= Gouy balance
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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 $32\,\mathrm{mm}$. <Foundational Magnetic Susceptibility by TeachSpin>[TeachSpin, Foundational Magnetic Susceptibility, Instrument: The Gouy Method and “Weighing the magnet” method; Specifications: magnet and sample dimensions]{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: $|F_z|=iLB_0$. 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. <Foundational Magnetic Susceptibility by TeachSpin>[TeachSpin, Foundational Magnetic Susceptibility, Instrument: Absolute calibration method and Diamagnetism]{ref}
For weak reversible linear response, a uniform sample displacing a weakly magnetic surrounding medium has, to leading order, downward <force>
$$
F_\downarrow\simeq\frac{A(\chi_v-\chi_{\rm medium})}{2\mu_0}
\left(B_{0,\rm bottom}^2-B_{0,\rm top}^2\right).
$$
This follows by integrating the induced-dipole force density $(\chi_v-\chi_{\rm medium})\partial_z(B_0^2)/(2\mu_0)$ 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, $F_\downarrow=-g\Delta m_{\rm magnet}$ 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 $A\int_{B_{0,\rm top}}^{B_{0,\rm bottom}}M(B_0)\,dB_0$ instead of a constant-$\chi_v$ endpoint square. A balance reading then measures an integrated response, not directly the local <differential magnetic susceptibility>.
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