Oil drop experiment

The oil drop experiment infers an oil drop's electric charge from its motion under gravity, buoyancy, drag and a calibrated electric field. Repeated changes of a single drop's electric charge reveal discrete steps; see the model, experimental procedure and limitations in elementary charge.

Robert Andrews Millikan's 1911 protocol alternated electrically driven ascent with timed descent after short-circuiting the plates; a changed ascent speed signalled ion capture.^[ref]

His 1913 refinements checked voltage before and after runs, calibrated optical distance and the clock, compared independent air-dynamic viscosity measurements, and varied pressure to test slip.^[ref] His 1924 lecture describes suppressing convection, switching the electric field, neutralizing drops, and reproducing smallest observed electrical speed increments.^[ref]

Harvey Fletcher's posthumous account recalls building the first oil apparatus and measuring with Robert Andrews Millikan; priority for suggesting oil remains uncertain.^[ref]

To interpret the readout, keep one drop and one electric field polarity. Let $u_i$ be its signed upward terminal velocity in charge state $i$, and let $v_f>0$ be its field-off falling speed. Use the linear drag coefficient $k$ already defined in elementary charge, with any required gas slip correction to Stokes drag included. Provided the drop's radius, surrounding air and electric field remain fixed, its electrical speed increment isThese relations apply while the electric force points upward in every state. A drop may still descend, with $u_i<0$, when that force is too weak to support it. A held drop has $u_i=0$ and therefore $S_i=v_f$, rather than zero electrical response. A neutral drop has $S_i=0$: its signed terminal velocity agrees with field-off descent. Thus neither raw upward speed nor the ability to hold a drop is itself a count of electrons.

The ratios cancel a fixed $k/E$, even if its value is poorly known. This is why relative charge quantization and absolute measurement calibration are different questions. A changed field-off $v_f$ warns that evaporation, air motion or another change has spoiled comparison of charge states. A stable $v_f$ is useful evidence but cannot exclude every compensating change. Repeat field-off checks around a proposed jump, allow the terminal velocity under linear drag to settle after switching, and propagate measurement uncertainty in both the field-on and field-off readings. Reusing a single estimate of $v_f$ gives the different $S_i$ a shared covariance.

An integer pattern must survive the stated measurement uncertainty. For an original simulated example, choose a reference electric charge $Q_{\ast }>0$, define $y_i=|q_i|/Q_{\ast }$, and stipulate state meansHere each ${\sigma }_i$ is an independent standard uncertainty of a state mean, before shared calibration effects. The generator constructs repeated noisy observations with these means; the numbers are deliberately chosen, not historical data. There is no conversion of $Q_{\ast }$ to coulombs and no measured value of the elementary charge in this example.

For a candidate spacing $a>0$ in units of $Q_{\ast }$ and proposed integers $n_i$, use weighted least squares:The last two expressions follow by differentiating the quadratic in $a$ and propagating the independent errors. They are conditional on fixed, correct integer assignments and the stated error model. They do not include uncertainty from choosing those assignments. For $\mathbf{n}=(2,3,5,4)$, they give $\hat{a}=1$, $u(\hat{a})\simeq 0.01089$, and ${\chi }^2=0.46875$. Such a small residual is compatible with this chosen error scale; it is not evidence that the errors vanish. A chi-squared distribution interpretation additionally needs the appropriate normal distribution error model and allowance for the fitted parameter; searching many assignments changes that interpretation.

The obstruction is identifiability: replacing $(a,n_i)$ by $(a/m,m{n}_i)$ for any positive integer $m$ leaves every predicted reading unchanged. In particular, an $e$-spaced pattern also fits an $e/2$ lattice using even indices. Even the coarsest well-fitting spacing need not be the physical unit: a dataset containing only even charge states can have twice that unit. Worse, if arbitrarily fine spacings and unrestricted integers are allowed, rounding each $y_i/a$ givesConsequently a finite fit cannot prove a minimal transferable electric charge. Reproducible smallest observed jumps supply additional evidence; interpreting them as single-carrier transfers requires independent physical support. One visibly abrupt transition could contain several unresolved transfers. In the simulation the successive differences are $0.93$, $2.06$, and $-1.03$, each with standard uncertainty $\sqrt{2}(0.08)\simeq 0.1131$. These support changes near $a,2a,-a$ conditional on that unit; they equally support $2,4,-2$ units of $a/2$. Neighboring differences share a state reading and are therefore correlated, even though the state means were stipulated independent.

A shared scale error can preserve every apparent integer relation. To first order, an independent fractional scale standard uncertainty $s$ adds $s^2{a}^2{n}_i{n}_j$ to the covariance matrix of the $y_i$. For the simulation, stipulating $s=0.02$ changes the conditional uncertainty of $\hat{a}$ to approximately $\sqrt{0.01089^2+0.02^2}=0.02277$, despite the same residual pattern. Repeating timings cannot average away this common scale contribution. In the uncorrected continuum model in elementary charge, holding the other measured inputs fixed gives $|q|\propto {\eta }^{3/2}$, so a small fractional error in air dynamic viscosity causes $3/2$ times that fractional error in inferred electric charge. The gas slip correction to Stokes drag requires its own propagation. Pattern agreement, drop-stability controls and absolute measurement calibration together make the inference persuasive within resolution.