Here’s a derivation of the minimum detectable magnetic field fluctuation we will be able to detect as a function of our measured angular deflection. There are some limitations to this derivation:

  1. I’ll assume that we’ve perfectly aligned the torsional zero with the external magnetic field first; i.e. at t=0, the magnetic dipole moment is precisely aligned with the torsional zero.
  2. I’ll assume that the magnetic field fluctuation is completely perpendicular to the residual initial magnetic field B_0.

Within the limits of these two assumptions, this derivation is exact. So, here we go—the figure below shows the geometry of our situation.

Initially, the magnetic dipole moment \vec{\mu} is aligned with the torsional zero, and a residual magnetic field B_0 exists; then a perpendicular field component \Delta B is applied. The dipole moment experiences a magnetic torque \vec{\mu}\times B which tries to align with the net field B, but this torque is thwarted in its effort by the restoring torque from the fiber -\kappa \delta. Hence, the equilibrium position is defined by

    \[\mu B \sin\theta = \kappa \delta\]

but, since \Delta B is perpendicular to B_0, we see that

    \[\sin\phi = \frac{\Delta B}{B},\]

and by inspection, \theta = \phi - \delta, so that we have

    \[\mu B \sin(\phi - \delta) = \kappa \delta,\]

and therefore

    \[\mu B \sin(\arcsin(\frac{\Delta B}{B}) - \delta ) = \kappa \delta.\]

By standard angle addition identities for the sine, we then have

    \[\mu B (\frac{\Delta B}{B} \cos\delta - \frac{B_0}{B} \sin\delta  ) = \kappa \delta\]

simplifying and solving for \Delta B, we have

    \[\Delta B = \frac{\frac{\kappa}{\mu} \delta + B_0 \sin\delta}{\cos\delta}\]

Clearly, for a given measured shift in orientation \delta, we obtain the smallest \Delta B when the residual field B_0 is as small as possible.

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