Quantum evidence / Charge

Oil-Drop Charge Measurement

How can the way an oil drop moves reveal its electric charge? Compare its speed with the field off and on. With the field off, the drop falls at a steady speed where air drag balances gravity minus buoyancy, and that speed gives its size. With the field on, the speed changes; the change gives the electric force, and from it the charge. Brownian motion in the air scatters each estimate a little.

Track one negatively charged oil drop with the electric field off and then on. Separate the effects of gravity, buoyancy, viscous drag and random thermal motion, then estimate the charge from the recorded motion as a continuous value.

The drop is a sphere of mineral oil in simulated overdamped motion. The model uses a simple slip correction that depends on pressure, an ideal uniform field and a calibrated camera. The charge can be set to any value, and the estimator does not assume it comes in whole units. This is a model of the measurement. It does not reconstruct Millikan’s historical records, and it does not independently prove that charge is quantized.

Plate gap: 5 mm. Drop radius greatly enlarged. Upward height follows the latest camera record; this tracking camera is idealized. Light is scattered sideways into the microscope.

Observe and keep the record

One drop. Two tracks.

Independent field-off and field-on tracks each start at the mid-plane. Physical time advances in 0.1 s increments; playback is accelerated. A plate contact ends that track and withholds the estimate.

Ready. Predict, then acquire.

Camera height in micrometers

Drift beneath the wandering.

$t/\mathrm{s},\quad z/\mathrm{\mu m}$

Teal: field off. Gold: field on. Dots and solid lines: camera records. Fine dashes: source mean drift. Purple dashes: endpoint drift estimates. Gray: saved observations.

Inference without the source parameters

A continuous charge estimate.

Inferred charge magnitude

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Inferred radius

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Stokes-only charge

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Off drift · estimate

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On drift · estimate

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Recorded frames

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$\Delta z/\mathrm{\mu m}$

The lower plot shows raw frame-to-frame displacements, including camera noise. Position errors along a Brownian track are correlated. Drift readouts come with plug-in standard errors; they are not a confidence interval for the charge.

Compare with a saved run

Latest six raw observations
#	Phase	Time / s	Height / µm

The CSV includes the raw observations, seed, units and the source and instrument settings. Source settings are simulator metadata and are not used in the inverse estimate. One saved run lasts for this page session.

The model and the evidence behind it

What this measurement leaves out.

\Delta z=v\Delta t+\sqrt{2D\Delta t}\,\xi,\quad \xi\sim\mathcal N(0,1)

The drop is a sphere that does not evaporate, in still air at 293.15 K. Oil density is 886 kg per cubic meter, air density 1.2, viscosity 0.0000182 Pa s and slip coefficient 0.0082 Pa m. Temperature, viscosity, density, gap and voltage calibration are treated as exact. There are no air currents, evaporation, transient inertia, charge jumps, image blur or field-edge effects. Camera noise is 0.3 micrometers per frame. The raw thermal paths are generated by the simulation; they are not historical timings.

PASCO · AP-8210A manual · apparatus, slip model and nominal oil density; Brownian camera acquisition is our explicit extension.

Read the historical evidence Compare magnetic moments

Start with the essentials

Focus question: How can the way an oil drop moves reveal its electric charge?
One-sentence intuition: Keep what the simulation assumes separate from what you estimate from the recorded data.

Core mathematical model

Drift speed and diffusion of the drop

\begin{aligned}v&=\frac{QU/d-W}{\zeta}\\W&=\frac{4\pi a^3}{3}(\rho_o-\rho_a)g\\D&=\frac{k_BT}{\zeta}\end{aligned}

Height is measured upward. The drop carries a negative charge, so the formula multiplies its magnitude by the voltage of the top plate relative to the bottom plate. A positive voltage gives an upward electric force.

Drop radius with slip correction

\begin{aligned}c&=b/p,\quad R=\frac{9\eta v_f}{2(\rho_o-\rho_a)g}\\a&=\sqrt{c^2/4+R}-c/2\\\zeta&=\frac{6\pi\eta a}{1+c/a}\end{aligned}

The radius comes from the fall speed with the field off. Pressure in this formula is in pascals. The analysis that uses Stokes drag alone works from the same records.

Charge from the two drifts

\widehat Q=\widehat\zeta\,(\widehat v_{\rm on}-\widehat v_{\rm off})\frac dU,\quad \widehat v=\frac{z(T)-z(0)}T

The estimate never reads the input charge and is not rounded to a whole number. Each drift speed comes from the two end points of its track, which respects the correlation between Brownian positions.

Common difficulties

The simulation already contains the law

Typical misconceptionIf I recover the parameter the simulation used, I have independently confirmed the law.

Better mental modelThese simulated records test the method of measuring and inferring. They are not historical observations, they are not a confidence interval for the parameter, and they do not count as outside scientific certification.

Run the experiment

01
Record both tracks
Acquire a field-off track and a field-on track. Each one starts again from the mid-plane.
What to observe: Compare which way the drop drifts in each track, and how much it wanders from thermal motion.
02
Turn off the slip correction
After acquiring, turn off the slip correction and keep the records.
What to observe: With a different drag model, the same motion gives a different radius and a different charge.
03
Try a continuous charge
Save a run, choose the Continuous control preset and acquire again.
What to observe: The estimate follows the continuous charge built into the simulation. A simulated result cannot establish that charge is quantized.

Oil-Drop Charge Measurement

One drop. Two tracks.

Drift beneath the wandering.

A continuous charge estimate.

What this measurement leaves out.