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Millikan Oil Drop Experiment

Professor Jeff Filippini 401 Spring 2020 Today’s Topics

1. History: Measuring the charge of the

2. Theory of the Experiment

3. Laboratory Setup

4. Data Analysis

Physics 401 2 Motivation: The Dawn of Elementary Particles • 1750s: Benjamin Franklin proposes is flow or surplus/deficit of single electrical fluid • 1881: Hermann von Helmholtz argues for elementary positive and negative electric charges (“atoms of electricity”) • 1897: J.J. Thompson shows particle nature of “cathode rays” emitted in tubes • Universal negatively-charged particles from vacuum tubes, radioactivity, … • Measured charge-to- ratio (e/m) via magnetic deflection • Mass <1/1000 of hydrogen ion (actually ~1/1836)

Physics 401 3 Robert Millikan and the Oil Drop Experiment

The in Physics 1923. Robert A. Millikan "for his work on the of electricity and on the ".

ROBERT ANDREWS MILLIKAN 1868-1953

University of Chicago Moved to Caltech in 1921 Physics 401 4 Robert Millikan and the Oil Drop Experiment

Work with Ph.D. student – not a 1868-1953 coauthor on famous paper. “The father of stereophonic sound” First functional electronic hearing aid

Physics 401 5 Oil Drop Apparatus

Motivation: 1. Measure the magnitude of the electron charge Apparatus precision ± 3% Millikan original ± 0.1%

2. Demonstrate that is quantized!

Physics 401 6 Oil Drop Experiment atomizer oil drops ∅~1µ

telescope +

d V Vg 500 V - ����

Forces acting on the oil drop: 1. 2. Electric force on a charged oil drop

Physics 401 8 The Role of Air: Buoyancy and

Lowering an object through a fluid means Motion through a fluid opposed lifting an equal volume by drag: friction with and within of the surrounding fluid the surrounding fluid

Objects eventually reach at which drag balances accelerating forces

Physics 401 9 Oil Drop Experiment atomizer oil drops ∅~1µ

telescope +

d V Vg 500 V - ����

Forces acting on the oil drop: 1. Gravity 2. Electric force on a charged oil drop 3. Buoyant force (weight of air displaced by oil drop) 4. Drag force on a moving oil drop

Physics 401 10 Oil Drop Apparatus: Schematic Layout

Light source

telescope

135-170o

atomizer

oil eyepiece source oil mist

P1 P1 Q,m

P2

500±1V spacer oil drop scale

Physics 401 11 Oil Drop Apparatus: Schematic Layout

Physics 401 12 How the Measurement Looks Allow drop to “undershoot” here before starting next rise time experiment

… and stops here Fall time measurement starts here…

Fall time tg

Rise time trise x x = fall distance = rise distance Must be the same for all drops!

… and stops here Rise time measurement starts here…

Physics 401 13 Balance of Forces: Newton’s Second Law

� Forces on the oil drop: �⃗ � 1. Gravity + buoyancy �� = −�� � 2. Drag force ����� = −���� � ⃗ � 3. Electric force � = � � �⃗ �

a: radius of drop �� � = � = �� + ����� + �� ρ: net density ρoil - ρair ��

Drop v: velocity of oil drop Q: charge of oil drop Terminal velocity: speed at which = 0 E: E=V/d V: across plates �� + ����� + �� = � η: of air Achieved within ~10-5 s by a 1 μm particle

Apparatus g: gravitational constant

Physics 401 14 Modifications to Stokes’ Law

����� = −���� � For particles of small radii (� ≲ 15 ��), Stokes’ Law must be corrected for the particle nature of air. This correction was derived by E. Cunningham � George Gabriel Stokes � = −��� � Ebenezer Cunningham ���� (1881-1977) (1819-1903) �� Here: • a = particle radius � � � = 1 + � + � � , � = 1.246; � = 0.42; � = 0.78 • λ = mean free path of gas � � "# molecules ! .× � ≈ 1 + � = 1 + ≈ 1.1 for � = 10 �, � = 760 ���� [] � = 65.3 �� �

Physics 401 15 What We Measure: Rise and Fall Times Allow drop to “undershoot” here before starting next rise time experiment

… and stops here Fall time measurement starts here…

Fall time tg

Rise time trise x x = fall distance = rise distance Must be the same for all drops!

… and stops here Rise time measurement starts here…

Physics 401 16 Measuring the Temperature Project: T measurement.opj

Physics 401 17 Solving Newton’s Law: �(�, �)

� The correction factor � can be found from Newton’s Law �⃗ � in the case of � = 0 (falling drop)

⃗ � �⃗ �⃗ + �⃗ = � �� − 6�� �⃗ = 0 ! a: radius of drop ρ: net density ρoil- ρair v: velocity of oil drop See write-up for derivation Drop Q: charge of oil drop

E: electric field E=V/d V: voltage across plates 1 � 2�� 6.18×10 � ≈ 1 − ; � = ; � = η: viscosity of air / � ��� � [����] g: gravitational constant �

Apparatus x: drift distance

Measured tg, trise: fall, rise times Physics 401 18 Solving Newton’s Law: �(�, �) � �⃗ 1 � 2�� 6.18×10 � ≈ 1 − ; � = ; � = � / � ��� � [����] � �⃗ ⃗ � Putting it all together: a: radius of drop ρ: net density ρ - ρ oil air � � v: velocity of oil drop � ��� �� � � � � Drop � = �� = + Q: charge of oil drop �/� � �� � � � �� � � ���� E: electric field E=V/d V: voltage across plates η: viscosity of air F: Stokes’ law correction g: gravitational constant : Stokes’ law force balance Apparatus x: drift distance S

Measured tg, trise: fall, rise times T: Your timing measurements Physics 401 19 Origin Projects: Data Collection Project: Millikan_raw data.opj Location: \\engr-file-03\PHYINST\APL Courses\PHYCS401\Students\1. Millikan Oil Drop experiment\Millikan_raw data.opj

Physics 401 20 Origin Projects: Data Analysis Location: \\engr-file-03\PHYINST\APL Courses\PHYCS401\Students\1. Millikan Oil Drop experiment\

Please make a copy (don’t move!) of Millikan1.opj and Millikan_raw data.opj in your personal folder and work with your personal copy of the project

Physics 401 21 Origin Data Analysis Project: Millikan1.opj

Your data Constants goes here

Calculations

Experimental Parameters Depend on exact setup and 6.18×10%& � � = environmental conditions $ � [����] P[mmHg] ⟹ Col(“Par”)[7]

Physics 401 22 Origin Data Analysis Project: Millikan1.opj

Physics 401 23 Origin Data Analysis Project: Millikan1.opj

æö19phdx 233 11æö 1 QFST=••=ç÷32 ç÷ + fVgr ç÷ tt èøctg èø g rise

Follow correct dependency sequence of calculations: �$ → �' → �, �, � → � → � Physics 401 24 Origin Data Analysis Project: Millikan1.opj

• Indices for parameters in Col(“Par”) • Actual air viscosity should be calculated manually before any other calculation

Physics 401 25 Origin Charge Calculation

n 7

6

5

4

3 n=Q/e

2

1

0 0 1 2 3 4 5 6 7 8 9 10 11 Drop number

Physics 401 26 Expected Results

4.5 50 Data: Bin1_Counts1 Model: Gauss Equation: y=y0 + (A/(w*sqrt(PI/2)))*exp(-2*((x-xc)/w)^2) 4.0 Weighting: y No weighting

40 Chi^2/DoF = 34.42645 3.5 R^2 = 0.81797

y0 0 ±0 xc1 0.92997 ±0.01884 w1 0.27612 ±0.0381 3.0 A1 13.76998 ±1.65786 30 xc2 1.87091 ±0.05159 w2 0.60549 ±0.11582 A2 16.60884 ±2.56952 2.5

n=Q/e 2.0 20 counts

1.5 10 1.0

0.5 0 0 1 2 3 4 5 6 7 8 9 10 11 1 2 Drop number Q/e

Physics 401 27 Effect of Choice of Oil Drops n=5 60 55 n=4 50 45 40 n=3 35 30

(s) 25

r t 20 15 n=2 n=1 Difficult to separate 10 5 n=3, 4, 5, … 0 0 5 10 15 20 25 30 35 40 45 50 55 60

tg (s) Physics 401 28 Modern Experiments

•Drop generation rate: 1 Hz •Fluid: Dow Corning silicon oil •Number of drops: 17 million •Mass: 70.1 milligrams •Duration: 8 months

Physics 401 29 Modern Experiments Summary as of January 2007 • Total mass throughput for all experiments: 351.4 milligrams of fluid • Total drops measured in all experiments: 105.6 million • No evidence for fractionally charged particles was found.

Physics 401 30 Measuring the Electron Charge

• 1909: Oil drop experiment, Robert A. Millikan e=1.5924(17)×10−19 C (0.1% error) • 1918: Shot-noise proposed by Walter H. Schottky Poisson fluctuations in current across a barrier appear as noise Measure noise variance � = 2�� ∆� over a bandwidth ∆� • In terms of Avogadro’s constant and Faraday constant ( per mole, from electrolysis) � = �/�; best uncertainty ~1.6 ppm • From superconducting junctions (Josephson constant, � = ) and quantum Hall effect (von Klitzing constant, � = ) "

Physics 401 31 Modern Definition

• SI base units redefined on May 20, 2019 1967 • Move to ground all 7 SI base units in terms of exact defined values of the physical 1983 2019 constants (exactly reproducible in lab!) • Response to the “crisis” of infinitesimal variations in the standard kilogram mass 2019 2019 • Modern exact definition: � = 1.602176634×10 ������� 2019 1979

Physics 401 32