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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A mock-up of our pendulum hanging in our enclosure. The magnets are gold plated Neodymium magnets (1" diameter), and they sandwich the fiber between to identical aluminum disks. The final pendulum will have 4 magnets and the outer magnets will have 50mm diameter front surfaced mirrors super-glued in place; the aluminum disks are designed so that the pendulum will have equal moments of inertia about each principle axis.

Our machinist (Steve Innes) has finished the main enclosure. Now what’s left is for me to give him plans for the optical system. I’ll be designing two different optical systems. The first will mount on the rotating plate visible in the photograph and will be used for calibrating the pendulum (finding out the torsion constant and the moment of inertia — simultaneously; but that’s a story for a future post). This internal system will be removed after the calibration is complete, and replaced with an optical system on the exterior plate (removed so that you can see the interior).
Now, I’m awaiting a new quadrant photodiode, as I fried the old one by back-biasing the diode accidentally. If there’s good news in this, it is that I found a source that sells the same detector for a cheaper price — Pacific Silicon Sensor.

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