Explain quantitatively the order of magnitude difference between the diamagnetic susceptibility of N₂ (sim 5 ×

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\text{N}_2

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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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The huge difference arises because diamagnetic susceptibility depends on the number density of atoms and the size of the electron orbits. In a gas like N₂ at STP, atoms are far apart (low density), while in a solid metal like Cu, atoms are tightly packed (high density). Additionally, copper has more electrons per atom and larger effective orbital radii, giving a much larger induced magnetic moment per atom. The combined effect yields a factor of about $10^4$ — exactly the observed gap.

Why this approach works

Diamagnetism is a universal property: when an external magnetic field is applied, it slightly alters the orbital motion of electrons, inducing a tiny magnetic moment that opposes the field. The size of this induced moment per atom is proportional to the square of the orbital radius and the number of electrons. But the bulk susceptibility $\chi$ also depends on how many atoms are packed into a given volume — the number density.

So the order-of-magnitude difference between N₂ gas and solid Cu comes from two separate factors:

Number density — how many atoms per cubic metre.
Atomic diamagnetic response — how large the induced moment is per atom.

Let’s quantify each.

Step-by-step calculation

1. Number density at STP vs. in a solid

For an ideal gas at STP (0 °C, 1 atm), one mole occupies 22.4 L = $2.24 \times 10^{-2} \,\text{m}^3$ .

Number of molecules per mole is Avogadro’s number $N_A = 6.02 \times 10^{23}$ .

So number density of N₂ molecules:

n_{\text{N}_2} = \frac{6.02 \times 10^{23}}{2.24 \times 10^{-2}} \approx 2.69 \times 10^{25} \,\text{m}^{-3}

For copper: density $\rho = 8.96 \,\text{g/cm}^3 = 8960 \,\text{kg/m}^3$ , atomic mass $M = 63.5 \,\text{g/mol} = 0.0635 \,\text{kg/mol}$ .

Number density of Cu atoms:

n_{\text{Cu}} = \frac{\rho N_A}{M} = \frac{8960 \times 6.02 \times 10^{23}}{0.0635} \approx 8.5 \times 10^{28} \,\text{m}^{-3}

Ratio of number densities:

\frac{n_{\text{Cu}}}{n_{\text{N}_2}} \approx \frac{8.5 \times 10^{28}}{2.69 \times 10^{25}} \approx 3.2 \times 10^{3}

So just from packing, Cu has about 3000 times more atoms per unit volume than N₂ gas.

Tip

This factor alone already accounts for most of the difference — but not all. The remaining factor comes from the atomic diamagnetic response.

2. Atomic diamagnetic susceptibility per atom

The classical Langevin formula for diamagnetic susceptibility per atom (or molecule) is:

\chi_{\text{atom}} = -\frac{\mu_0 e^2}{6 m_e} \sum \langle r^2 \rangle

where $\sum \langle r^2 \rangle$ is the sum of mean-square orbital radii of all electrons in the atom/molecule.

For a diatomic N₂ molecule, each nitrogen atom has 7 electrons, so 14 electrons total. But the electrons are tightly bound in small orbitals (first-row element). A typical $\langle r^2 \rangle$ for a 2p electron in N is about $(0.5 \,\text{Å})^2 = 0.25 \times 10^{-20} \,\text{m}^2$ . Summing over all electrons gives roughly:

\sum \langle r^2 \rangle_{\text{N}_2} \sim 14 \times 0.25 \times 10^{-20} \approx 3.5 \times 10^{-20} \,\text{m}^2

For copper (atomic number 29), the inner electrons (up to 3d) have smaller radii, but the outer 4s electron and especially the 3d electrons have larger orbits. A typical $\langle r^2 \rangle$ for a 3d electron in Cu is about $(1.0 \,\text{Å})^2 = 1.0 \times 10^{-20} \,\text{m}^2$ , and there are 10 such d-electrons. The 4s electron has an even larger orbit, but it contributes less because it’s only one electron. A rough sum:

\sum \langle r^2 \rangle_{\text{Cu}} \sim ( \text{core electrons: small} ) + 10 \times 1.0 \times 10^{-20} \approx 1.0 \times 10^{-19} \,\text{m}^2

That’s about 3 times larger than for N₂. …

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Q.Explain quantitatively the order of magnitude difference between the diamagnetic susceptibility of $\text{N}_2$ ( $\sim 5 \times 10^{-9}$ ) (at STP) and Cu ( $\sim 10^{-5}$ ).

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)

Why this approach works

Step-by-step calculation

1. Number density at STP vs. in a solid

2. Atomic diamagnetic susceptibility per atom

Method: Estimating Order-of-Magnitude Differences in Diamagnetic Susceptibility

Steps

Q.Explain quantitatively the order of magnitude difference between the diamagnetic susceptibility of N2\text{N}_2N2​ (∼5×10−9\sim 5 \times 10^{-9}∼5×10−9) (at STP) and Cu (∼10−5\sim 10^{-5}∼10−5).

Why this approach works

Step-by-step calculation

1. Number density at STP vs. in a solid

2. Atomic diamagnetic susceptibility per atom

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Q.Explain quantitatively the order of magnitude difference between the diamagnetic susceptibility of $\text{N}_2$ ( $\sim 5 \times 10^{-9}$ ) (at STP) and Cu ( $\sim 10^{-5}$ ).

Unlock everything free for 14 days