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Differential magnetic susceptibility is the local slope at a specified bias magnetic field intensity, holding temperature and other stated constraints fixed. It equals volume magnetic susceptibility only for a linear response through the origin. For example, $M=M_s\tanh (H/H_s)$ has $M/H=(M_s/H_s)\tanh (x)/x$, but $dM/dH=(M_s/H_s)/{\cosh }^2(x)$, where $x=H/H_s$. At $x=1$ these are about $0.762$ and $0.420$ times their common zero-field slope. A finite sweep chord is another quantity, not automatically this derivative.

If the differentiated field is instead the internal magnetic flux density, the longitudinal chain rule applied to $B_{\mathrm{int}}={\mu }_0(H_{\mathrm{int}}+M)$ givesThis differs from the vacuum/applied-$B_0$ conversion in volume magnetic susceptibility. It assumes a differentiable single branch with locally invertible $B_{\mathrm{int}}(H_{\mathrm{int}})$.

In vacuum, magnetization is zero and magnetic flux density obeys $\mathbf{B}={\mu }_0\mathbf{H}$. The applied quantities ${\mathbf{H}}_{\mathrm{app}}$ and ${\mathbf{B}}_0={\mu }_0{\mathbf{H}}_{\mathrm{app}}$ mean the field of the external sources at the sample position with the sample absent. The internal magnetic field intensity includes the sample's demagnetizing field; its internal magnetic flux density additionally includes ${\mu }_0\mathbf{M}$. A field quoted in the tesla is a $B$ quantity, even when an instrument informally calls it “$H$”.

For a uniformly magnetized ellipsoid, along a principal axis with demagnetizing factor $N_d$ in the International System of Units,For a sphere, $N_d=1/3$. The axial value approaches zero in the long slender ellipsoid limit and the normal value approaches one in the thin flat ellipsoid limit. Thus a stipulated intrinsic volume magnetic susceptibility ${\chi }_v=3$ would yield apparent values $1.5$ in a sphere and approximately $3$ in a long axial sample. These are calculated geometry effects, not measured material values. For fixed $N_d$, implicit differentiation gives the same conversion for local differential magnetic susceptibility, $dM/d{H}_{\mathrm{app}}={\chi }_{\mathrm{diff}}/(1+N_d{\chi }_{\mathrm{diff}})$.

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.

Magnetic susceptibility describes how readily a magnetic field changes a material's magnetization. The field variable and normalization matter: volume magnetic susceptibility in the International System of Units is dimensionless, whereas a derivative of total magnetic moment with respect to $B$ has units of $\mathrm{J}{\mathrm{T}}^{-2}$. A slope measured against the applied magnetic field intensity also depends on the sample's demagnetizing field.

For an isotropic, reversible linear response through the origin, writeHere $\mathbf{M}$ is magnetization, or magnetic moment per volume; ${\mathbf{H}}_{\mathrm{int}}$ is the internal magnetic field intensity; ${\mathbf{B}}_{\mathrm{int}}$ is the internal magnetic flux density; and ${\mu }_0$ is the vacuum permeability. Both $M$ and $H$ have units $\mathrm{A}{\mathrm{m}}^{-1}$. The magnetic permeability is ${\mu }_0(1+{\chi }_v)$. ^[ref]

The independent-moment magnetic response below shows why one number cannot describe every field: magnetization approaches a saturation magnetization, while its differential magnetic susceptibility approaches zero. Anisotropy requires a magnetic susceptibility tensor; hysteresis requires specifying the preparation and sweep branch as well as the bias magnetic field, temperature, and timescale. The Gouy balance turns a small response into a measurable force without presuming that its readout is already the intrinsic volume magnetic susceptibility.

Write the two allowed moment projections as $\pm \mu$, where $\mu =g{\mu }_B/2$ for spin one-half. Their Zeeman energies in the prescribed flux density are $\mp \mu B$. The single-particle partition function and the mean moment along the field areFor noninteracting moments, magnetization is moment per volume:For electron-like spin-only moments $g\simeq 2$, these become $M_s=N{\mu }_B/V$ and $\eta ={\mu }_B/k_B$.

Differentiate with respect to the specified $B$ to obtainThis $B$ derivative is not a dimensionless SI volume susceptibility. In the weak-feedback approximation $B\simeq {\mu }_{0}H$, the usual dimensionless differential susceptibility is ${\chi }_{\mathrm{diff}}\simeq {\mu }_0{\chi }_B$. The small-field limit gives Curie's law ${\chi }_0={\mu }_{0}N{\mu }^2/(V{k}_{B}T)$. At finite bias a secant response $M/H\simeq {\mu }_{0}M/B={\chi }_0\tanh (x)/x$ differs from the differential slope ${\chi }_0{\mathrm{sech}}^{2}x$. For an internal $B={\mu }_0(H+M)$ retained self-consistently, the exact local conversion is ${\chi }_H={\mu }_0{\chi }_B/(1-{\mu }_0{\chi }_B)$; demagnetizing geometry can further affect an applied-field measurement. Stating the field convention avoids a hidden factor of ${\mu }_0$ or confusion between a slope and a ratio.

For $B=5$ T, the same spin-only two-state probabilities apply, but the low-temperature argument is no longer small. At 300 K, $x=0.0111952$; at 2 K, $x=1.67928$. HenceThe 2 K sample is about $93.3\%$ of its saturation magnetization, with about $96.6\%$ of the moments in the aligned state. The two percentages describe different observables. A linear extrapolation would predict $M/M_s\simeq x=1.679$, exceeding saturation and therefore contradicting the bounded two-state population.

For completeness, at this 2 K, 5 T bias the dimensionless secant susceptibility is approximately ${\chi }_0\tanh x/x=0.01199$, while the differential susceptibility is ${\chi }_0{\mathrm{sech}}^{2}x=0.002805$ under the same prescribed-field approximation. Neither equals the zero-field Curie slope $0.02159$. Thus saturation reduces the incremental response even while the magnetization itself is large.

Saturation magnetization is the limiting magnetization when the responding magnetic moments are fully aligned within the stated model or experimental regime. In independent-moment magnetic response, $M_s=n\mu$. Reaching a large magnetization does not imply a large differential magnetic susceptibility: the remaining slope can be almost zero.