Diamagnetism

Diamagnetism is the weakest form of magnetism, exhibited by materials that develop a slight opposition to an applied external magnetic field. Unlike ferromagnetic materials such as iron, which are strongly attracted to magnets, diamagnetic materials — including water, copper, gold, and bismuth — are gently repelled. This repulsion is so faint that it goes unnoticed in everyday life, but it can be measured precisely in the laboratory and has remarkable consequences, including the levitation of small objects such as frogs and strawberries using powerful superconducting magnets. The origin of diamagnetism lies at the quantum mechanical level: when an external magnetic field is applied, it slightly alters the orbits of electrons in every atom, inducing tiny magnetic moments that oppose the field. Because this effect is present in all materials, every substance is at least weakly diamagnetic. However, in paramagnetic and ferromagnetic materials, stronger effects overwhelm the diamagnetic contribution, making it effectively invisible. Only in materials with no unpaired electrons — so that there is no permanent atomic magnetic moment — does diamagnetism dominate. Understanding diamagnetism is important not only for classifying magnetic materials, but also for applications such as magnetic levitation, magnetic resonance imaging (MRI) contrast, and the study of superconductors, which are perfect diamagnets expelling all magnetic flux from their interior — a phenomenon known as the Meissner effect. High school physics students who grasp diamagnetism gain a deeper understanding of how matter interacts with electromagnetic fields at the atomic scale.

Key takeaways
  • Electrons as Tiny Current Loops
  • Applying an External Field — Lenz's Law at the Atomic Scale
  • Induced Magnetisation Opposes the Field
  • Force on the Material
  • Temperature Independence

Electrons as Tiny Current Loops

Every electron orbiting a nucleus is equivalent to a tiny current loop, producing a small orbital magnetic dipole moment μ⃗\vec{\mu}. In most atoms, these moments cancel in pairs due to the Pauli exclusion principle, leaving no net permanent magnetic moment — a key condition for diamagnetic behaviour.

Applying an External Field — Lenz's Law at the Atomic Scale

When an external magnetic field B⃗\vec{B} is switched on, it changes the magnetic flux through each electron's orbital loop. By Faraday's law of induction, this induces an EMF that drives a change in the orbital motion. By Lenz's law, the induced change opposes the cause: the electron speeds up or slows down so that its new magnetic moment partially cancels the applied field. This is the microscopic origin of the diamagnetic effect.

Induced Magnetisation Opposes the Field

The collective effect of all atoms producing opposing moments gives the material a net magnetisation M⃗\vec{M} that points antiparallel to B⃗\vec{B}. The magnetisation is proportional to the applied field: M⃗=χmH⃗\vec{M} = \chi_m \vec{H}, where χm\chi_m is the magnetic susceptibility. For diamagnets, χm\chi_m is small and negative, typically in the range −10−6-10^{-6} to −10−5-10^{-5}. The negative sign encodes the opposition to the applied field.

Force on the Material

A magnetic field exerts a force on a magnetised body. For a diamagnet in a non-uniform field, the force is F⃗=∇(m⃗⋅B⃗)\vec{F} = \nabla(\vec{m} \cdot \vec{B}). Because m⃗\vec{m} is antiparallel to B⃗\vec{B}, the force pushes the material toward regions of lower field strength — away from the magnet. This is why diamagnets are repelled from the poles of a permanent magnet.

Temperature Independence

Unlike paramagnetism, which follows Curie's law (χm∝1/T\chi_m \propto 1/T) and weakens with increasing temperature, diamagnetism is essentially temperature-independent. The induced effect depends only on the electronic structure of the atom and the applied field, not on thermal alignment of pre-existing moments. This makes diamagnetic susceptibility a robust, stable property of a material.

The Key Equation

$$\chi_m = -\frac{\mu_0 N Z e^2}{6 m_e} \langle r^2 \rangle
- **\chi_m** — Magnetic susceptibility (dimensionless): Ratio of magnetisation to applied magnetic field intensity; negative for diamagnets. - **\mu_0** — Permeability of free space (T\cdot m/A): Fundamental constant equal to $4\pi \times 10^{-7}$ T\cdot m/A. - **N** — Number density of atoms (m⁻³): Number of atoms per unit volume of the material. - **Z** — Atomic number (dimensionless): Total number of electrons per atom contributing to the diamagnetic effect. - **e** — Elementary charge (C): Charge of one electron, approximately $1.602 \times 10^{-19}$ C. - **m_e** — Electron mass (kg): Rest mass of an electron, approximately $9.109 \times 10^{-31}$ kg. - **\langle r^2 \rangle** — Mean-square orbital radius (m²): Quantum mechanical average of the squared distance of electrons from the nucleus. ## Worked Example **Problem.** A sample of bismuth has a magnetic susceptibility of $\chi_m = -1.66 \times 10^{-4}$. It is placed in a uniform external magnetic field of intensity $H = 1.20 \times 10^5$ A/m. (a) Calculate the magnetisation $M$ of the bismuth sample. (b) State whether bismuth is attracted to or repelled from the magnet, and explain why.

M = \chi_m H = (-1.66 \times 10^{-4})(1.20 \times 10^5) \approx -19.9 \text{ A/m}

**Answer.** The magnetisation of the bismuth sample is approximately $-19.9$ A/m (antiparallel to the applied field), and the sample is repelled from the magnet because its negative susceptibility means the induced moment opposes the external field.
Figure
Diagram
Diamagnetism: Induced Moments Oppose Applied FieldNSExternal field B (left to right)DiamagneticMaterialInduced M (opposing B)FFrepulsionrepulsionSusceptibility: χ_m ≈ −10⁻⁵ (small, negative)Magnetisation M = χ_m H (antiparallel to H)Net force pushes material away from both poles
Origins · 1 minPaul Langevin and how this was discovered
PL
Paul Langevin
1872–1946 · French

Paul Langevin developed the classical theory of both diamagnetism and paramagnetism in 1905, deriving the expression for diamagnetic susceptibility — now called the Langevin formula — by treating electrons as classical orbiting charges subject to Lenz's law. He also derived the Langevin function describing paramagnetic saturation. His framework was later refined by quantum mechanics but remained conceptually foundational. Langevin was one of Albert Einstein's closest friends in Europe, and he was among the first physicists to fully understand and champion the theory of special relativity to French scientific audiences. He was also arrested by the Nazis during the occupation of France and placed under house arrest for his anti-fascist activities.

  • Paramagnetism — Paramagnetism is the contrasting behaviour where materials with unpaired electrons are weakly attracted to a field; comparing it with diamagnetism clarifies how electron configuration determines magnetic response.
  • Ferromagnetism — Ferromagnetism represents the strongest magnetic ordering and is directly contrasted with the tiny, field-induced, negative susceptibility of diamagnetism.
  • Faraday's Law of Electromagnetic Induction — The atomic-scale mechanism of diamagnetism is a direct application of Faraday's law: a changing magnetic flux induces an opposing EMF and hence an opposing current (electron orbital change).
  • Lenz's Law — Lenz's law is the physical principle that explains why the induced orbital moment opposes the applied field, making it the conceptual heart of diamagnetism.
  • Magnetic Susceptibility — Magnetic susceptibility χm\chi_m is the quantitative measure used to classify diamagnets (negative χm\chi_m), paramagnets (small positive χm\chi_m), and ferromagnets (large positive χm\chi_m).