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A Gouy balance detects the force exerted by a nonuniform magnetic field on a sample of uniform cross-section extending between high- and low-field regions. TeachSpin's implementation reverses the usual arrangement: a rare-earth permanent-magnet assembly sits on a one-pan electronic balance, while a separate holder brings the sample into the field. The magnet's apparent mass change measures the opposite of the sample's vertical force, by Newton's third law. The described pole faces have diameter $25\mathrm{mm}$ and separation $15\mathrm{mm}$; samples occupy cuvettes of cross-section $(10\mathrm{mm}{)}^2$ and height about $32\mathrm{mm}$. ^[ref]

The same balance calibrates magnetic flux density by measuring the reaction force on the magnet when a wire of known horizontal length $L$ carries known electric current $i$ perpendicular to the field: $|F_z|=iL{B}_0$. The manufacturer describes water being repelled by the high-field region, identifying negative magnetic susceptibility. This observed sign supplies a qualitative test, not a measured Curie's law curve. ^[ref]

For weak reversible linear response, a uniform sample displacing a weakly magnetic surrounding medium has, to leading order, downward forceThis follows by integrating the induced-dipole force density $({\chi }_v-{\chi }_{\mathrm{medium}}){\partial }_z(B_0^2)/(2{\mu }_0)$ along a downward coordinate. $A$ is the cross-sectional area, and the two $B_0$ values are applied magnetic flux densities. The force depends on the endpoint fields; the entire field-gradient profile need not be measured under these assumptions. In the reversed apparatus, $F_{\downarrow }=-g\Delta m_{\mathrm{magnet}}$ after subtracting the holder background, where $g$ is gravitational acceleration. This sign must be reversed before inferring volume magnetic susceptibility.

Useful controls are an empty-holder reading at the same position, a distant-sample baseline, repeated wire-current reversals during field calibration, and stable temperature. They separate holder response, balance drift, force offsets, and thermal changes; these are proposed checks for the stated inference. Finite endpoint fields, nonuniform cross-section, uneven powder packing, stray forces and a demagnetizing field limit the simple formula. If magnetization is nonlinear, the weak-response force involves $A{\int }_{B_{0,\mathrm{top}}}^{B_{0,\mathrm{bottom}}}M(B_0)d{B}_0$ instead of a constant-${\chi }_v$ endpoint square. A balance reading then measures an integrated response, not directly the local differential magnetic susceptibility.

Consider a stipulated dilute collection of independent two-level systems, each with longitudinal magnetic moment $s\mu$, $s=\pm 1$, and energy $E_s=-s\mu B_0$. The prescribed vacuum/applied magnetic flux density is $B_0={\mu }_0{H}_{\mathrm{app}}$. Ignore interactions, orbital magnetic response, and the small sample-field correction. This is the magnetic version of the existing independent spin-one-half two-level system, with alignment probabilities rather than an interacting Curie–Weiss model.

For temperature $T>0$, the canonical ensemble gives the one-moment partition function $Z_1=2\cosh x$, with $x=\mu B_0/(k_{B}T)$. Taking the expected value over its two Boltzmann weights givesHere $n$ is the number density, $k_B$ the Boltzmann constant, and $\tanh$ the hyperbolic tangent. For $|x|\ll 1$, its small-argument expansion of the hyperbolic tangent gives $M\simeq {\mu }_{0}n{\mu }^2{H}_{\mathrm{app}}/(k_{B}T)$, a Curie's law response in the applied-$H$ convention. For large $|x|$, $M$ approaches the saturation magnetization $n\mu$ with sign set by the field, and its differential magnetic susceptibility falls towards zero.

Both plots use stipulated $n=10^{26}{\mathrm{m}}^{-3}$, $\mu =9.2740101\times 10^{-24}\mathrm{J}{\mathrm{T}}^{-1}$, $k_B=1.380649\times 10^{-23}\mathrm{J}{\mathrm{K}}^{-1}$, and ${\mu }_0=1.25663706\times 10^{-6}\mathrm{T}\mathrm{m}{\mathrm{A}}^{-1}$. The moment is approximately one Bohr magneton; the chosen vacuum permeability is a rounded model input. At $T=2\mathrm{K}$, $M_s=927.401\mathrm{A}{\mathrm{m}}^{-1}$ and ${\chi }_{\mathrm{model}}(0)=3.91410\times 10^{-4}$. The linear approximation overestimates $M$ by less than $3\%$ for $|x|\le 0.3$, corresponding to $|{\mu }_0{H}_{\mathrm{app}}|\le 0.89324\mathrm{T}$ or $|H_{\mathrm{app}}|\le 0.71082\mathrm{MA}{\mathrm{m}}^{-1}$. These parameters keep the modeled response weak; neglected magnetic feedback is of relative order ${\chi }_{\mathrm{model}}(0)$, rather than an explanation of a material's ordered phase.

At $B_0=0$, cooling increases the zero-field slope as $1/T$. At fixed $B_0\ne 0$, cooling instead gives ${\chi }_{\mathrm{model}}\sim 4{\mu }_{0}n{\mu }^2{e}^{-2\mu |B_0|/(k_{B}T)}/(k_{B}T)$: a saturated moment can hardly change further. The order of the zero-field and zero-temperature limits matters. This independent model has no finite-temperature collective phase transition; the diverging zero-field slope alone does not establish one.

Volume magnetic susceptibility is the dimensionless ratio of longitudinal magnetization to internal magnetic field intensity in a linear regime through the origin. Positive response gives paramagnetism. Always distinguish magnetization $M=\mathcal{M}/V$ from total magnetic moment $\mathcal{M}$: dividing by volume changes the normalization. Mass-normalized and mole-normalized values likewise require the mass density or the volume per amount of substance before comparison with ${\chi }_v$.

In the existing Pauli paramagnetism formula, $M$ denotes total magnetic moment, and $g(E_F)$ is the total spin-summed density of states. Relabel that response asHere ${\mu }_B$ is the Bohr magneton. If its $B$ is a prescribed vacuum/applied magnetic flux density $B_0={\mu }_0{H}_{\mathrm{app}}$, the dimensionless apparent slope is ${\mu }_0{\chi }_B^{\mathrm{vol}}$. It agrees approximately with intrinsic ${\chi }_v$ only when the demagnetizing field correction and magnetic feedback are negligible. The existing Curie's law formula also uses a total-magnetic moment, $B$-derivative convention; its $N$ counts moments rather than giving a number density.