Millikan's oil-drop experiment

Substituting the more accurate expression for $\eta$ into the expression $-m'g - 6\pi\eta a v_g$ gives:

-\frac{4}{3}\pi a^3\rho'g = \frac{6\pi\eta_0av_g}{1+\frac{b}{ap}},

which rearranges to a quadratic equation in the droplet radius, $$a$$ :

a^2 + \frac{b}{p}a + \frac{9\eta_0v_g}{2\rho'g} = 0.

Taking the positive root gives the following expression for $$a$$ (recall that $$v_g < 0$$ ):

a = -\frac{b}{2p} + \frac{1}{2}\sqrt{\frac{b^2}{p^2} - \frac{18\eta_0v_g}{\rho'g}},

which is used in the function get_a below.

millikanoildropimproved

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

df = pd.read_csv("eg10-millikan-data.txt", sep=r"\s+", index_col=[0, 1])

# Air viscosity parameters, eta0 (kg.m-1.s-1) and b (Pa.m).
eta0 = 1.869e-5
b = 7.88e-3
# Air density, oil density, density difference (kg.m-3).
rho_air = 1.17
rho_oil = 917.3
rhop = rho_oil - rho_air
# Air pressure (Pa), acceleration due to gravity (m.s-2).
p = 100.82e3
g = 9.803
# Fall/rise distance (m), electric field (V.m-1).
d = 11.09e-3
E = -322.1e3


def get_eta(a):
    """Return the viscosity adjusted for droplet radius."""
    return eta0 / (1 + b / a / p)


def get_a(tg):
    """Get the droplet radius from fall time."""
    # Droplet fall terminal velocity.
    vg = -d / tg
    c = b / p
    a = (-c + np.sqrt(c**2 - 18 * eta0 * vg / rhop / g)) / 2
    return a


for drop in df.index.levels[0]:
    drop_df = df.loc[drop].T
    # Rename the index to simple 'tg' and 'te'.
    drop_df.index = drop_df.index.str.slice(0, 2)
    # Average over all tg values.
    tg = drop_df.loc["tg"].values.mean()
    # Average te for each experiment.
    te = drop_df.loc["te"].mean()
    a = get_a(tg)
    eta = get_eta(a)
    q = 6 * np.pi * eta * a * d / E * (1 / tg + 1 / te)
    df.loc[drop, "q"] = q.values

sorted_q = sorted(-df.loc[:, "q"])
dq = np.diff(sorted_q)
e_estimate = dq[(dq > 1.0e-19) & (dq < 2.0e-19)].mean()
df["N"] = (df["q"] / e_estimate).astype(int)
e = (df["q"] / df["N"]).mean()
print(f"Estimated value for e = {e} C")

The output is a slightly better estimate of $$e$$ :

Estimated value for e = 1.5976041889076869e-19 C

P9.4.2Millikan's oil-drop experiment