= Vibrating-sample magnetometer
{wiki}
A <vibrating-sample magnetometer> measures a specimen's <magnetic dipole moment> through the induced <electromotive force> as it moves relative to stationary pickup coils. Dividing that <magnetic dipole moment> by the specimen's independently determined <volume> gives its average <magnetization>.
In <Simon Foner>'s 1959 apparatus, the specimen was weighed and glued to a mount. A loudspeaker drive moved it perpendicular to an applied uniform <magnetic field> between magnet poles; stationary coils detected the induced <electromotive force>. A reference magnet on the same vibrating assembly generated a second coil <electromotive force>. Adjusting the reference <voltage divider> to an electrical null compared the specimen signal with this reference. The <measurement calibration> substituted a magnetically saturated pure <nickel> sphere of known <magnetic dipole moment>; a specimen-removed reading corrected the mount's contribution. The calibrated signal inferred the specimen's <magnetic dipole moment>, not its local <magnetization> map. <Measurement uncertainty> in the standard, positioning, finite specimen size, coil noise, and mechanical coupling limited the inference.<Versatile and Sensitive Vibrating-Sample Magnetometer by Simon Foner>[Versatile and Sensitive Vibrating-Sample Magnetometer, pp. 549–550, §I and Figs. 1–2; p. 553, §IV and Fig. 9; p. 554, §§V.A–B; p. 556, §§V.G and VI.A.1]{ref}
The readout follows <Faraday's law of induction>. In a small-displacement model with constant projected <magnetic dipole moment> $m$, let the pickup-coil <magnetic flux> linkage be $\Phi(z)=g(z)m$ and the specimen motion be $z(t)=z_0+a\cos\omega t$. Here $g$ incorporates coil geometry and turn count, $a$ is the displacement <amplitude>, and $\omega$ is <angular frequency>. To first order in displacement, with $g'(z_0)$ the <derivative> at the mean position,
$$
V(t)=-\frac{d\Phi}{dt}\simeq m\,g'(z_0)a\omega\sin\omega t.
$$
This is an original idealized readout relation: a larger induced <electromotive force> can mean larger <magnetic dipole moment>, greater motion, or a changed coil position. A synchronously moving reference allows common motion factors to cancel in the ratio of induced <electromotive forces>. The <measurement calibration> and specimen-removed control are therefore integral to the measurement. A single <amplitude> reading cannot by itself distinguish microscopic alignment from a change in individual <magnetic dipole moments>.
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