Separate magnetic field geometry from material response

Map a vector field, not an isolated pole

A magnetic field assigns a vector to each observation point and can predict forces on moving charges, currents and magnetic dipoles. In this introductory electromagnetic model there are no isolated magnetic monopole sources: closed-surface magnetic flux is zero. This is Gauss’s magnetic law, one of the four Maxwell equations; here it constrains the signed flux sum rather than requiring a zero field at every point. Bar-magnet field lines continue through the material to form loops, outward from the north end and inward at the south end outside. Cutting a magnet produces smaller dipoles, not a separated north pole. Closed-loop diagrams are a representation, not a claim that every general mathematical field line is a finite simple circle.

Use the supplied far-field model

For a point-dipole moment m along z, the supplied extension is B = mu0[3(m dot rhat)rhat − m]/(4pi r³). It describes the distant field of a localized dipole, not its interior. The interactive map excludes the singular source and adds a separately assigned uniform x-directed background. On the dipole axis, doubling distance reduces the pure dipole field by a factor of eight. Earth’s field is approximately dipolar on a global first model, not exactly this point-source field at every local site.

Test zero closed flux with nonzero local field

In a separate mathematical packet B = (2x,−2y,0) microtesla with coordinates in metres and coefficients in microtesla per metre, a unit cube has x-face fluxes 0 and +2, y-face fluxes 0 and −2, and zero z fluxes, all in microtesla m². Their sum is zero even though many local fields are nonzero. Increasing both gradient magnitudes equally preserves this balance; increasing only one would violate the stipulated divergence-free magnetic-source model.

Connect dipoles to microscopic material structure

Current loops and atomic electron angular momentum contribute magnetic moments. Intrinsic electron spin is quantum angular momentum, not a literal classical charged sphere spinning in place. A compass dipole tends to align with the local field; like labeled poles repel and unlike poles attract in the familiar bar-magnet comparison. Permanent and induced magnetism involve different alignment histories, not different kinds of isolated poles.

Compare material evidence rather than labels alone

Iron, nickel and cobalt are common ferromagnetic examples: domains can align and retain remanent magnetization. Aluminum, titanium and magnesium are examples of weak paramagnetic responses that do not retain that alignment when the applied field is removed. All materials have an electronic diamagnetic contribution; that response opposes the applied field and can be masked by stronger responses. In an original supplied packet, sample A has normalized moments 0, +0.02, 0 at applied-field states 0, +1, 0; sample B has 0, +0.8, +0.3; sample C has 0, −0.01, 0. A is consistent with reversible paramagnetism, B with remanence and C with diamagnetism, but these three observations do not identify chemical composition uniquely. Predict the extra evidence needed to separate a calibration offset from B’s retained moment.

State the permeability limitation

Vacuum permeability sets the reference. Material permeability describes a constitutive response and can depend on field strength, temperature, orientation and magnetic history; it is not always one constant multiplier. A supplied linear weak-response approximation B = mu0(1 + chi)H predicts a 1.02 factor for chi = 0.02 at fixed H, whereas a remanent nonlinear sample cannot be reduced to that single slope through the origin.

Worked original case

With m = 2 A m² and a 1 microtesla uniform x background, the point (1,1) m has Bx ≈ 1.10607 microtesla and Bz ≈ 0.0353553 microtesla. Pure dipole axial field is 0.4 microtesla at 1 m and 0.05 microtesla at 2 m. The uniform background must be subtracted before testing the inverse-cube ratio.

Investigation

Compare pure dipole and background-added component maps, retaining an equal-scale direction sketch. Independently sum the six flux faces. Classify the supplied three material histories with a stated alternative explanation and predict a discriminating zero-field recheck; do not infer composition from one response.

Changed-case practice

Try a new practice case

For the pure dipole on its axis, what is B at distance 2r divided by B at r?

Numerical result in the requested units Check this practice result

This public, unsaved check is not a secure assessment. Explain the source integral, symmetry or charge boundary in the retained course evidence workspace.

Sources and model boundary

Original teaching checked 31 October 2026 against the current official framework, Unit 12. The displayed magnetic calculations use the stipulated introductory approximation mu0 = 4pi×10⁻⁷ SI, not a claim that this is exact in modern SI. All apparatus and records are virtual, supplied or generated. No real magnet handling, electrical current setup, particle apparatus or material operation is requested. Public feedback and retained reports are practice evidence, not independent mastery or performed laboratory time.

Supplemental primary teaching reference: Magnetism in matter. The supplied cases, derivations and diagrams here are original; the source’s older exact-mu0 convention and classical spin language are not adopted literally.