A U-tube differential manometer contains a connected column of liquid with one gas connection above each liquid meniscus. Write $p_L,p_R$ for the left and right pressures, and $z_L,z_R$ for the meniscus heights on an upward vertical coordinate. Assume constant mass density $\rho$ and downward gravitational acceleration of magnitude $g$; neglect the gas column's mass density and differences in capillary pressure. The signed vertical length difference $\Delta h=z_R-z_L$ then givesLarger left pressure means a lower left liquid meniscus: $p_L>p_R$ implies $z_L<z_R$. Equal pressures give equal meniscus heights.

The balance follows directly from hydrostatic pressure. At a common height $z_0$ below both surfaces, the two points in the connected homogeneous liquid have equal pressure:Subtracting cancels the common bottom depth and leaves $p_L-p_R=\rho g(z_R-z_L)$. The same-height rule follows from hydrostatic equilibrium, $\nabla p=-\rho g{\mathbf{e}}_z$: a horizontal displacement changes no pressure, and for constant $\rho$ the quantity $p+\rho gz$ is constant throughout the connected liquid. Equal height alone is insufficient for unrelated vessels or for fluid flow with significant acceleration.

For an original constructed example, choose water of stipulated mass density $\rho =1000\mathrm{kg}{\mathrm{m}}^{-3}$, gravitational acceleration $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$, and vertical length difference $\Delta h=0.060\mathrm{m}$. ThenThe SI pressure unit is the pascal: the product has units $\mathrm{kg}{\mathrm{m}}^{-3}\times \mathrm{m}{\mathrm{s}}^{-2}\times \mathrm{m}=\mathrm{kg}{\mathrm{m}}^{-1}{\mathrm{s}}^{-2}=\mathrm{Pa}$. Swapping the two connections gives $-588.6\mathrm{Pa}$ and reverses which liquid meniscus is lower. In equal-bore arms, volume conservation makes each surface move $30\mathrm{mm}$ from its initial equal-height position; using only one surface's displacement would halve the inferred pressure. With unequal arm areas $A_L,A_R$, a left fall $a$ and right rise $b$ instead obey $A_{L}a=A_{R}b$, while the vertical length difference is still $a+b$.

If the right port is open to the local atmosphere, $p_L-p_R$ is the left gauge pressure. Only with that reference and known atmospheric pressure can one obtain absolute pressure fromFor an explicitly stipulated atmospheric pressure of $100000\mathrm{Pa}$, the example gives $100588.6\mathrm{Pa}$ absolute pressure. If the right port is connected to another vessel or a pressure tapping, the same reading is a differential pressure and supplies no absolute pressure without knowing that reference.^[ref]

A magnetic circuit approximates a narrow core and its gaps by sections carrying a common magnetic flux $\Phi$. With negligible leakage and approximately uniform magnetic flux density across each section, Gauss's law for magnetism gives $B=\Phi /A$. The magnetostatic Ampère law then gives $NI=\int Hdl=\Phi \int dl/({\mu }_0{\mu }_{r}A)$. This requires an appropriate material relation; constant magnetic permeability is an approximation for an actual magnetic core.

An infinite solid cylinder with radius $R$, prescribed uniform volume charge density $\rho$, and angular velocity $\omega \hat{\mathbf{z}}$ has current density $\mathbf{J}=\rho \omega r\hat{\mathbf{\varphi }}$. Treat a shell at radius $a$ and thickness $da$ as a solenoid carrying surface current $\rho \omega ada$. With no applied background magnetic field and permeability $\mu$,The same result follows from Ampère's circuital law, since $-d{B}_z/dr=\mu \rho \omega r$ inside.

Ferrimagnetism has opposing, generally unequal ordered magnetic sublattices, with net magnetization $|M_A-M_B|$ in a collinear model. Louis Néel used it to explain magnetic oxides.^[ref]

The counting example in magnetic sublattice gives $3(2m_0)-3(m_0)=3m_0$. Unequal populations also suffice: four sites carrying $+m_0$ and two carrying $-m_0$ give $2m_0$, even with identical individual magnitudes. Thus one must compare total sublattice contributions per common volume, rather than merely asking whether two atomic species carry different magnetic moments.

Both ferromagnetism and ferrimagnetism can produce a macroscopic magnet. Their internal organization differs: the latter retains opposing contributions. This makes ferrimagnetism a useful bridge between net magnetization and microscopic magnetic order: reducing one opposing contribution can increase the net sum, even though the total sum of moment magnitudes decreases.

Antiferromagnetism is ordered compensation of magnetic moments. In the collinear two-sublattice model, ${\mathbf{M}}_A=-{\mathbf{M}}_B\ne 0$: net magnetization vanishes but staggered magnetization remains. Louis Néel developed this possibility.^[ref]

For an original local energy check, perturb the antiparallel pair in exchange interaction by writing $\theta =\pi -\delta$. ThenFor $J>0$, a small relative rotation costs positive energy. This verifies a pair's preference; it does not prove a bulk ordering temperature. A whole crystal must satisfy many bonds simultaneously, and thermal fluctuations and quantum effects matter. The Heisenberg antiferromagnet and antiferromagnetic Ising model specify different assumptions about those collective degrees of freedom.

Why is the distinction useful? A measurement of zero magnetization cannot decide between an ordered alternating arrangement, a disordered ensemble, and oppositely magnetized magnetic domains. A spatial probe such as magnetic neutron diffraction tests the pattern directly. The manganese monoxide neutron diffraction experiment is a particularly clear example: the order adds a new periodic scattering signal even though the net magnetic moment cancels.

The exchange interaction is an effective interaction between spins originating in the quantum mechanics of electrons. Its sign convention must accompany any formula. For the illustrative classical fixed-length spin pair used here, writeWith this plus-sign convention, $J>0$ favors antiparallel spins: $E(\pi )=-J{S}^2$, compared with $E(0)=+J{S}^2$. The energy difference is $2J{S}^2$. If instead the Hamiltonian is written with an overall minus sign, the same physical preference has a negative coupling. This prevents a sign convention in the Ising model from being mistaken for a disagreement about the ordering.

Magnetic order is a reproducible spatial organization of microscopic magnetic moments. A net magnetization describes their sum; a staggered magnetization describes an alternating pattern. These are different order parameters. For example, an alternating row of equal up and down arrows has zero sum but a definite pattern, whereas randomly oriented arrows can have a small sum without that pattern.

A magnetometer measures a magnetic field or a specimen's magnetic dipole moment. A vibrating-sample magnetometer determines the latter from the electromotive force induced in pickup coils by specimen motion.

Diamagnetism is a magnetic response in which the induced magnetization opposes the applied magnetic field intensity. In a linear isotropic description, $\mathbf{M}=\chi \mathbf{H}$ with negative magnetic susceptibility $\chi$.

Interlayer exchange coupling is an interaction that favors particular relative orientations of magnetization in different magnetic layers. In a metallic multilayer, spacer electrons can mediate coupling even when a nonmagnetic layer separates the magnets.

A simple model writes energy per area as $E/A=-J{\hat{\mathbf{m}}}_1\cdot {\hat{\mathbf{m}}}_2$. Positive $J$ favors parallel alignment and negative $J$ favors antiparallel alignment. Spacer material and thickness affect the coupling; magnetic-field sweeps can reveal the field needed to overcome it. Correlated interface roughness can instead produce orange-peel coupling, so a preferred alignment alone does not uniquely establish exchange mediation. ^[ref]

The magneto-optic Kerr effect is a change in the polarization of an electromagnetic wave upon reflection from a magnetized material. The magnetic response can change both polarization orientation and ellipticity. It differs from the Faraday effect, which is observed in transmission.

A polarizer, reflective sample, analyzer and photodetector turn a small polarization change into an intensity signal. Sweeping an applied magnetic field while recording the signal can trace changes of magnetization and switching. The probed component depends on optical and magnetic geometry, so interpreting the trace requires calibration rather than identifying arbitrary brightness with total magnetization. ^[ref]

Magnetic shielding reduces an unwanted magnetic field in a specified region. Residual fields and fields generated inside the shield still need to be assessed in a magnetometer experiment.

Magnetic anisotropy is a directional dependence of a material's magnetic properties. A preferred direction must be distinguished from an externally imposed magnetic field.

A magnet is a material object or current arrangement that produces a magnetic field. Permanent magnets retain magnetization, while electromagnets use a supplied electric current in a winding.

For an ideal infinitely long solenoid with interior magnetic field $B_0\hat{\mathbf{z}}$ and radius $R$, an axisymmetric Coulomb gauge representative of the magnetic vector potential isThis follows from $2\pi r{A}_{\varphi }=\Phi (r)$ by Stokes theorem. The exterior has zero curl but nonzero circulation around the enclosed magnetic flux. The Coulomb gauge condition alone does not eliminate every residual gauge transformation; symmetry and regularity select this customary representative.

For $r,b>0$, minimization of the homogeneous Landau free energy gives $2rm+2b{m}^3=B^3$. The unique minimum near zero has $m=B^3/(2r)+O(B^9)$. Therefore its first two derivatives with respect to $B$ vanish at zero, whereas its third derivative is $3/r$. With $r=a(T-T_c)$ the first nonzero response diverges as $3/[a(T-T_c)]$. This is response to the variable $B$ through a cubic source, so it differs from the conventional third-order coefficient for a linearly coupled magnetic field. The missing factor $3!$ would give an incorrect susceptibility if one confused a series coefficient with a derivative.

One noninteracting spatial orbital with two spin energies $ϵ\pm {\mu }_{B}B$, in the grand canonical ensemble with mean occupation one, has chemical potential $\mu =ϵ$ for every positive temperature. Its spin population difference is $-\tanh [{\mu }_{B}B/(2k_{B}T)]$, giving total-moment response $\chi ={\mu }_B^2/(2k_{B}T)$ at zero field. Empty and doubly occupied orbital states are permitted. Fixing exactly one electron instead leaves only the two singly occupied states and gives $\chi ={\mu }_B^2/(k_{B}T)$. The factor of two is an ensemble effect in this small system.

An electromagnet produces a magnetic field using an electric current. Winding an electrical conductor into a coil combines the fields from its turns; a long solenoid is an idealized example. Far from its ends, an air-core solenoid has approximatelywhere $N$ turns carry electric current $I$ over length $L$. Reversing $I$ reverses the axial magnetic field. ^[ref]

A core with ferromagnetism can concentrate the magnetic flux, but its response depends on magnetic saturation and magnetic hysteresis. The constant-permeability air-core formula cannot simply be extrapolated to arbitrarily large core fields. A magnetic compass near a powered coil provides a qualitative test: switching or reversing the electric current changes the compass orientation.