Magnetisation M defined
A circulating electron in an atom carries a magnetic moment. In a bulk sample these atomic moments add up vectorially, and the result may be a non-zero net moment. NCERT defines the magnetisation $\mathbf{M}$ of a sample as its net magnetic moment per unit volume:
$$\mathbf{M} = \frac{\mathbf{m}_{\text{net}}}{V}$$
Magnetisation is a vector. Its dimensions are $[\text{L}^{-1}\text{A}]$ and it is measured in ampere per metre ($\text{A m}^{-1}$) — the same unit as the moment ($\text{A m}^2$) divided by volume ($\text{m}^3$). A sample with no net alignment of atomic moments, such as an ordinary piece of iron before magnetisation, has $M = 0$.
The definition is deliberately framed per unit volume rather than per atom. Two rods of the same material but different size, both fully aligned, carry different total moments but share the same magnetisation, because $M$ characterises the state of the material itself rather than the size of the specimen. This is the magnetic counterpart of how electric polarisation $P$ is dipole moment per unit volume in electrostatics, and it lets us write field equations that hold point by point inside any sample.
Atomic moments point randomly when unmagnetised, giving zero vector sum. Under alignment they add up, producing a net moment and hence a non-zero magnetisation per unit volume.
Magnetic intensity H
Consider a long solenoid of $n$ turns per unit length carrying current $I$. The interior field is $B_0 = \mu_0 n I$. If the interior is now filled with a material of non-zero magnetisation, the field grows beyond $B_0$. NCERT writes the net field as $B = B_0 + B_m$, where the additional contribution from the core is $B_m = \mu_0 M$, proportional to the magnetisation.
To isolate the part of the field that comes from external factors — here the solenoid current — NCERT introduces the magnetic intensity $\mathbf{H}$, defined by:
$$\mathbf{H} = \frac{\mathbf{B}}{\mu_0} - \mathbf{M}$$
$\mathbf{H}$ has the same dimensions as $\mathbf{M}$ and is also measured in $\text{A m}^{-1}$. In the solenoid, $H = nI$ — it depends only on the winding and current, not on the core material. This is the key conceptual split: $\mathbf{H}$ tracks the external cause, $\mathbf{M}$ tracks the material's response.
B and H are not the same field
Students often treat $B$ and $H$ as interchangeable because both describe "the magnetic field." They differ in meaning and in unit. $B$ (tesla) is the total field, set by both external currents and the material. $H$ ($\text{A m}^{-1}$) carries only the external contribution and is independent of the core material.
In a solenoid, $H = nI$ stays fixed whether the core is air or iron — but $B$ jumps by a factor of $\mu_r$ when iron is inserted.
The master relation B = μ0(H + M)
Rearranging the definition of $\mathbf{H}$ gives the central equation of this section, relating the total field to the external intensity and the material's magnetisation:
$$\mathbf{B} = \mu_0(\mathbf{H} + \mathbf{M})$$
This expression partitions the total field into two parts. The first, through $\mathbf{H}$, is due to external factors such as the solenoid current. The second, through $\mathbf{M}$, is due to the specific nature of the magnetic material. The constant $\mu_0 = 4\pi \times 10^{-7}\ \text{T m A}^{-1}$ is the permeability of free space, the same constant that appears in the Biot–Savart law.
The total field $B$ inside the core is the sum of the externally-driven part $\mu_0 H$ and the material-driven part $\mu_0 M$. Removing the core ($M = 0$) leaves only $B_0 = \mu_0 H = \mu_0 nI$.
Susceptibility and permeability
The magnetisation a material develops is influenced by the applied intensity. For most materials this relationship is linear, expressed through the magnetic susceptibility $\chi$:
$$\mathbf{M} = \chi\,\mathbf{H}$$
$\chi$ is a dimensionless number measuring how strongly a material responds to an external field. It is small and positive for paramagnetic materials, small and negative for diamagnetic materials (where $\mathbf{M}$ and $\mathbf{H}$ point in opposite directions), and large and positive for ferromagnetic materials. Substituting $M = \chi H$ into the master relation:
$$\mathbf{B} = \mu_0(\mathbf{H} + \chi\mathbf{H}) = \mu_0(1 + \chi)\mathbf{H} = \mu_0 \mu_r \mathbf{H} = \mu\,\mathbf{H}$$
Two new constants emerge. The relative magnetic permeability $\mu_r = 1 + \chi$ is a dimensionless number — the magnetic analogue of the dielectric constant. The permeability of the substance $\mu = \mu_0\mu_r = \mu_0(1 + \chi)$ shares the units and dimensions of $\mu_0$. The three quantities $\chi$, $\mu_r$ and $\mu$ are interrelated: knowing any one fixes the other two.
The chain $\chi \to \mu_r \to \mu$ is the single most exam-relevant relationship in this section, because it converts a measured material property into the permeability that governs the field. For a paramagnetic substance $\chi$ is a small positive number, so $\mu_r$ is just above unity; for a diamagnetic substance $\chi$ is slightly negative and $\mu_r$ falls just below one. For ferromagnetic materials $\chi$ runs into the hundreds or thousands, so $\mu_r$ is correspondingly large and the core dramatically intensifies the field. Whenever a problem hands you a susceptibility and asks for permeability, the route is always $\mu_r = 1 + \chi$ followed by $\mu = \mu_0\mu_r$ — exactly the calculation NEET 2020 demanded.
The sign and size of $\chi$ is exactly what sorts materials into three families. See Dia-, Para- and Ferromagnetism for the full classification.
χ has no units — and only diamagnets make it negative
Because $\chi = M/H$ is a ratio of two quantities with identical units ($\text{A m}^{-1}$), it is dimensionless and unitless. Do not assign it $\text{A m}^{-1}$ in an answer. Equally, of the three magnetic classes only diamagnetic materials carry a negative susceptibility — a point NEET 2016 tested directly.
Diamagnetic: $\chi < 0$. Paramagnetic: $0 < \chi < \varepsilon$ (small). Ferromagnetic: $\chi \gg 1$.
Units at a glance
Mixing up which quantity is dimensionless and which carries $\text{A m}^{-1}$ is the most common source of error in this topic. The table below collects every quantity from §5.4 with its defining relation and SI unit.
| Quantity | Symbol | Defining relation | SI unit |
|---|---|---|---|
| Magnetisation | $M$ | M = m_net / V | A m⁻¹ |
| Magnetic intensity | $H$ | H = B/μ₀ − M | A m⁻¹ |
| Total magnetic field | $B$ | B = μ₀(H + M) | tesla (T) |
| Magnetic susceptibility | $\chi$ | χ = M/H | dimensionless |
| Relative permeability | $\mu_r$ | μr = 1 + χ | dimensionless |
| Permeability of substance | $\mu$ | μ = μ₀μr | T m A⁻¹ |
| Permeability of free space | $\mu_0$ | 4π × 10⁻⁷ | T m A⁻¹ |
Worked example: solenoid core
This is NCERT Example 5.5 worked through in full. It threads together $H$, $B$ and $M$ in a single numerical setting and mirrors the style of NEET numericals on the topic.
A solenoid has a core of relative permeability $\mu_r = 400$. The windings carry a current of $2\ \text{A}$, with $1000$ turns per metre. Find (a) $H$, (b) $M$, (c) $B$.
(a) $H$ depends only on the current and turns: $H = nI = 1000 \times 2.0 = 2 \times 10^{3}\ \text{A m}^{-1}$.
(c) The total field is $B = \mu_r \mu_0 H = 400 \times 4\pi \times 10^{-7} \times 2 \times 10^{3} = 1.0\ \text{T}$.
(b) Magnetisation: $M = \dfrac{B - \mu_0 H}{\mu_0} = (\mu_r - 1)H = 399 \times 2 \times 10^{3} \approx 8 \times 10^{5}\ \text{A m}^{-1}$.
Notice the structure: $H$ is fixed by the coil, $B$ is amplified $\mu_r$-fold by the core, and $M$ — which is responsible for the amplification — is enormous compared with $H$ because $\mu_r \gg 1$ for a ferromagnetic core.
Magnetisation and magnetic intensity in one screen
- Magnetisation $M = m_{\text{net}}/V$ is net moment per unit volume; unit $\text{A m}^{-1}$.
- Magnetic intensity $H = B/\mu_0 - M$ carries the external contribution; same unit $\text{A m}^{-1}$.
- Master relation: $B = \mu_0(H + M)$, with $B$ in tesla and $\mu_0 = 4\pi \times 10^{-7}\ \text{T m A}^{-1}$.
- Susceptibility $\chi = M/H$ is dimensionless; negative only for diamagnetic materials.
- Relative permeability $\mu_r = 1 + \chi$; permeability $\mu = \mu_0\mu_r$, in $\text{T m A}^{-1}$.