Millikan attempt

Typical experimental setup as used today by school and university students

The Millikan experiment is an experiment with which the American physicists Robert Andrews Millikan and Harvey Fletcher succeeded in 1910 in determining the size of the elementary charge much more precisely than was previously possible. However, only Robert Millikan received the Nobel Prize in Physics for this measurement in 1923 , as he and Harvey Fletcher contracted prior to the publication of the work that only Millikan's name would be given for the work in the journal Science . Fletcher, on the other hand, was named as the only author in an article in Physical Review about the confirmation of Brownian molecular motion and was able to use this for his dissertation . Harvey Fletcher made the agreement with Robert Millikan public in a text that was not printed in Physics Today until after his death in 1982 . ${\displaystyle e}$

At the suggestion of Robert Millikan, Harvey Fletcher relied on experiments previously carried out by Harold Albert Wilson , Joseph John Thomson and other researchers, which he significantly improved , as part of his doctoral thesis . His most important improvement was to replace the previously used substances water and alcohol with low-volatility liquids such as oil and mercury. In order to determine the elementary charge, the rate of descent of electrically charged oil droplets was measured in the presence of an electric field compared to the case without an electric field. The value of the elementary charge determined in this way was: ${\displaystyle e}$

${\displaystyle e=1{,}592\cdot 10^{-19}}$ C .

Experimental setup and basic procedure

Schematic experimental setup of the Millikan experiment using the "two-field method"

Recording of the oil droplets (light points), the scale has a total length of 0.98 millimeters, two graduation marks are ≈0.033 millimeters (33 µm) apart

Play media file

Video recording of the observation. To see: rising and falling charged oil droplets due to the polarity reversal of the capacitor, as well as the droplets "wandering out" backwards or forwards out of the focus of the microscope and readjustment of the same.

Electrically charged droplets in the field of a plate capacitor

With an atomizer, very fine oil droplets are first generated, which are so small (about 0.5 µm in diameter) that they can no longer be observed directly with a conventional microscope . In order to follow them nonetheless, the dark field method is used, in which the oil droplets are illuminated at an angle of about 150 ° to the viewer, i.e. from almost the opposite direction, and the resulting diffraction disks are followed in the microscope (although it should be noted that the microscope exchanged above and below, so you can see the diffraction disks of sinking oil droplets moving upwards and vice versa). At least some of the oil droplets must be electrically charged, which was achieved in Millikan's experimental setup using an X-ray tube whose ionizing radiation electrostatically charged the oil droplets. As a rule, however, the friction of the oil droplets against one another during atomization or in the air is sufficient to charge them sufficiently.

The droplets then get between the plates of a plate capacitor . The oil droplets are so small that the air acts like a viscous liquid to them. For a long time they float in the air as an aerosol . However, the electric field of the capacitor exerts a force on charged oil droplets that far exceeds gravity. The Coulomb force pulls the positively charged droplets to the negatively charged plate of the capacitor. The resulting movement can be observed as the movement of the diffraction pattern recognizable with the microscope.

The inaccessible limbo

If the plates of the capacitor are mounted horizontally one above the other, an electrical force can be exerted on the droplets by applying a suitable voltage to the plates, which compensates for the sum of the first two forces, i.e. the downward weight force is equal to the sum of the electrical force Force as well as the buoyancy force holds and the oil droplet in question thus floats in principle. ${\displaystyle F_{\mathrm {G} }}$ ${\displaystyle F_{\mathrm {E} }}$ ${\displaystyle F_{\mathrm {A} }}$

By solving the equation , the charge of the oil droplets could now be determined in principle - but in practice this fails because the diffraction disks in the microscope do not allow any conclusions to be drawn about the radius of the oil droplets, so the right side of the equation remains indefinite. In addition, due to the Brownian movement , the state of suspension is difficult to identify exactly. ${\displaystyle F_{\mathrm {E} }=F_{\mathrm {G} }-F_{\mathrm {A} }}$

Indirect determination of the droplet radius via Stokes friction

In order to determine the radius of the droplets nonetheless, one can use the fact that an equilibrium of forces is established not only through the electric field in the capacitor, but also through the influence of the speed-dependent Stokes friction force, but not in the form of a floating state of the oil droplets concerned, but a constant speed of their falling or rising. ${\displaystyle v}$

In practice, there are two different methods for doing this: With the "single field method" , after a selected oil droplet has reached an approximate state of suspension, its speed of fall is measured solely on the basis of gravity; with the "two-field method", on the other hand, the oil droplet is initially let down from the field of the condenser and (after reversing the polarity of the field) then pull it up again, recording the rate of descent and rate of rise of the droplet. ${\displaystyle v_{1}}$ ${\displaystyle v_{2}}$

Deriving the connections

Sink in the field

Climb in the field

There are two variants of the experiment, the floating (or single field) and the constant field method (or two field method). In the hovering method, one speed is selected to be zero and the second speed and the voltage required for standstill are measured. With the two-field method, the amount of the voltage is fixed and the two speeds are measured when the electrical field is reversed. The two-field method is the more common.

When the oil droplet moves, the following forces occur, which are graphically illustrated in the pictures:

1. Weight force of a spherical oil droplet: ${\displaystyle F_{G}=mg={\frac {4}{3}}\pi r^{3}\rho _{O}g}$
2. Buoyancy force of a ball in air: ${\displaystyle F_{A}={\frac {4}{3}}\pi r^{3}\rho _{L}g}$
3. Coulomb force in the electric field : ${\displaystyle F_{E}=qE={\frac {qU}{d}}}$
4. Stokes friction force : ${\displaystyle F_{\mathrm {R} }=6\pi \cdot r\cdot \eta \cdot v}$

The sizes occurring therein are defined as follows:

• ${\displaystyle \pi }$= Circle number
• ${\displaystyle \rho _{O}}$= Density of the oil
• ${\displaystyle \rho _{L}}$ = Density of air
• ${\displaystyle g}$= Acceleration due to gravity
• ${\displaystyle U}$ = Voltage on the plate capacitor
• ${\displaystyle d}$ = Plate spacing of the plate capacitor
• ${\displaystyle \eta }$= Viscosity of the air
• ${\displaystyle r}$ = Radius of the oil droplet
• ${\displaystyle v_{1}}$ = Amount of the rate of descent of the oil droplet
• ${\displaystyle v_{2}}$ = Amount of the rate of rise of the oil droplet

The levitation method

A selected oil droplet is brought to an approximate standstill (floating) by varying the capacitor voltage and its speed of fall is then measured with the electric field switched off . As soon as there is a balance between friction, buoyancy and weight when the oil droplet falls, the following applies:

${\displaystyle F_{R}=F_{G}-F_{A}}$

Inserting the known relationships results in:

${\displaystyle 6\pi \eta v_{1}r={\frac {4}{3}}r^{3}\pi \left(\rho _{O}-\rho _{L}\right)g}$

Change to results in: ${\displaystyle r}$

${\displaystyle r={\sqrt {\frac {9v_{1}\eta }{2\left(\rho _{O}-\rho _{L}\right)g}}}}$

This means that the radius of the oil droplets can be determined solely from the measurable falling speed. In order to get to the charge, the state of suspension is now considered. In this, the equation applies analogously to the above formula:

${\displaystyle F_{E}=F_{G}-F_{A}}$

because now the electrical force is in equilibrium with weight and buoyancy. Inserting the relationships for the forces results in:

${\displaystyle {\frac {qU}{d}}={\frac {4}{3}}r^{3}\pi \left(\rho _{O}-\rho _{L}\right)g}$

This can be reshaped after loading:

${\displaystyle q={\frac {\left(\rho _{O}-\rho _{L}\right){\frac {4}{3}}\pi r^{3}gd}{U}}}$

Inserting the equation for into the equation for yields an equation for the charge, which only depends on the measurable quantities voltage and speed of fall: ${\displaystyle r}$ ${\displaystyle q}$

${\displaystyle q={\frac {9\pi d}{U}}{\sqrt {\frac {2\eta ^{3}v_{1}^{3}}{\left(\rho _{O}-\rho _{L}\right)g}}}}$

The charge and the radius can thus be calculated directly from the measured variables.

The constant field method

With a given capacitor voltage, the rate of descent of a selected oil droplet, which is initially moving downwards, and then, after reversing the polarity of the electric field with the capacitor voltage being maintained, its rate of rise is determined. In case of sinking: ${\displaystyle v_{1}}$ ${\displaystyle v_{2}}$

${\displaystyle F_{G}+F_{E}=F_{R,1}+F_{A}}$

In the case of climbing:

${\displaystyle F_{G}+F_{R,2}=F_{E}+F_{A}}$

Subtracting the two equations from each other, inserting the known relationships and solving for yields: ${\displaystyle r}$

${\displaystyle r={\frac {qU}{3\pi d\eta \left(v_{1}+v_{2}\right)}}}$

Substituting this equation into one of the two force equations yields an equation for the charge

${\displaystyle q={\frac {9\pi d}{2U}}{\sqrt {\frac {(v_{1}-v_{2})\eta ^{3}}{\left(\rho _{O}-\rho _{L}\right)g}}}(v_{1}+v_{2})}$

Inserting the equation for into the equation for yields for the radius ${\displaystyle q}$ ${\displaystyle r}$

${\displaystyle r={\frac {3}{2}}{\sqrt {\frac {(v_{1}-v_{2})\eta }{\left(\rho _{O}-\rho _{L}\right)g}}}}$

Now both the radius and the charge can be determined solely by measurable quantities.

Cunningham correction

Since the size of the oil droplets is in the range of the mean free path of air, the Cunningham correction should also be taken into account for the Stokes friction . The frictional force is extended by a term that can normally be neglected for large bodies:

${\displaystyle F_{\mathrm {R} }\rightarrow F_{\mathrm {R} }\left(1+{\frac {1{,}63\cdot \lambda }{r}}\right)^{-1}}$

where is the mean free path of air and the droplet radius. The equations now have to be solved again and become a bit more complex, but also significantly more precise. ${\displaystyle \lambda }$ ${\displaystyle r}$

Determination of the elementary charge

Measurement results of the droplet charge in the Millikan experiment

Since every oil droplet consists of a larger number of atoms and can carry not just one but also several charges, every calculated charge of an oil droplet is an integral multiple of the elementary charge . If the charge distribution of many experiments is plotted in a graph, the result is no continuous distribution. It turns out that only multiples of the elementary charge can occur. ${\displaystyle q}$ ${\displaystyle e=1{,}602\cdot 10^{-19}\,\mathrm {C} }$

A single elementary charge on a particle can only be observed if the voltage is high enough to keep any visible oil droplets with an elementary charge at least in suspension. This is not the case in most of the experimental setups.

Since 1910, much more precise methods for determining the elementary charge have been developed, including using the quantum Hall effect . Since the redefinition of the International System of Units in 2019, the units ampere and coulomb have been defined via the elementary charge. The elementary charge is therefore no longer a constant to be determined experimentally, but has been set to an exact value:

${\displaystyle e=1{,}602\,176\,634\cdot 10^{-19}\;\mathrm {C} }$.

Web links

Commons : Millikan experiment  - collection of images, videos and audio files

• On the Elementary Electrical Charge and the Avogadro Constant - information on Robert Millikan including the original publication, published in Physical Review in 1913
• Remotely Controlled Lab (RCL) (real experiment, remotely controllable via the Internet, see there under "RCLs")
• Millikan experiment on the Saarland University website (PDF file; 116 kB)
• Millikan oil droplet experiment at school level ( LEIFI )
• Materials for the Millikan experiment compiled by T. Unkelbach
• Description of the Millikan experiment at Lern-Online.net (PDF file; 250 kB)

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1. ↑ ^{a b} Millikan, RA (1911): The isolation of an ion, a Precision Measurement of its charge, and the Correction of Stokes's Law in: Physical Review (Series 1) Vol 32, Issue 4, April 1911, pp 349th -397 ( doi: 10.1103 / PhysRevSeriesI.32.349 ), filed November 1910
2. ^ Robert Millikan in Britannica Online
3. ^ Heinrich Zankl: Nobel Prizes: explosive affairs, controversial decisions . John Wiley & Sons, 8 March 2012, ISBN 978-3-527-64145-1 , pp. 23f.
4. ↑ Harvey Fletcher: My Work with Millikan on the Oil-drop Experiment. Archived from the original on March 27, 2014. Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. In: Physics Today . June 1982, p. 43. Retrieved December 21, 2013. 
5. ↑ ^{a b} Millikan, RA (1913): On the Elementary Electrical Charge and the Avogadro Constant in: Physical Review (Series 2), Volume 2, 109–143, Issue 2 - August 1913 ( doi: 10.1103 / PhysRev.2.109 )
6. ↑ Test instructions from Saarland University ( Memento of the original from November 11, 2014 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF; 118 kB) 
7. ^ Resolution 1 of the 26th General Conference on Weights and Measures (2018) ( online , accessed December 14, 2019), English. "The SI is the system of units in which [...] the elementary charge e is 1.602 176 634 × 10 ⁻¹⁹  C"
8. ↑ CODATA Recommended Values. National Institute of Standards and Technology, accessed July 21, 2019 .  Value for the elementary charge. The value is exact, i.e. H. with no measurement uncertainty.