Physics307L F07:People/Joseph/Formal Lab Report2

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Determination of the Charge of a Single Electron

Nikolai Joseph University of New Mexico Albuquerque, NM njoseph@unm.edu

Abstract

The Millikan Oil Drop Experiment was a landmark experiment in physics because it was the first time the charge of the electron was experimentally found. The experiment consists of measuring the time it takes for one drop of oil to fall some distance under the influence of gravity and then the time it takes to rise when subjected to an electric potential. With that data and measurements of the environment around the experiment (temperature, pressure, viscosity) , the charge of the electron can be found. We took the data and measured the charge of the electron to be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle e=1.56(5)\times10^{-19} C} , which differs slightly from the accepted value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 1.602\times10^{-19}C} 1, possibly due to inexact environmental considerations and delays in the recording equipment.

Introduction

The electron sits high in the world of particles and is fundamental to understanding the physical universe. Understanding the charge gives an explanation of how particles interact and the structure of matter. Determining the charge allows accurate calculations of fundamental interactions between matter and electro-magnetic fields. Determining the charge can be done by allowing droplets of oil to fall through an electric potential and measuring how much the potential moves the droplet.2 This is not the only method of determining the charge of the electron3This ought give an integer value as the measured charges change. There is has been strong evidence for the existence of fractional charges as well4, but the fundamental charge is what we are interested in here. Even though the currently accepted value was not determined this way, when Millikan first performed this experiment he brought the world closer than anyone before. It has been said that Millikan's mistake was the values he used for air viscosity5. I hope that using more refined values for the viscosity and the air pressure I'll find the fundamental charge with a good degree of accuracy. I plan on doing this by using a computer applet to time the drops as they fall, and then record the times as they rise in the presence of an electric field. Those measurements, along with several determined values will allow the fundamental charge to be determined.


Method

To measure the charge of the electron the influence on the electron by a known electric potential was used. We measured the times that it took for the droplet of oil to fall a known distance (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle l=1.0\times10^{-3}m} ) solely under the influence of gravity. We then applied the electric potential (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V=500 volts} ) and measured the time it took for the droplet to rise over that same distance. Doing this multiple times generated the data necessary to compute the charge of the electron.

Equipment

The experiment centered around the use of the Millikan Oil Drop Apparatus (Pasco AP-8210). See fig 1 or look at the [manual]). There was also:

    1. TEL-Atomic 50V-500V power source: connected to the Millikan Apparatus via BNC cables.
    2. Fluke III True RMS multimeter #1: connected to the Millikan Apparatus via BNC cables to show what voltage was being applied to the plates that when a acurrent was applied, would cause the electron to rise
    3. Fluke III True RMS multimeter #2: : connected to the Millikan Apparatus via BNC cables indicated the voltage through the thermistor
    4. stopwatch or stopwatch applet: for recording the rise and fall times
    5. atomizer: for spraying the oil in a fine mist into the oil chamber
    6. mineral oil: the oil that would be divided into the actual drops that were being watched

Setup

The apparatus needs to be on a level surface to remove as much uneven gravitational influence as possible. The apparatus was placed atop several textbooks, to provide a more suitable viewing height, and was leveled using the adjustable screw legs and a bubble level on the device. The power supply was connected to the ports labeled "Plate voltage" on the apparatus and multimeter was plugged into the power supply to allow the voltage through the plates to be known. The light source was plugged into the wall, and it had no switch; the light was on only if the cable was plugged in. BNC cables connected the port labeled "Chamber Temperature" to a multimeter that displayed the voltage through the thermistor. A table on the apparatus gave corresponding temperatures to voltages. The chamber could be disassembled by removing the outer housing, the droplet hole cover, the capacitor plates, and the spacer within it. The plates were cleaned and reassembled. The viewing reticle had to be calibrated and to do required the chamber be opened up and and the viewing pin needed to be inserted into the hole where the droplet hole cover was. With the light on, someone looked through the reticle and focus the reticle, using the focusing knobs, until the pin was as clearly visible as possible. Once this was done, the pin was removed and droplet hole cover was replaced, and the chamber was reassembled.

Ap8210.jpg

This picture show the main components of the AP8210 Oil Drop Apparatus.

Procedure

The first data to take was recording the thickness of the spacer. This was done with a micrometer and the attained value was Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle d=7.59\times 10^-3 m} . That value has use later on. The next step was to reassemble the chamber setup for data taking. Allow the power supply, light source, and thermistor all to warm up for a few minutes before beginning. The atomizer was filled with mineral oil of density Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \rho=8.86\times 10^2 \frac{kg}{m^3}} (given by the manufacturer on the bottle). The ionization control was set to Spray Droplet position and the initial temperature of the thermistor was recorded. All overhead lighting was killed and one person looked through the viewing reticle. Before the oil was introduced all that was visible was the grid of the viewing reticle against a dim yellow-orange background from the light source. With a person looking, the other person put the atomizer so its tip was in the hole in the top center of the out housing. The person with the atomizer squeezer the bulb very quickly one time and allowed it to refill with air. Then another, much slower squeeze on the atomizer bulb, not allowing it to refill with air while it was still in the hole on the housing. During the second squeeze, the person viewing should see tiny, golden droplets fill the viewing area beyond the grid. This technique requires a degree of finesse and may take a few tries! During all of this the charging plate should be in the Plates Grounded position Once the droplets were all visible, the viewer would change the ionization source to Off. The viewer selects a droplet that is falling slowly, and is located so that the grid does not interfere with the visibility of the drop. The viewer starts the timer as the drop passes one grid line and will stop timing when it passes the next lower one. Following the same drop, the viewer applies a voltage from the charging plates, and records the time it takes for the droplet to rise. The ionization source can be turned to the On position and the thorium-232 source within the chamber will bombard the droplet with alpha particles, charging the charge on the droplet. This will causes the droplet to rise much quicker when a voltage is applied. It is very important that the same drop be followed. Collecting data for various droplets doing various things produces inconsistent and useless data. For the final calculations we need the air pressure. Unfortunately, published air pressures are always adjusted to sea level, whereas we are about 5300 feet above sea level. Making the conversion equation as: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{P_1}{P_2}=\frac{P_x}{P_3}} With: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle P_1} =standard pressure at 5300 ft=Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 833.756 hPA} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle P_2} =standard pressure at sea level=Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 1013.56 hPA} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle P_3} =sea level pressure in Albuquerque on 12/5=Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 1014.8 hPa} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle P_3} =actual air pressure in Albuquerque on 12/5 Finding P_1 took a trick. I used the standard air pressure at 5000 feet, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 843.08 hPA} , and the standard air pressure at 6000 feet, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 812 hPA} . I added them and multiplied by .3. I then subtracted that total value from the pressure at 5000 feet. After everything I got the number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle p=834.78 hPA}


Data

A total of 8 droplets were followed, we determined it only relevant to use the data from 2 of the droplets. This was not reckless abandonment of data though, There was a strong systematic tendency for the drops to drift behind the coordinate grid and they became very difficult to see. So difficult in fact, that it was determined that any measurements recorded whilst the droplet was obscured from sight was far too unreliable to be of any worthwhile contribution. All of the discarded data is listed in Appendix A, and won't be used in our calculations. Below is the listed useful data. The data is structured as follows:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V} : The potential that was applied to create the repulsive electric field, volts (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V} )
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle T} : The temperature within the viewing chamber, in degrees Celsius (°C)
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle t_f} (s): The fall time of the droplet, in seconds (s)
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle t_r} (s): The rise time of the droplet, in seconds (s)

Here are a few data points, the rest are contained in Appendix B.

Droplet 1

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V=500.2 +/- 1 } V

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle T=22.6} °C

Fall time (s) 11.5 12.1 12.8 12.1 12.25
Rise time (s) 2.7 2.8 2.9 2.8 2.9

Droplet 2

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V=500 +/- 1} V

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle T=25 +/-.1} °C

Fall Time (s) 9 9.8 9.7 9.8 9.7
Rise Time (s) 2.3 2.6 2.1 2.2 2.2

To test whether or not more charge could be accumulated the drops were ionized and data was taken afterwards. That data is in Appendix B, and the distribution of the how many times the droplets fell in given time brackets.

Risevfall.jpg



Calculations and Results

The target value to be determined is the charge of the electron, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle q} . Here are the necessary equations and values used in doing the calculations for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle q} :

Values

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \rho=8.86\times 10^2 \frac{kg}{m^3}} (density of the mineral oil)
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle g=9.8 \frac{m}{s^2}} (gravitational acceleration)
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle p=8.3478\times10^{-4} PA} (air pressure)
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle l=1.0\times10^{-3} m} (length droplet will be measured over)
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle b=8.20\times10^{-3} Pa\cdot m} (proportion constant)

Determined

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle T} (temperature from thermistor in °C)
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V} (Voltage between plates in viewing chamber in volts)
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle t_f} (fall time in seconds)
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle t_r} (rise time in seconds)
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle d=7.59\times 10^{-3} m} (width of the plastic spacer)


Equations

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \eta} (viscosity of air as a function of T found in a table in Pa*s)
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle v_f=\frac{l}{t_f}} (average velocity of oil droplet falling in no field in m/s)
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle v_r=\frac{l}{t_r}} (average velocity of oil droplet rising in a field in m/s)
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle a=\sqrt{\left(\frac{b}{2p}\right)^2+\frac{9\eta v_f}{2g\rho}}-\frac{b}{2p}} (radius of droplet in meters)

Which all contribute to: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle q=\frac{4}{3}\pi\rho g d\frac{a^3}{V}\frac{\left(v_r+v_f\right)}{v_f}} (charge on oil droplet in Coulombs)


Droplets are listed by number. An "A" signifies 1 second of ionization, a "B" signifies 2 seconds of ionization, and "C" signifies 3 seconds of ionization.

Droplet/Charge Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \eta\left(\times10^{-5} Pa\cdot s\right)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle v_f\left(\times10^{-5} \frac{m}{s}\right)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle v_r\left(\times10^{-4} \frac{m}{s}\right)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle a\left(\times10^{-7} m\right)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \left| q\right|\left(\times10^{-19} C\right)}
1 1.83 4.179(1) 1.78(8) 5.83 5.76
1A 1.83 4.150(4) 1.25(2) 5.8 4.33
2 1.85 5.030(5) 2.19(9) 6.44 7.89
2B 1.85 5.040(1) 1.60(5) 6.44 6.16
2C 1.85 4.776(1) 2.00(5) 6.26 7.02


Where the uncertainty is in a value is in parenthesis.

From here I found the averages of the similar suspected Q values, and plotted those against the suspected multiples of Q. There would be a relationship of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle Q=n*e} , with n being the suspected multiple and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle e} being the fundamental charge. Once I plotted this and found the uncertainty in the fitting, the linear slope was:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle e=1.56(5)\times 10^{-19} C}

With a standard error of about 5.3%

This shows the suspected multiples plotted against the actual Q values.  Excel rounds data off, but the slope of the linear fit has a standard error of about 5.3%

All of these calculations can be found in the attached Excel file.

Conclusion

This error is most likely from the lack of control in the environment. The exact air pressure in the room was not known, and the EXACT viscosity of the air was not known. There were also possible temperature fluctuations that weren't accounted for. The most reasonable was the timing. Regardless of how well someone moves, there is a delay between when someone sees an action and can hit a button to record it. This delay would influence the times recorded, which make a world of difference experiment.

Yet a value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle e=1.56(5)\times 10^{-19} C} with a standard error of about 5.3% is quite reasonable, and I am confident that under the right conditions that 5% could be done away with. The value is actually quite close to the value Millikan himself got5! Unfortunately, for now a 5% error is not enough for any reasonable person to bank on. A 5% error in current density might kill a man, and no one would want that. This experiment can be done better in the future, and I am sure that as time goes on a great many aspiring physicists will narrow this error down until it exists no longer.


References

1. [1]

2. [2] and 1

3. [3]

4. 1977 or 2007

5. [4] [PASCO Scientific Model AP-8210] <reference />


Acknowledgements

I'd like to thank the following:

  • Bradley Knockel, my lab for the last 2 years and expert theorist who helped me sort out everything I needed to know from everything I already knew. Respectful and brilliant, he is not just a colleague but a good friends as well.
  • Dr. Steven Koch, thank you for the help with the experiment and comments on my report. Thank you for putting our work into context and applying meaning to it on a weekly basis. Without you everything would be a whole lot worse.
  • Kyle Martin, thank you for the great idea on data analysis.
  • UNM, for providing us with all of the equipment.

Appendices

Appendix A


Old data that was not taken with enough information at the time, but nonetheless of some interest:</br>

Droplet 1

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V} =+503V
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle T} =23°C
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle t_f} 47 49 51.3 45.5 43.9
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle t_r} 4.5 4.6 4.8 4.9 4.8


Droplet 2

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V} =-503V
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle T} =26°C
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle t_f} 59.2 60.1 69.9 62.6
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle t_r} 9.6 9.3 9.3 9.1


Droplet 3

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V} =-504V
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle T} =27°C
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle t_f} 42.3 47.2 50.8 47.1
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle t_r} 12.1 12.1 12.9 13.5


Droplet 4

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V} =+505V
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle T} =27°C
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle t_f} 57.5 63.5 63.0
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle t_r} 10.0 9.7 9.1

Appendix B

File:Millikan.xls