An ideal bismuth compass would eventually point along the lines of constant Earth magnetic field strength, which are typically roughly East-West, but:
- Every measurement would take at least a hundred thousand years since that is minimum settling time needed for the needle to align itself East-West, and
- The measurement would be ambiguous, since you wouldn't know which direction was East and which direction was West.
It would be much faster to just wait for a sunny day and determine East and West from where the Sun rises and sets, or to simply use a regular ferromagnetic compass.
Settling Time
Bismuth is strongly diamagnetic, so a small piece of bismuth with volume $V$ is repelled by a magnetic field $\vec{B}$ with a force: $$ \vec{F}=\frac{\chi}{\mu_0}V\left(\vec{B} \cdot \vec{\nabla}\right)\vec{B} $$ where $\chi=−1.66\times 10^{-4}$ is bismuth's volume magnetic susceptibility and $\mu_0$ is the permeability of vacuum. If the magnetic field and gradient are aligned (e.g. in the $z$ direction), this simplifies to: $$ F=\frac{\chi}{\mu_0}V B \frac{\partial B}{\partial z} $$ If my quick calculations are correct, for a needle with mass (per volume) density $\rho$, mass $M$, and length $L$, the torque around its midpoint will be $$ \tau=\frac{\chi\,ML^2}{24\,\mu_0\,\rho}\,\left(\frac{\partial B}{\partial z}\right)^2\, \sin{2\theta} $$ where $\theta$ is the angle of the needle with respect to the East-West direction. For a thin needle whose radius is much less than its length, the angular acceleration due to this torque will be: $$ \ddot{\theta}=\frac{\chi}{2\,\mu_0\,\rho}\,\left(\frac{\partial B}{\partial z}\right)^2\, \sin{2\theta} $$
Near 45° North, the magnetic field strength is about $45\,\mathrm{\mu T}$, with a field gradient $\sim 20\,\mathrm{pT/m}$ in altitude and $\sim 5\,\mathrm{pT/m}$ in latitude. This would give the bismuth needle an angular acceleration:
$$\ddot{\theta} \sim 3\times 10^{-25}\ \mathrm{radians/s^2}$$
So if the needle starts off randomly oriented at $\theta\sim 45$° to the East-West direction, it will take about
$$t_{settling} \sim \sqrt{2 \theta/\ddot{\theta}} \sim 100,000\ \mathrm{years} $$
for the needle to align and settle in the East-West direction.
This is a rather long wait, and we haven't even considered that movements of the Earth's magnetic poles and seismic, atmospheric, thermal, human, and other possible disturbances would likely mean the needle would never settle.
Ambiguity
Although the needle might eventually align itself to the East-West direction, either end of the needle is equally likely to point East or West. To distinguish East from West, you'd need more information such as where the Sun rises and sets. But if you can see where the Sun rises and sets daily, you don't need a bismuth compass that takes thousands of years to settle.
