Estimate of the Earth's magnetic field

Problem

Estimate the Earth’s magnetic field at the surface, knowing that a compass orients itself towards magnetic North (the deflection of a precision compass around a wire carrying a small current, placed horizontally, can help estimate the value).

Solution

Idea behind the method: a horizontal wire is placed above a compass and a known current $i$ is passed through it. The wire generates a field $B_{filo}$ perpendicular to the Earth’s field $B_T$. By increasing the current (or bringing the wire closer) one finds the threshold at which the compass deflects by exactly $45^\circ$: at that point the two fields have the same magnitude, $B_{filo} = B_T$.

At that point it is enough to calculate the field of the wire with the trial values:

$B_T = B_{filo} = \frac{\mu_0\, i}{2\pi r}$

With typical laboratory values (current of a few amperes, distance of a few centimetres) one obtains a value of the correct order of magnitude.

$\ev{B_T \sim 10^{-5}\;\text{T}}$

Precise measurements give $B_T \approx 5\cdot 10^{-5}$ T, in excellent agreement with the estimate.

Links

Topics: Magnetismo Concepts: Campo magnetico Skills: Stime di Fermi