Magnetism and matter

What you will understand

For NEET UG students: tell zero torque from stable alignment, calculate dipole energy changes, and distinguish weak diamagnetic or paramagnetic response from ferromagnetic domains.

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Quick answer: angle controls torque and energy differently

For a fixed magnetic dipole moment $\mathbf m$ in a uniform external field, $\boldsymbol\tau=\mathbf m\times\mathbf B$ and $U=-\mathbf m\cdot\mathbf B=-mB\cos\theta$. Torque magnitude is $mB\sin\theta$. Here $\theta$ is the angle between m and B. Torque vanishes both when aligned and when opposed, but the energy is minimum at alignment and maximum at opposition. The opposed position is unstable, even though its instantaneous torque is zero.

Identify the moment direction before the angle

For a current loop, $\mathbf m=NI A\hat{\mathbf n}$ with $A$ the area of one turn and $\hat{\mathbf n}$ fixed by curling right-hand fingers with conventional current. For a bar magnet, the dipole moment points from its south end to its north end. The unit of m is $\mathrm{A\,m^2}$. A loop plane at $30^\circ$ to B has its normal at $60^\circ$ or $120^\circ$ to B: torque magnitude is the same, but the energy sign needs the directed normal. A uniform field produces no net translational force on an ideal dipole, while it may produce torque.

Compare three orientations in one uniform field

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Magnetic flux and the dipole approximation

Magnetic field lines form closed loops; for a closed surface $\oint\mathbf B\cdot d\mathbf A=0$. Zero net magnetic flux does not imply zero B at every point. In the textbook bar-magnet model, cutting a magnet produces smaller magnets with both poles rather than an isolated pole. Far from a small dipole in vacuum, at equal centre-distance r, $B_\mathrm{axial}=\mu_0(2m)/(4\pi r^3)$ and $B_\mathrm{equatorial}=\mu_0m/(4\pi r^3)$ in magnitude. The axial field is along m and the equatorial field opposes m. These are far-field point-dipole approximations, not exact near-pole formulas for a finite bar.

Read the material response before assigning a formula

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Weak material response: distinguish H, M and B

Magnetisation $\mathbf M$ is net magnetic moment per unit volume, measured in $\mathrm{A/m}$. In SI, $\mathbf B=\mu_0(\mathbf H+\mathbf M)$. For a linear isotropic material in the specified range, $\mathbf M=\chi_m\mathbf H$, so $\mu_r=1+\chi_m$. Susceptibility is dimensionless. A small negative $\chi_m$ means induced magnetisation opposite to H (diamagnetic response); a small positive value means weak magnetisation along H (paramagnetic response). H is the internal magnetic field intensity in these relations; a specimen’s shape can affect its internal field, so do not substitute an external-field value without a justified model.

Ferromagnetism needs domains and history

In a ferromagnetic material, many atomic magnetic moments can order together in domains. An unmagnetised specimen can have substantial domain moments that cancel in the overall sample. Applied fields can change domain populations and alignment, giving a large response. The resulting magnetisation can be nonlinear and depend on earlier field exposure; residual magnetisation can remain after the applied field is removed. Do not treat a constant positive susceptibility over all fields as a universal ferromagnetic law. The weak linear examples in the diagram are illustrative numbers, not measurements of iron or another named substance.

Try it: a perpendicular moment is not an equilibrium

A dipole has $m=0.30\,\mathrm{A\,m^2}$ in a uniform $0.20\,\mathrm T$ field. Compare torque magnitude and energy at $90^\circ$ and $180^\circ$. Which position has the larger energy, and is either stable?

Try it: read a negative susceptibility

A hypothetical homogeneous linear isotropic material has $\chi_m=-2.0\times10^{-5}$. Its internal $H=1000\,\mathrm{A/m}$ points right. Find the magnetisation and relative permeability. Does the small negative susceptibility make the total B point left?

NEET recall: compare the three behaviours

Rebuild a short table from memory: diamagnetic → small induced response opposite H; paramagnetic → weak response along H; ferromagnetic → collective domain behaviour that can be large and history dependent. Then label aligned, perpendicular and opposed dipoles by both torque and energy. A material-class question and a dipole-orientation question require different reasoning.

Come back tomorrow: explain the tempting wrong answer

Explain why “zero torque means stable” is false using the opposed dipole. Recompute the rotation example from $180^\circ$ back to $0^\circ$; the ideal quasi-static external work becomes $-0.20\,\mathrm J$. Finally, explain why zero closed-surface magnetic flux does not mean no magnetic field. Then attempt broad chapter practice below; these are chapter pools rather than dedicated dipole or material-only sets.

Sources and review

Technical review: Codex (automated). Primary textbook sources checked on 23 October 2026. Diagrams, numerical examples and self-checks are original teaching material. Human subject review is not claimed.

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