Millikan Terminal Drop Charge Balance

worked-example

Consider a dimensionless sphere with viscosity η = 1, density difference times gravity gΔρ = 1, drop radius r = 0.5, and an applied upward electric field E = 1. The charge on the drop is q = 1.

First, calculate the effective weight W:

W = (4/3) * π * r3 = (4/3) * π * (0.5)3 ≈ 0.5235987755982988

Calculate the field-off terminal speed v_(fall), where drag balances weight:

v_fall = -W / (6 * π * r * η) = -0.5235987755982988 / (6 * π * 0.5 * 1) ≈ -0.05555555555555555

Calculate the terminal upward speed v with the electric field on, where electric force, weight, and drag balance:

v = (qE - W) / (6 * π * r * η) = (1 * 1 - 0.5235987755982988) / (6 * π * 0.5 * 1) ≈ 0.050547739839041345

Finally, using the measured speeds to infer the charge:

q = 6 * π * r * η * (v - v_fall) / E = 6 * π * 0.5 * 1 * (0.050547739839041345 - (-0.05555555555555555)) / 1 ≈ 1.0

experiment

Explore the relationship between charge, drop radius, electric field, and terminal velocities in the interactive module:
https://riano.app/module/millikan-terminal-drop-charge-balance

1. Set Drop radius to 0.5 and Charge units to 1.
2. Set Selected field to 1 and Field mode to "Selected".
3. Observe the calculated Terminal upward speed (approx 0.0505) and Inferred charge (1.0).
4. Change the Field mode to "Hold if possible". Notice that the Terminal upward speed becomes exactly 0 as the field adjusts to exactly balance gravity.

limitations

This idealized model operates strictly in the Stokes flow regime where drag is proportional to velocity. It explicitly omits complexities found in the real experiment, such as:

• Brownian noise: Small droplets undergo random walks due to thermal collisions with air molecules, which adds noise to real speed measurements.
• Slip correction: When droplets are comparable in size to the mean free path of gas molecules, Stokes' law must be modified (Cunningham correction factor).
• Quantized charge assumption: This module synthetically assumes integer charge units for the simulation; it is a pedagogical model and not a raw data proof of charge quantization.

references

Primary reference for the historical context and original experimental details:

• Robert A. Millikan's Nobel Lecture (1924): "The electron and the light-quant from the experimental point of view", detailing the oil drop method.