Is it correct to say: Writing U = mB cosθ, dropping the minus sign?

U = −mB cosθ. The aligned position (θ = 0°) has the lowest energy, −mB, and is the stable one.

Is it correct to say: Using the dipole field formulas at distances comparable to the magnet's length?

B_axial = (μ₀/4π)(2m/r³) and B_eq = (μ₀/4π)(m/r³) hold only when r is much larger than the magnet (a short magnet).

Is it correct to say: Treating H and B as the same quantity?

B is the total field in tesla; H is the magnetic intensity in A m⁻¹ set by free currents. They are linked by B = μ₀(H + M).

Is it correct to say: Thinking magnetic field lines start at the N-pole and end at the S-pole?

Outside the magnet they run from N to S, but inside they continue from S to N. They form closed loops, which is why the flux through any closed surface is zero.

Is it correct to say: Saying diamagnetic materials have negative permeability?

χ is negative, but μr = 1 + χ is positive and slightly less than 1. A superconductor is a perfect diamagnet with χ = −1 and μr = 0.

Is it correct to say: Using temperature in °C in Curie's law?

χ ∝ 1/T with T in kelvin. For example, 27 °C = 300 K.

📄 Formula sheet (PDF)

Magnetism and Matter

Bar magnet as a magnetic dipole, dipole in a uniform field, Gauss's law for magnetism, magnetisation, magnetic intensity, susceptibility and dia-, para- and ferromagnetism — NCERT Class 12 Physics Ch 5

📐 12 formulas✏️ 4 examples🎯 6 practice⚖️ Units III–IV: 17 marks (with Ch 4, 6 and 7)🏫 CBSE📚 Class 12✓ 2025–26 syllabus

💡

Board Exam Tips

→Use the electrostatic analogy: replace p by m and 1/(4πε₀) by μ₀/4π, and the dipole field formulas of Ch 1 carry over directly to a bar magnet.
→On the equatorial line the field is antiparallel to m and half the axial value at the same distance. State the direction, not just the magnitude.
→Stable equilibrium is θ = 0° (U = −mB); unstable is θ = 180° (U = +mB). The work needed to turn a magnet from stable to unstable is 2mB.
→Prepare a comparison of diamagnetic, paramagnetic and ferromagnetic materials: sign and size of χ, value of μr, behaviour in a non-uniform field and effect of temperature.
→Curie's law needs absolute temperature. Convert °C to K before using χ ∝ 1/T.

📐 Formulas(12)

Magnetic Field on the Axis of a Bar Magnet★ Board fav

B_{a x ia l} = \frac{μ _{0}}{4 π} \frac{2 m}{r ^{3}}

Symbol	Meaning
$B_{a x ia l}$	Magnetic field on the axis (T)
$m$	Magnetic dipole moment (A m² or J T⁻¹)
$r$	Distance from the centre of the magnet (m)
$μ_{0}$	Permeability of free space = 4π×10⁻⁷ T m A⁻¹

Magnetic Field on the Equatorial Line of a Bar Magnet

B_{e q} = - \frac{μ _{0}}{4 π} \frac{m}{r ^{3}}

Symbol	Meaning
$B_{e q}$	Magnetic field on the equatorial line (T)
$m$	Magnetic dipole moment (A m²)
$r$	Distance from the centre of the magnet (m)

Torque on a Magnetic Dipole in a Uniform Field★ Board fav

τ = m \times B, τ = m B sin θ

Symbol	Meaning
$τ$	Torque (N m)
$m$	Magnetic dipole moment (J T⁻¹)
$B$	Uniform magnetic field (T)
$θ$	Angle between m and B

Potential Energy of a Magnetic Dipole★ Board fav

U_{m} = - m \cdot B = - m B cos θ

Symbol	Meaning
$U_{m}$	Magnetic potential energy (J)
$θ$	Angle between m and B

Work Done in Rotating a Magnetic Dipole

W = m B (cos θ_{1} - cos θ_{2})

Symbol	Meaning
$W$	Work done (J)
$θ_{1}, θ_{2}$	Initial and final angles between m and B

Gauss's Law for Magnetism

\oint_{S} B \cdot d S = 0

Symbol	Meaning
$B$	Magnetic field (T)
$d S$	Area element of the closed surface (m²)

Magnetisation

M = \frac{m _{n e t}}{V}

Symbol	Meaning
$M$	Magnetisation (A m⁻¹)
$m_{n e t}$	Net magnetic moment of the sample (A m²)
$V$	Volume of the sample (m³)

Magnetic Intensity in a Solenoid

H = n I

Symbol	Meaning
$H$	Magnetic intensity (A m⁻¹)
$n$	Number of turns per unit length (m⁻¹)
$I$	Current in the winding (A)

Field Inside a Magnetised Material★ Board fav

B = μ_{0} (H + M), H = \frac{B}{μ _{0}} - M

Symbol	Meaning
$B$	Magnetic field (T)
$H$	Magnetic intensity (A m⁻¹)
$M$	Magnetisation (A m⁻¹)

Magnetic Susceptibility

M = χ H

Symbol	Meaning
$χ$	Magnetic susceptibility (no unit)

Permeability and Susceptibility★ Board fav

B = μ_{0} μ_{r} H = μ H, μ_{r} = 1 + χ

Symbol	Meaning
$μ_{r}$	Relative permeability (no unit)
$μ$	Permeability of the material (T m A⁻¹)

Curie's Law (Paramagnetism)

χ = \frac{C μ _{0}}{T}

Symbol	Meaning
$C$	Curie constant of the material
$T$	Absolute temperature (K)

✏️ Solved Examples

1Solved Exampleeasy3 steps

A bar magnet of magnetic moment 0.40 J T⁻¹ is placed in a uniform magnetic field of 0.25 T with its axis at 30° to the field. Find (a) the torque on it and (b) its potential energy.

Torque on a dipole in a uniform field

τ = m B sin θ = 0.40 \times 0.25 \times sin 3 0^{\circ} = 0.40 \times 0.25 \times 0.5

2Solved Exampleboard3 steps

A short bar magnet has a magnetic moment of 0.60 J T⁻¹. Find the magnitude and direction of its magnetic field at a distance of 15 cm from its centre (a) on its axis and (b) on its equatorial line. Take μ₀/4π = 10⁻⁷ T m A⁻¹.

Convert the distance to metres

r = 0.15 m, r^{3} = 3.375 \times 1 0^{- 3} m^{3}

3Solved Exampleboard5 steps

A long solenoid with 800 turns per metre carries a current of 1.5 A. Its core is made of a material of relative permeability 500. Find (a) the magnetic intensity H, (b) the susceptibility and magnetisation of the core and (c) the magnetic field B inside the core. Take μ₀ = 4π×10⁻⁷ T m A⁻¹.

H depends only on n and I

H = n I = 800 \times 1.5 = 1.2 \times 1 0^{3} A m^{- 1}

4Solved ExampleHOTS4 steps

A short bar magnet held with its axis at 30° to a uniform magnetic field of 0.30 T experiences a torque of 0.060 N m. Find (a) its magnetic moment, (b) the work an external agent must do to turn it from its stable position to a position perpendicular to the field, (c) the work needed to turn it from its stable position to its unstable position and (d) the torque on it in the unstable position.

Magnetic moment from τ = mB sin θ

m = \frac{τ}{B s i n θ} = \frac{0.060}{0.30 \times s i n 3 0 ^{\circ}} = \frac{0.060}{0.15} = 0.40 J T^{- 1}

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Writing U = mB cosθ, dropping the minus sign
✓U = −mB cosθ. The aligned position (θ = 0°) has the lowest energy, −mB, and is the stable one.
2
Using the dipole field formulas at distances comparable to the magnet's length
✓B_axial = (μ₀/4π)(2m/r³) and B_eq = (μ₀/4π)(m/r³) hold only when r is much larger than the magnet (a short magnet).
3
Treating H and B as the same quantity
✓B is the total field in tesla; H is the magnetic intensity in A m⁻¹ set by free currents. They are linked by B = μ₀(H + M).
4
Thinking magnetic field lines start at the N-pole and end at the S-pole
✓Outside the magnet they run from N to S, but inside they continue from S to N. They form closed loops, which is why the flux through any closed surface is zero.
5
Saying diamagnetic materials have negative permeability
✓χ is negative, but μr = 1 + χ is positive and slightly less than 1. A superconductor is a perfect diamagnet with χ = −1 and μr = 0.
6
Using temperature in °C in Curie's law
✓χ ∝ 1/T with T in kelvin. For example, 27 °C = 300 K.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
A magnet of moment 2.5 J T⁻¹ is held at 60° to a uniform field of 0.20 T. Find its potential energy.
Q2
The susceptibility of a paramagnetic salt is 3.0×10⁻⁴ at 27 °C. Find its susceptibility at −73 °C.
Q3
A material has χ = −2.6×10⁻⁵. Classify it and find its relative permeability.
Q4
An iron rod of volume 2.0×10⁻⁵ m³ has a net magnetic moment of 50 A m². Find its magnetisation.
Q5
For a short bar magnet, find the ratio of the distances on the axis and on the equatorial line at which the field has the same magnitude.
Q6
What is the net magnetic flux through a closed surface that encloses only the N-pole end of a bar magnet?

📝 Notes

Magnetism and Matter

This chapter treats a bar magnet as a magnetic dipole and then asks how different materials respond when placed in a magnetic field.

The bar magnet is a magnetic dipole

A bar magnet behaves like a current-carrying solenoid of the same moment $m$ . Far from the magnet its field has exactly the form of an electric dipole's field, with $p \to m$ and $1/\varepsilon_0 \to \mu_0$ :

On the axis: $B = \dfrac{\mu_0}{4\pi}\dfrac{2m}{r^3}$ , along $\vec{m}$ .
On the equatorial line: $B = \dfrac{\mu_0}{4\pi}\dfrac{m}{r^3}$ , opposite to $\vec{m}$ .

The big difference from electrostatics is Gauss's law: $\oint \vec{B}\cdot d\vec{S} = 0$ , because there are no magnetic monopoles.

Dipole in a uniform field

Torque $\tau = mB\sin\theta$ and energy $U = -mB\cos\theta$ come as a pair. The torque is largest ( $mB$ ) at $\theta = 90^\circ$ and zero at both $0^\circ$ (stable) and $180^\circ$ (unstable). To flip a magnet from $\theta = 0^\circ$ to $180^\circ$ needs $W = 2mB$ .

Describing a magnetised material

Work through four quantities in order:

$H = nI$ — set by the current in the winding.
$M = \chi H$ — the material's response.
$B = \mu_0(H + M) = \mu_0\mu_r H$ .
$\mu_r = 1 + \chi$ .

Dia-, para- and ferromagnetism

Diamagnetic (e.g. bismuth, copper, water): $\chi$ small and negative, $\mu_r$ just below 1; weakly pushed from stronger to weaker field regions.
Paramagnetic (e.g. aluminium, sodium, oxygen): $\chi$ small and positive, $\mu_r$ just above 1; follows Curie's law $\chi = C\mu_0/T$ .
Ferromagnetic (e.g. iron, cobalt, nickel): $\chi \gg 1$ because of domains; becomes paramagnetic above the Curie temperature $T_C$ .