E/M, PH221 version

Hands-On Lab 8

Charge-to-Mass Ratio of the Electron

The Problem

Determine the charge-to-mass ratio of electrons and determine the uncertainty of your result.

Constraints

Work in groups of 2.
If you are in Advanced Modern Physics, your method for measuring e/m must be entirely different from the method that you will use for your lab in that course.

Cautions

This experiment uses high voltage and high current power supplies. Be careful around these! Do not plug in or unplug any wires or touch any wires unless you are sure that the electric power to the wires is turned off.
Do not operate the high voltage power supply above 4.5 kV, as this will burn out the filament that supplies the electron beam.
Do not operate the high current power supply above 10 Amps, as this will blow the fuse in the ammeter.

Experimental Design

Devise a method. (Will you use just a B-field, just an E-field, or both?) Derive the necessary formula(s). Here are design questions to consider:

What effect might the Earth’s magnetic field have on this experiment? Do you think it will be important? How should the experiment be designed to minimize the effect of Earth’s magnetic field?
How will you measure the magnetic field of the Helmholtz coils in the region of interest and determine the uncertainty of the measurement? The value of the field can be calculated using current and coil geometry. (See Ch.35.) Should you trust the ammeter reading on the current supply? How will you take measurements for coil diameter and separation, given the finite thickness of the coil? How will you sample the field directly using a Hall probe?
How will you estimate the uncertainty in the values of the accelerating potentials?
If your method involves projecting the electron beam in a circular path, how will you determine the radius of the curvature of the path? If the method involves a parabolic path, how will you collect position information? If the method involves a straight line, how will you account for the nonuniformity of the field. The latter prevents the line from being truly straight.
Making believable estimates of the uncertainties of your measurements usually requires that you demonstrate the reproducibility of your measurement technique. How will you do this?

Preliminary measurements

In order to assess the quality of your experimental design, take a quick set of preliminary measurements. For this assessment, it’s not necessary to estimate uncertainties or make multiple measurement trials. Some measurements, like radius of curvature, can be eyeballed. The idea is to make sure that you haven’t overlooked something in your design and to get a better idea of what the experimental limitations will be.

In your lab book, you should have already recorded experiment title, goal, partner, table of contents entry. Now add the following:

A Theory section, so labeled, providing a derivation of the equation that you will use to calculate e/m of the electron, given your experimental measurements. Include a half-page diagram with relevant quantities labeled.
A complete Legend of Symbols, so labeled.
A Preliminary Data section, so labeled, in which you list all preliminary measurements. Use symbols that you defined in the Legend of Symbols. Each person must obtain a different set of measurements. Therefore, make sure that you change something between sets (such as accelerating potential or current).
A Preliminary Calculation section, so labeled, in which you show a complete calculation of e/m, given your theoretical equation. Show all substitutions with units, and provide a complete units analysis.

Data

After your preliminary analysis has been checked, return to the lab to:

Record all data, completely labeled, with units and significant figures, in your lab book. Make notes and sketches about your methods of measurement. This is very important, as it provides evidence of the care that you took in your measurements.

Each person in the group must obtain a complete and unique set of measurements. For each measurement, give the estimated uncertainty range for later use in estimating error. For example, current = 0.125 ± 0.005 A

Analysis

Once you have your data, show a complete calculation of e/m. Each person must do this!
Make a quantitative estimate of the total uncertainty in your measurement. You will need to used your estimated measurement uncertainties in order to come up with the total. For this estimate, apply the following rules. (These rules are not based on rigorous statistical methods, but they are adequate for this experiment. They will tend to overestimate the uncertainty.)

When measurements are multiplied or divided, add the uncertainties of the measurements as percentages.

When you take the square root of a measurement, the percentage uncertainty is halved.

When measurements are added or subtracted, add the absolute uncertainties.

State your final result for e/m ± your total estimated uncertainty.
Calculate the percentage difference between your value of e/m and the known value.

Discussion and Conclusion

Summarize your method and results as always. Note whether the total estimated uncertainty in your measurement of e/m is comparable to the percentage difference between your measurement and the known value. Discuss qualitative sources of error which were not included in uncertainty calculation but which could have a significant contribution.