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 .

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 is aligned with the torsional zero, and a residual magnetic field exists; then a perpendicular field component is applied. The dipole moment experiences a magnetic torque which tries to align with the net field , but this torque is thwarted in its effort by the restoring torque from the fiber . Hence, the equilibrium position is defined by but, since is perpendicular to , we see that and by inspection, , so that we have and therefore By standard angle addition identities for the sine, we then have simplifying and solving for , we have Clearly, for a given measured shift in orientation , we obtain the smallest when the residual field is as small as possible.

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