Precision in the Quantum World: Revisiting the Millikan Oil Drop Experiment

Abstract

By observing and measure the oil drops’ movement while they are in free fall and in an electric field with set condition, it could determine how many electric charges are carrying by oil drops. Therefore, the result could demine the fundamental value of single electron. while measuring out different charges that carried by different amount of oil. Oil drops should all carry certain numbers of electrons and that number should always be whole. By collecting large amount of data that indicate how much electric is carrying by the oil drops.

All data should be a multiple of one base value and integral multiple toward one and other. In this lab we used a light source to illuminated the oil drops and measure out they are rise and fall times. A radioactive source is also use in this experiment to increase the chance that a single oil drop could change its charge during the observation.

Introduction

Electron is a fundamental unit of charges that an object is carrying, which has a standard value as e=1,602 E-19 C.

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This become one of the fundamental constant in physics. In 1912, by produced microscopic oil drop by spray bottle, which only carry a few electrons. Millikan is able to establish Accurate measurement of an electron’s charges by first time. The oil drop experiment is conducted between two plates and space was separate by a spacer, a radioactive source (in this case the Thorum 232 is acting as such) is also with the camber in order to change the charges during single drop observation.

Oil drops with the chamber are subject to three different forces, which are viscous, electric and gravitational forces. By analyzing these three forces though expression which can be drive to calculate the charges in an oil drop.

Theory

The theoretical framework for the experiment revolves around the equilibrium of forces: gravitational, viscous, and electric, acting on the oil drops. The analysis incorporated modifications to Stokes' law to account for the reduced velocities observed, leading to a refined formula for calculating the charge on an oil drop. This formula incorporates known constants and measurements taken during the experiment, such as the oil drop's velocity in the presence and absence of an electric field.

Set oil drop mass a m and charges as q, spacer distance as d, k is the coefficient of friction between air and the drop, E is the electric intensity. when oil drop travel down (in terminal velocity):

mg=k* v(f)

And when the oil drop goes upward. We have qE=mg+k v(r)

Both side become q=mg((v(f)+v(r)))/(Ev(f)), when took out k.

Eliminate m from equation above, gets mg=4/3 πa^3 Ƿg, where a is radius of the oil and Ƿ is the density of the oil.

From stokes’ law, we understand the relationship between spherical body and its terminal velocity, which express by F(f)=6πŋav(f)  and from equation 4 gets a=(9ηv/2gρ)^(1/2).

But in this experiment the oil drop is moving too slow and stokes’ law is not correct in such condition, it require the viscosity to be multiplied as following expression as:

n=(1/(1+b/pa))ŋ, with substituting n in to equation 6’s ŋ and put back the result in equation 5.

The overall expression could be found as q=4/3 πρg(√((b/2p)^2+9ŋv(f)/2gρ)-b/2p)^3 (v(f)+v(r))/(Ev(f)),

As π,p,ρ,g,p,ŋ,b are pre-set measurement (known constant) in this experiment. First, v(f) and v(r) are independent variable dependent on time and traveled distant.

Experimental Method

The setup involved a chamber placed between two plates, where oil droplets were introduced and subjected to an adjustable electric field. Key parameters, including pressure, temperature, and resistance, were meticulously recorded to ensure consistency. The experiment's novelty was highlighted by the dual observation of oil drop movement under natural fall and when influenced by an electric field, enabling precise calculations of charge based on the droplets' acceleration or deceleration.

Results and Analysis

Base on the data, droplet 2 and 3 has a very close value, which different by 0.19×(10)^(-19)c and droplet 3 has the smallest value of q in all 10 droplets.

Assume droplet 3 as q(0) and q(0)=e.

From observation. Droplet 2 and 3 has a very close q. but 4 out of 10 sets of droplet electric has a “whole and half” multiplication relationship to droplet 3. Which means 1.42×(10)^(-19)C is not the correct value of single electron. But noticing droplet 3 has a close multiple of 2 and 3 with droplet 1, 5, 6 and 9.

It means the constant value of e must be less than 2C.

Base on the observation, the electric charges could have a higher value than 1.42×(10)^(-19)C, in which I set up 2 more q(0) value which are q(0)=q(droplet 2)=q(e) and q(0)=q((droplet 2+droplet 3)/2)=1.515×(10)^(-19)C.

Noticing there is an increasing improvement as assumed electron’s value going toward 1.61×(10)^(-19)C. in fact, with ±.2e error, all result relates to droplet 2 are within the acceptable range. The electron’ electric charges value should within ±.2C of 1.61×(10)^(-19)C.

Using line to graph all different assumed q(0)=q(e), then put in all averaged observation data into the graph. After draw a trend line and display the trend line equation, which is y=1.5935x+.1668. the final calculation of electron’s charges value is 1.5935±0.2 ×(10)^(-19)C. The error mainly is systematic error which cause by my co-partner has a 0.17 to 0.28 reflection time on pushing the bottom on the stop watch, and my delay on reporting droplet reach radical. Minor random error is cause by equipment and background heat was constantly changing the temperature of the dark room and sound and movement vibration also brings effect on oil drop’s free fall. However, after calculation and analysis, the lab result has a (1.594-1.602)/1.602×100%=-0.53% error.

Conclusion

This meticulous reevaluation of the Millikan Oil Drop Experiment not only reaffirms the quantized nature of electric charge but also enhances the precision of the electron's charge measurement. The slight deviation from Millikan's original finding underscores the experiment's sensitivity to systematic and random errors, such as reaction times in measurement and environmental fluctuations. Despite these challenges, the experiment solidified the fundamental understanding of electron charge quantization, with a calculated value remarkably close to the accepted standard, showcasing the enduring relevance of classical experiments in the quantum realm.