Editing Oil drop experiment (section)

==Experimental procedure==

===Apparatus===
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[[Image:Simplified scheme of Millikan’s oil-drop experiment.svg|right|thumb|Simplified scheme of Millikan's oil drop experiment|576x576px]]
[[Image:Millikan’s oil-drop apparatus 1.jpg|right|thumb|Oil drop experiment apparatus|335x335px]]
Millikan's and Fletcher's apparatus incorporated a parallel pair of horizontal metal plates.  By applying a potential difference across the plates, a uniform electric field was created in the space between them. A ring of insulating material was used to hold the plates apart. Four holes were cut into the ring, three for illumination by a bright light, and another to allow viewing through a microscope.

A fine mist of oil droplets was sprayed into a chamber above the plates.  The oil was of a type usually used in [[vacuum]] apparatus and was chosen because it had an extremely low [[vapour pressure]]. Ordinary oils would evaporate under the heat of the light source causing the mass of the oil drop to change over the course of the experiment. Some oil drops became electrically charged through friction with the nozzle as they were sprayed.<ref name=Pekola-2013/>{{rp|p=1423|q=In those experiments, the oil droplets were, however, charged randomly by an uncontrollable process of absorption of ions which exist normally in air.}}

Alternatively, charging could be brought about by including an ionizing radiation source (such as an [[X-ray tube]]). The radiation source can remove electrons from the droplet, as well as the air surrounding it. Some of the droplets get negatively charged by receiving a net excess of electrons that were ionized out of the air molecules around it. Positively charged drops will be pulled downwards both by gravity and the electric field, and will not reach mechanical equilibrium. The negatively charged droplets are the important ones for the experiment, because they can reach mechanical equilibrium.{{cn|date=March 2025}}

The droplets entered the space between the plates and, because they were charged, could be made to rise and fall by changing the voltage across the plates.

===Method===
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[[Image:Scheme of Millikan’s oil-drop apparatus.jpg|thumb|372x372px|Diagram in Millikan's original 1913 paper]]
Initially the oil drops are allowed to fall between the plates with the electric field turned off. They very quickly reach a [[terminal velocity]] because of friction with the air in the chamber.  The field is then turned on and, if it is large enough, some of the drops (the charged ones) will start to rise. (This is because the upwards electric force ''F''<sub>E</sub> is greater for them than the downwards gravitational force ''F''<sub>g</sub>, in the same way bits of paper can be picked by a charged rubber rod). A likely looking drop is selected and kept in the middle of the field of view by alternately switching off the voltage until all the other drops have fallen. The experiment is then continued with this one drop.

The drop is allowed to fall and its terminal velocity ''v''<sub>1</sub> in the absence of an electric field is calculated. The [[drag (physics)|drag]] force acting on the drop can then be worked out using [[Stokes' law]]:

:<math>F_{u} = 6\pi r \eta v_1, \,</math>
where ''v''<sub>1</sub> is the terminal velocity (i.e. velocity in the absence of an electric field) of the falling drop, ''η'' is the [[viscosity]] of the air, and ''r'' is the [[radius]] of the drop.

The weight '''''w''''' is the volume ''D'' multiplied by the density ''ρ'' and the acceleration due to gravity '''''g'''''. However, what is needed is the apparent weight. The apparent weight in air is the true weight minus the [[upthrust]] (which equals the weight of air displaced by the oil drop). For a perfectly spherical droplet the apparent weight can be written as:

:<math>\boldsymbol{w}=\frac{4\pi}{3}r^3(\rho-\rho_\textrm{air})\boldsymbol{g}.</math>

At terminal velocity the oil drop is not [[acceleration|accelerating]]. Therefore, the total force acting on it must be zero and the two forces ''F'' and <math>{w}</math> must cancel one another out (that is, <math>F = {w}</math>). This implies
:<math>r^2 = \frac{9 \eta v_1}{2 g (\rho - \rho_\textrm{air})}. \,</math>

Once ''r'' is calculated, <math>{w}</math> can easily be worked out.

Now the field is turned back on, and the electric force on the drop is
:<math>F_E = q E, \,</math>
where ''q'' is the charge on the oil drop and ''E'' is the electric field between the plates. For parallel plates
:<math>E = \frac{V}{d}, \,</math>
where ''V'' is the potential difference and ''d'' is the distance between the plates.

One conceivable way to work out ''q'' would be to adjust ''V'' until the oil drop remained steady.  Then we could equate ''F''<sub>''E''</sub> with <math>{w}</math>. Also, determining ''F''<sub>''E''</sub> proves difficult because the mass of the oil drop is difficult to determine without reverting to the use of Stokes' Law.  A more practical approach is to turn ''V'' up slightly so that the oil drop rises with a new terminal velocity ''v''<sub>2</sub>.  Then
:<math>q\mathbf{E}-\mathbf{w}=6\pi\eta(\mathbf{r}\cdot \mathbf v _2)=\left|\frac{\mathbf v_2}{\mathbf v_1}\right| \mathbf{w}. </math>

=== Comparison to modern values ===
Effective from the [[2019 revision of the SI]], the value of the elementary charge is ''defined'' to be exactly {{physconst|e}}.
Before that, the most recent (2014) accepted value<ref>{{cite web |url=http://physics.nist.gov/cuu/Constants/archive2014.html |title=2014 CODATA Values: Older values of the constants |date=25 June 2015| work=The NIST Reference on Constants, Units, and Uncertainty |publisher=[[National Institute of Standards and Technology|NIST]] |access-date=2019-08-19}}</ref> was {{val|1.6021766208|(98)|e=-19|u=[[Coulomb|C]]}}, where the (98) indicates the uncertainty of the last two decimal places. In his Nobel lecture, Millikan gave his measurement as {{val|4.774|(5)|e=-10|u=[[statcoulomb|statC]]}},<ref>{{cite speech | title = The electron and the light-quant from the experimental point of view | first = Robert A. | last = Millikan | author-link = Robert Millikan | date = May 23, 1924 | location = Stockholm | url = http://nobelprize.org/nobel_prizes/physics/laureates/1923/millikan-lecture.html | access-date = 2006-11-12}}</ref> which equals {{val|1.5924|(17)|e=-19|u=C}}. The difference is less than one percent, but is six times greater than Millikan's [[standard error]], so the disagreement is significant.

Using [[X-ray]] experiments, Erik Bäcklin in 1928 found a higher value of the elementary charge, {{val|4.793|0.015|e=-10|u=statC}} or {{val|1.5987|0.005|e=-19|u=C}}, which is within uncertainty of the exact value. [[Raymond Thayer Birge]], conducting a review of physical constants in 1929, stated "The investigation by Bäcklin constitutes a pioneer piece of work, and it is quite likely, as such, to contain various unsuspected sources of systematic error. If [... it is ...] weighted according to the apparent probable error [...], the weighted average will still be suspiciously high. [...] the writer has finally decided to reject the Bäcklin value, and to use the weighted mean of the remaining two values." Birge averaged Millikan's result and a different, less accurate X-ray experiment that agreed with Millikan's result.<ref>{{cite journal |last1=Birge |first1=Raymond T. |title=Probable Values of the General Physical Constants |journal=Reviews of Modern Physics |date=1 July 1929 |volume=1 |issue=1 |pages=1–73 |doi=10.1103/revmodphys.1.1|bibcode=1929RvMP....1....1B }}</ref> Successive X-ray experiments continued to give high results, and proposals for the discrepancy were ruled out experimentally. [[Sten von Friesen]] measured the value with a new [[electron diffraction]] method, and the oil drop experiment was redone. Both gave high numbers. By 1937 it was "quite obvious" that Millikan's value could not be maintained any longer, and the established value became {{val|4.800|0.005|e=-10|u=statC}} or {{val|1.6011|0.0017|e=-19|u=C}}.<ref>{{cite journal |last1=von Friesen |first1=Sten |title=On the values of fundamental atomic constants |journal=Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences|date=June 1937 |volume=160 |issue=902 |pages=424–440 |doi=10.1098/rspa.1937.0118|bibcode=1937RSPSA.160..424V |doi-access=free }}</ref>