What are the dimensions of $\chi$, the magnetic susceptibility? Consider an H-atom. Guess an expression for $\chi$, upto a constant by constructing a quantity of dimensions of $\chi$, out of parameters of the atom: $e$, $m$, $v$, $R$ and $\mu_0$. Here, $m$ is the electronic mass, $v$ is electronic velocity, $R$ is Bohr radius. Estimate the number so obtained and compare with the value of $\chi \sim 10^{-5}$ for many solid materials.

What are the dimensions of chi, the magnetic susceptibility? Consider an H-atom. Guess an expression for chi,

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Concept understanding — Magnetic Materials Magnetization

From a Paperclip to a Magnet: The Intuition

You already know that a magnet can pick up iron nails. But what is actually happening inside that nail when it gets near the magnet? And why does a plastic comb, rubbed on hair, pick up tiny bits of paper — but never iron filings?

The answer lies in magnetization — the process by which a material becomes magnetic.

Think of a piece of iron as a chaotic crowd of tiny compass needles. Each needle is an atomic magnetic moment (a tiny magnet, arising from the spin of electrons). In unmagnetized iron, these needles point in random directions. Their magnetic effects cancel out, so the iron as a whole shows no net magnetism.

Now bring a strong magnet close. Its magnetic field acts like a command: "Line up!" The tiny compass needles inside the iron start rotating, aligning themselves with the external field. The more they align, the stronger the iron's own magnetic field becomes. This alignment is magnetization.

Note

Magnetization is not the same as inducing a current. It is a purely magnetic reorientation of atomic dipoles inside a material.

The Precise Definition

Magnetization ( $\vec{M}$ ) is the net magnetic dipole moment per unit volume of a material. It tells you how strongly a material is magnetized — how many tiny atomic magnets are aligned, and in which direction.

If a material has $N$ atoms per unit volume, each with an average magnetic moment $\vec{\mu}_{\text{avg}}$ , then:

\vec{M} = N \vec{\mu}_{\text{avg}}

The SI unit of $\vec{M}$ is amperes per metre (A/m). Why? Because a magnetic dipole moment has units of A·m², and dividing by volume (m³) gives A/m.

\vec{M} = \frac{\text{total magnetic dipole moment}}{\text{volume}}

How Magnetization Connects to the Magnetic Field

When a material gets magnetized, it produces its own magnetic field. The total magnetic field $\vec{B}$ inside the material is the sum of:

The external applied field $\vec{H}$ (caused by free currents, like the current in a solenoid)
The material's response — the magnetization $\vec{M}$

The fundamental relation is:

\vec{B} = \mu_0 (\vec{H} + \vec{M})

where $\mu_0 = 4\pi \times 10^{-7} \, \text{T·m/A}$ is the permeability of free space.

Watch out

Do not confuse $\vec{H}$ (magnetic field intensity, or "magnetizing field") with $\vec{B}$ (magnetic flux density). $\vec{H}$ is what you apply; $\vec{M}$ is what the material does; $\vec{B}$ is the total field you measure.

The Three Kinds of Magnetic Materials

Not all materials respond the same way to an external field. The magnetization $\vec{M}$ is proportional to $\vec{H}$ for most materials (at least for small fields):

\vec{M} = \chi_m \vec{H}

where $\chi_m$ is the magnetic susceptibility — a dimensionless number that tells you how easily a material magnetizes.

Material Type	$\chi_m$	Behaviour	Example
Diamagnetic	Small and negative ( $\approx -10^{-5}$ )	Weakly repelled by a magnet; $\vec{M}$ opposes $\vec{H}$	Water, copper, bismuth
Paramagnetic	Small and positive ( $\approx 10^{-5}$ to $10^{-3}$ )	Weakly attracted; $\vec{M}$ aligns with $\vec{H}$	Aluminium, oxygen gas
Ferromagnetic	Large and positive ( $\gg 1$ )	Strongly attracted; $\vec{M}$ can be huge and persists even after $\vec{H}$ is removed	Iron, nickel, cobalt

Why this formula?

Magnetic Materials & Magnetization: Why the Key Formulas Hold

Let's build this from the ground up — starting with what magnetization physically means, then deriving the formulas step by step.

1. What is Magnetization ( $\vec{M}$ )?

Magnetization is the net magnetic dipole moment per unit volume of a material.

Inside a material, atoms act like tiny magnetic dipoles (due to electron spin and orbital motion).
Without an external field, these dipoles point randomly → net $\vec{M} = 0$ .
When an external field $\vec{H}$ is applied, dipoles align partially → net $\vec{M} \neq 0$ .

Definition:

\vec{M} = \frac{\text{net magnetic dipole moment}}{\text{volume}}

Units: A/m (same as $\vec{H}$ ).

2. The Fundamental Relation: $\vec{B} = \mu_0 (\vec{H} + \vec{M})$

This is the master equation linking the three magnetic fields:

$\vec{B}$ = magnetic flux density (the total field inside the material)
$\vec{H}$ = applied magnetic field (due to free currents)
$\vec{M}$ = magnetization (response of the material)
$\mu_0$ = permeability of free space ( $4\pi \times 10^{-7}$ H/m)

Why this form?

Step 1: In vacuum, there is no material, so $\vec{M} = 0$ . Then:

\vec{B} = \mu_0 \vec{H}

Step 2: Inside a material, the dipoles themselves produce an additional field. The total $\vec{B}$ is the sum of:

The field due to free currents ( $\mu_0 \vec{H}$ )
The field due to bound currents (from aligned dipoles), which is $\mu_0 \vec{M}$

Hence:

\vec{B} = \mu_0 \vec{H} + \mu_0 \vec{M} = \mu_0 (\vec{H} + \vec{M})

Key insight: $\vec{M}$ is not an independent field — it's the material's response to $\vec{H}$ .

3. Magnetic Susceptibility ( $\chi_m$ ) and Permeability ( $\mu$ )

For linear, isotropic, homogeneous materials (most common in exams), magnetization is proportional to the applied field:

\vec{M} = \chi_m \vec{H}

$\chi_m$ = magnetic susceptibility (dimensionless)
$\chi_m > 0$ for paramagnetic materials
$\chi_m < 0$ for diamagnetic materials
$\chi_m \gg 1$ for ferromagnetic materials (but not linear!)

Derivation of relative permeability $\mu_r$ :

Substitute $\vec{M} = \chi_m \vec{H}$ into the master equation:

\vec{B} = \mu_0 (\vec{H} + \chi_m \vec{H}) = \mu_0 (1 + \chi_m) \vec{H}

Define:

\mu_r = 1 + \chi_m \quad \text{(relative permeability)}

\mu = \mu_0 \mu_r \quad \text{(absolute permeability)}

Thus:

\vec{B} = \mu \vec{H}

Why this matters: It shows that the material simply scales the applied field by a factor $\mu_r$ .

4. Why $\chi_m$ Has Different Signs (Physical Reasoning)

Material Type	$\chi_m$	Why?
Diamagnetic	$\chi_m < 0$ (small, ~ $10^{-5}$ )	Applied field induces opposing dipole moments (Lenz's law at atomic level). $\vec{M}$ opposes $\vec{H}$ .
Paramagnetic	$\chi_m > 0$ (small, ~ $10^{-3}$ )	Permanent atomic dipoles align partially with $\vec{H}$ . Thermal agitation fights alignment.

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$\chi$ is dimensionless. The only dimensionless combination of $\{e,m,v,R,\mu_0\}$ is $\chi \sim \mu_0 e^2/(mR)$ (the velocity $v$ drops out); numerically $\approx 6.7\times10^{-4} \sim 10^{-4}$ , about an order of magnitude larger than the observed $\sim10^{-5}$ .

Dimensions of $\chi$

Susceptibility is defined by $\vec{M} = \chi\,\vec{H}$ . Magnetisation and magnetising field are both in $\text{A/m}$ , so their ratio is a pure number:

[\chi] = \frac{[M]}{[H]} = \frac{\text{A·m}^{-1}}{\text{A·m}^{-1}} = 1 \quad(\text{dimensionless}).

Constructing $\chi$ from atomic parameters

Look for $\chi \sim \mu_0^{a} e^{b} m^{c} v^{d} R^{f}$ . In terms of $M,L,T,I$ :

[\mu_0]=MLT^{-2}I^{-2},\ \ [e]=IT,\ \ [m]=M,\ \ [v]=LT^{-1},\ \ [R]=L.

Requiring the product to be dimensionless gives

\begin{aligned} M:&\quad a + c = 0\\ I:&\quad -2a + b = 0\\ T:&\quad -2a + b - d = 0\\ L:&\quad a + d + f = 0 \end{aligned}

From the $I$ -equation $b = 2a$ ; substituting into the $T$ -equation gives $d = 0$ ; then $c = -a$ and $f = -a$ . The combination is therefore $\left(\dfrac{\mu_0 e^2}{m R}\right)^{a}$ , and the fundamental dimensionless quantity is

\chi \sim \frac{\mu_0 e^2}{m R}.

Tip

The velocity power came out $d=0$ : $v$ drops out entirely. The estimate depends only on the orbit size $R$ and the charge-to-mass ratio, not on how fast the electron moves.

Dimension check: $\dfrac{(MLT^{-2}I^{-2})(I^2T^2)}{(M)(L)} = 1.$

Numerical estimate

For the hydrogen atom (Bohr radius $R = a_0$ ):

\frac{\mu_0 e^2}{mR} = \frac{(4\pi\times10^{-7})(1.6\times10^{-19})^2}{(9.1\times10^{-31})(5.3\times10^{-11})}.

Numerator: $4\pi\times10^{-7} \times 2.56\times10^{-38} \approx 3.22\times10^{-44}$ .
Denominator: $9.1\times10^{-31} \times 5.3\times10^{-11} \approx 4.82\times10^{-41}$ . …

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Concept understanding — Magnetic Materials Magnetization

From a Paperclip to a Magnet: The Intuition

The Precise Definition

How Magnetization Connects to the Magnetic Field

The Three Kinds of Magnetic Materials

Why this formula?

Magnetic Materials & Magnetization: Why the Key Formulas Hold

1. What is Magnetization ( $\vec{M}$ )?

2. The Fundamental Relation: $\vec{B} = \mu_0 (\vec{H} + \vec{M})$

Why this form?

3. Magnetic Susceptibility ( $\chi_m$ ) and Permeability ( $\mu$ )

Derivation of relative permeability $\mu_r$ :

4. Why $\chi_m$ Has Different Signs (Physical Reasoning)

Dimensions of $\chi$

Constructing $\chi$ from atomic parameters

Numerical estimate

Method: Dimensional Analysis to Guess an Unknown Physical Quantity from a Set of Parameters

Steps

Dimensions of χ\chiχ

Constructing χ\chiχ from atomic parameters

Numerical estimate

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Dimensions of $\chi$

Constructing $\chi$ from atomic parameters

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