Millikan oil-drop experiment - 2D microscope view

What is modelled

Oil drops a micrometre across are sprayed between two horizontal plates 16 mm apart. With the field off a drop falls at its terminal speed; with the field on, a charged drop is pushed up. Timing the fall and the rise over the same 2 mm (the red lines in the microscope) gives the drop's radius and its charge. Collecting many drops shows that every charge is a whole multiple of one smallest charge.

Equations

• Equation of motion: m dv/dt = qE − (ρoil − ρair)(4/3)πa3g − k v with the drag coefficient k = 6πηa / (1 + b/(p a)). The factor in the denominator is the Cunningham slip correction (b = 8.2 mPa m): at 0.5 um the drag is about 14 % lower than plain Stokes. The relaxation time m/k is a few microseconds, so drops are always at terminal speed.
• Free fall vf = Fg/k, rise vr = (qE − Fg)/k with E = V/d. Eliminating the drop size: q = k (vf + vr) / E, and the radius follows from vf by iterating the corrected Stokes law.
• Brownian motion: D = kBT / k. Each drop takes a random walk on top of its drift, so a timed crossing of distance d is a first-passage time with an inverse-Gaussian distribution (mean d/v, variance 2D d / v³). For these drops that is about 3 % noise per timing.
• Air viscosity from Sutherland's law, air density from the ideal gas law; both change with the temperature and pressure sliders.
• The elementary charge is estimated by scanning candidate values between 1.2 and 2.2 x 10^-19 C for the smallest rms distance of q/e from whole numbers.

Simplifications

• Oil density 886 kg/m3 is fixed and the drops do not evaporate (that was the reason for using oil). The field is perfectly uniform and vertical; there is no convection in the air, no residual charge on the plates and no drift of charge between measurements.
• The timer starts and stops on the exact crossing of the lines, as if by a camera, so only Brownian noise limits the timing; a human with a stopwatch would add roughly 0.1-0.3 s of reaction error. The microscope view is a flat 2D projection.
• The automatic batch chooses the voltage so every drop rises at 0.8 of its fall speed; real experimenters tune the voltage by eye. The 3D chamber shows drops about 1000 times larger than real so they can be seen.
• Millikan's own 1913 value (1.592 x 10^-19 C) was about 0.6 % low, mainly because of the air viscosity he used; this model uses the modern viscosity and recovers the modern value of e.

Model checks

• Numerically integrated drop motion reaches the analytic terminal speed and follows the 1 − e^−t/τ approach.
• A noise-free round trip through the forward speeds and the Cunningham inversion returns radius and charge exactly.
• Sampled first-passage times reproduce the analytic mean and variance of drift-diffusion timing.
• Sixty simulated drops with Brownian timing noise give a fitted elementary charge close to the true value.