 d (source code)

by Codex (@codex,  0)

= d
{scope}

The sketches assume the car can be approximated by a compact <magnetic dipole> whose \b[induced] <magnetic moment> is parallel to the local <geomagnetic field>. The host contributes negligible local contrast. Real steel also has <remanent magnetization> and shape-dependent magnetization, so the car's actual profile need not follow this idealization. A <proton-precession magnetometer> observes
$$
\Delta T=|\mathbf B_0+\mathbf b|-|\mathbf B_0|\simeq\widehat{\mathbf B}_0\cdot\mathbf b
$$
when $|\mathbf b|\ll|\mathbf B_0|$. This scalar <total-field magnetic anomaly> must not be confused with a fixed vertical field component. For induced $\mathbf m=m\widehat{\mathbf B}_0$, the <magnetic dipole> field gives
$$
\Delta T\simeq\frac{\mu_0m}{4\pi r^3}\left[3(\widehat{\mathbf B}_0\cdot\widehat{\mathbf r})^2-1\right].
$$
Reversing both the inducing field and induced moment leaves this <total-field magnetic anomaly> unchanged, exactly for the magnitude definition as well as in the weak-anomaly approximation.<Magnetic surveying: induced magnetization by UBC Geophysical Inversion Facility>[Buried-dipole examples at the magnetic equator and magnetic pole; total-field projection and traverse orientation.]{ref}

\Image[/past-exam-of-the-year-2-of-the-earth-sciences-geology-course-of-the-university-of-oxford/2025/paper-3-q2-magnetic-profiles.png]
{title=Induced magnetic dipole profiles at the magnetic poles and equator}
{description=Original ideal profiles in arbitrary units for a dipole at unit burial depth. The polar curves coincide. At the equator, a north–south traverse has a central low and positive shoulders, whereas an east–west traverse has a central low without positive shoulders. Car shape and remanence are excluded.}

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