User:Brian P. Josey/Notebook/Junior Lab/2010/11/08

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e/m Ratio

This week, my partner, Kirstin, and I did the e/m ratio experiment. This experiment it used to determine the ratio of the charge of an electron to its mass. Combined with the Millikan oil drop experiment, two fundamental physical quantities can be determined, the mass and the charge of the electron. This quantities are significant in that they are very useful in quantum mechanics, and more importantly, the charge of the electron is the fundamental charge and the smallest amount of charge that can exist on its own. (Some other subatomic particles have fractional charge, but they cannot exist on their own.)

Historically, this experiment was first conducted by J. J. Thomson in 1897 using cathode ray tubes. This experiment, however, is a little more updated and has a different approach than the original groundbreaking experiment. In place of the cathode tubes, we have a glass tube full of a very dilute helium gas. This tube is surrounded by a Helmholtz coil that can supply a nearly uniform magnetic field throughout the whole tube. In this tube, we will release electrons from a heater plate, focus them into a nearly coherent beam, and apply a varying magnetic field. This varying magnetic field will change the trajectory of the electrons, which emit light from the collisions with the helium, so that it forms a complete circle. We then measured the dimensions of the circular path, and the voltages used to free the electrons to determine the ratio, e/m.

Set-up

The bulb containing helium surrounded by the Helmholtz coils.
The Gelman and HP power supplies.
The SOAR power supply for the heater.
The final set-up including the meters.

Equipment:

  • SOAR DC Power Supply Model PS-3630 - used to power the heater
  • e/m Experimental Apparatus -composed of a tube of dilute helium gas, an electron heater, focusing magnets, and a Helmholtz coil used to provide a constant magnetic field
  • Gelman Instrument Company: Deluxe Regulated Power Supply -powers the electrodes
  • Hewlett Packard 6384A DC Power Supply -powers Helmholtz coils

Connections:

  • SOAR output minus to lower plug on heater
  • SOAR output plus to upper plug on heater side
  • Gelman plugs on back into electrode plug on e/m device
    • Red to positive
    • Black to negative
  • Volt meter plugs onto multimeter
  • Helmholtz on e/m to HP power supply

Multimeter

We also used three different meters, one ammeter and two voltmeters, to monitor the currents and potentials that we fed into the experimental set up. They were:

  • ammeter -connected in series with the Helmholtz coil to measure the current going into the coils,
  • voltmeter -connected in parallel with to ensure the voltage on the heater does not exceed 6.3 V,
  • voltmeter -connected in parallel to the electrode to measure the voltage at which the electrons are fired at.

Procedure

After we set up all of the equipment, we began the experiment. First we turned on the heater for the electron gun and let it warm up for two minutes, after which, we raised its initial voltage to 6.3 V as measured from the voltmeter. This voltage was kept constant for the duration of the experiment. We then raised the acceleration voltage of the electrons and focused the beam so that it appeared as a single line firing to the right.

After the generating the electron beam, we set out to measure the current passing through the Helmholtz coils and the voltage of the electrons. By adjusting the current going into the Helmholtz coil, we were able to change the path of the electrons by bending it. A higher current resulted in a higher magnetic field acting on the electrons, and a tighter coil. Adjusting the accelerating potential on the electrons would also change their pattern, a higher voltage resulting in a wider circle. We then made twenty measurements. For the first ten, we kept the accelerating voltage constant near 200 V, and adjusted the current of the Helmholtz coil. The last ten measurements were conducted by maintaining a constant current, at 1.125 A, and adjusting the voltage.

To measure the radius of the beam, we aligned the beam with its reflection on the ruler behind the bulb, and recorded the marking that they crossed. While we measured the radius on both the left and right sides, we will only used the radius on the right. The reason for this is that the one on the left is more likely to change depending on the interactions that it goes through with the helium in the tube. It was only measured to serve as a check for the experiment, weeding out any discrepancies.

Data and Discussion

Here is a summary of the data that I collected from this experiment:

{{#widget:Google Spreadsheet

key=0AjJAt7upwcA4dFpiSk1kcXc2SFZTS3hSQkNXekVYa0E width=600 height=325

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As explained in the procedure, we measured the Helmholtz coil current, accelerating potential and radius twenty times. For the first ten, we maintained the accelerating voltage at a constant 200 V, and adjusted the current from approximately 1.05 A to 1.5 A in steps of 0.05 A. As the current increased, the ring formed by the moving electrons tightened. For the next ten, we kept the current at 1.254 A, as close as we could get it to the middle value of our first set, and then adjusted the accelerating potential from 150 V to 250 V in steps of 10 V. As the data illustrates the raising potential created wider rings as a result of the faster moving electrons. We were then able to analyze the data. This analysis is summarized in this table:

{{#widget:Google Spreadsheet

key=0AjJAt7upwcA4dGY0ZDloU3A2ZVBMUjE4UUR4QzBLTEE width=600 height=325

}}

To determine the magnetic field, we used the following formula:

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 = \frac {\mu R^2 N I}{(R^2+x^2)^{3/2}} }

where:

  • B - the magnetic field, in Telsa,
  • μ - permeability of free space, 4π * 10-7 weber/ amp-meter2,
  • R - radius of the rings, 0.15 m,
  • N - number of times the wires go around the Helmholtz coil, 130 times,
  • I - current measured in A,
  • x - the axis where the measurement is taken, in this case x=R/2,

Together, these terms give a simple factor to calculate the magnetic field from the current, namely:

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 = (7.8*10^{-4} \frac {weber}{amp-meter^2} )* I }

This value is then given in units tesla, which are equivalent to weber/meter2.

In order to find the e/m ratio from the data, we first need to find the relationship of the ratio of e/m to our data. Using the follow formulas we can determine the e/m ratio:

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 eV= \frac {mv^2}{2} }

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 F_B = qvB \, }

where:

  • e and q - the charges of the electron,
  • V - the acceleration potential,
  • v - the velocity the charges travel at,
  • FB - the force due to the magnetic field, and
  • B - the external magnetic field

Together, this equations give the e/m ratio:

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 {e} {m} = \frac {2V} {r^2 B^2} }

Now, to actually find the ratio, we could do it in two ways. The first is to plot the inverse of the radius versus the current at a constant accelerator voltage. This gives a simple linear relationship:

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 {1} {r} = \sqrt {\frac {(7.8*10^{-4})^2}{2V} \frac {e}{m}} I }

Using linear regression and the method of least squares, we can plot the data and find the best fit line. This is graphed below:

Inverse radius v I.png

This method gives us a slope of s=16.8179. Using the above equation, the e/m ratio can be found by plugging it into the follow equation:

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 {e}{m} = \frac {2s^2 V}{(7.8*10^{-4})^2} }

This ultimately gives a ratio of 1.10571*1010 C/kg.

The second method to find the ratio is to graph the radius squared versus the acceleration potential at a constant current. This linear relationship is given by:

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 r^2 = \frac {2V} {(7.8*10^{-4})^2 I} * \frac {m}{e} }

Plotting this data gives the following graph and best fit line:

Radius squared vs v.png

Here the best fit line has a slope of s=1.111*10-5. Using this slope, the e/m ratio can be found using the following relationship:

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 {e} {m} = \frac {2} {s(7.8*10^{-4} I)} }

This then gives a ratio of e/m =1.84*108 C/kg.

Conclusion

Between our two measurements of the e/m ratio, we were able to determine an average ratio of (5.6 ± 0.4)*109 C/kg.SJK 03:15, 21 December 2010 (EST)
03:15, 21 December 2010 (EST)
Does it really make sense to average two values that differ by two orders of magnitude? Hopefully you don't think that anymore :) My guess is that you have a calculation error for the second measurement, but I do not have time to find it now. The experiment is flawed, so it's not surprising that your first measurement is significantly off.
Unfortunately, this value is very low compared to the accepted value of the e/m ratio, which is 1.76 * 1011 C/kg. There are a couple of possible sources of error in this experiment, which we were warned would occur. While we can be confident in our measurements of the acceleration potential and Helmholtz current, there is some difficulty in measuring the radius of the curve in the experiment. The primary issue in measuring the curvature is that the ruler behind the bulb on the set up is subject to significant parallax error. This could have resulted in us mistakenly taking the wrong radii for our data points. It is also possible that there is an error in our calculations somewhere along the line, however, I have personally checked the calculations and was unable to find any errors. So the ultimate source of the discrepency between our data and the accepted result is a mystery.

Acknowledgments and References

As always, I want to thank my lab partner, Kistin. At the same time Dr. Koch and Katie the TA also helped a significant deal. And because I always rely on the other students for help and guidance, I want to thank Randy and Emran from this semester for helping Kirstin and I fix the set-up, and Alex Benedict, Alexandra Andrego and Anastasia Ierides for their superb notebooks that I followed. Here are the links to Alex B.'s and Alex A.'s notebooks.