint B \cdot d A = 0$ ) and reflects the non-existence of magnetic monopoles in nature so far.

Q10. What are magnetic domains?

In a ferromagnetic material, small regions (typically $1 0^{- 6}$ to $1 0^{- 4}$ m across) called domains have their atomic moments spontaneously aligned. In an unmagnetized sample, different domains point in random directions and cancel overall. An external field grows favourable domains and rotates others, producing a large net magnetization - the basis of ferromagnetism.

Previous year questions on Magnetism and Matter

9 questions from past papers, each with a step-by-step solution.

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1. Bar Magnet - Pole Strength and Dipole Moment

2. Bar Magnet as an Equivalent Solenoid

3. Magnetic Field due to a Bar Magnet (Short Dipole)

3.1 Axial (end-on) field

3.2 Equatorial (broadside-on) field

3.3 General point at angle θ from the axis

4. Torque, Energy, and Oscillation in Uniform Field

5. Magnetization, Magnetic Intensity, and Susceptibility

5.1 Magnetization Mm​

5.2 Magnetic intensity H and induction B

5.3 Susceptibility and permeability

6. Classification of Magnetic Materials

6.1 Diamagnetism

6.2 Paramagnetism

6.3 Ferromagnetism

7. Effect of Temperature

Common Mistakes to Avoid

Frequently Asked Questions

Q1. What is a magnetic dipole moment and its SI unit?

Q2. How is a bar magnet equivalent to a solenoid?

Q3. What is the magnetic field on the axis and equatorial line of a short bar magnet?

Q4. What are diamagnetic, paramagnetic, and ferromagnetic materials?

Q5. What is Curie's law?

Q6. What is the Curie temperature?

Q7. Why do diamagnetic materials get repelled by magnets?

Q8. What is the difference between magnetic field B and magnetic intensity H?

Q9. Can you isolate a north or south magnetic pole?

Q10. What are magnetic domains?

Previous year questions on Magnetism and Matter

Ready to master Magnetic Effects of Current and Magnetism?

3.3 General point at angle $θ$ from the axis

5.1 Magnetization $M_{m}$

5.2 Magnetic intensity $H$ and induction $B$

Q8. What is the difference between magnetic field $B$ and magnetic intensity $H$ ?