19 Measurement of the Electron Charge to Mass Ratio

Measurement of the Electron Charge to Mass Ratio

This lab is designed to align with AAOT science outcome #1: Gather, comprehend, and communicate scientific and technical information in order to explore ideas, models, and solutions and generate further questions.

Materials

  • digital device with spreadsheet program
  • digital device with internet access

ObjectiveS

  1. Explain the basic components of an e/m apparatus, including filament, accelerating gap, bulb, and Helmholtz coils.
  2. Apply voltage, charge, kinetic energy, circular motion, Lorentz Force, and electromagnet concepts to model the motion of charged particles in a uniform magnetic field.
  3. Experimentally measure the path radius of electrons for different acceleration voltages and Helmholtz currents.
  4. Analyze the experimental data to determine the charge to mass ratio of the electron (e/m) .
  5. Compare the experimental result to the accepted standard e/m value.
  6. Explain two separate ways that the charge of the electron can be measured in order to calculate the mass from the e/m ratio.

Methods

Experimental Methods

The following video demonstrates the experimental setup and the collection of all necessary data for this lab.

The e/m apparatus allows us to determine the ratio of the electron charge to electron mass. This measurement requires that electrons move through a constant magnetic field in order to follow a circular trajectory. For the case of the e/m apparatus, the magnetic field is generated using a set of Helmholtz coils. We won’t be able to measure the strength of the field during the e/m experiment, so we will need to calculate the field strength from the known current through the coils.  The strength of the magnetic field between the coils is given by:

B = \left( \frac{4}{5}\right) ^{3/2} \frac{\mu_0 NI}{R}

(See here if you are interested in a derivation of this equation).

1) Before beginning our analysis we want verify that the magnetic field is constant and that the equation above predicts the correct field. You will notice in the video that the field between the coils was measured for a particular Helmholtz current value. Record the current value, the mean magnetic field, and the standard deviation in the magnetic field.

 

2) Use the formula above to calculate the predicted field between the coils for the current shown in the video. Show your work.

 

3) Calculate a percent difference between the predicted magnetic field and that measured with the magnetic field probe. Show your work.

 

4) Does it appear that the magnetic field is reasonably predicted by the Helmholtz coil formula? Explain.

 

5) Next let’s verify that the field is relatively constant within the region that the electrons will move.  Use the mean and standard deviation values provided in the video to calculate the % variation in the magnetic field throughout the region between the coils. Show your work.

 

6) Does it appear that the magnetic field is relatively constant?

 

Modeling Methods

7) To model the circular path of the electrons, start by writing the expression for the centripetal acceleration during uniform circular motion. Use a for the acceleration, r for the path radius, and v for the speed.

8) Apply Newton’s 2nd Law to obtain an expression for the centripetal force. Use F for the force and m for the mass.

 

9) In this case the centripetal force is provided by the Lorentz Force. Equate the centripetal force you have above to the expression for the magnitude of the Lorentz force on an electron and symbolically solve for the speed of the electron. Use B for the magnetic field magnitude and e for the electron charge. Explain what happens to the sin(\Theta)

 

10) The kinetic energy (and thus speed) of the electron is determined by the accelerating voltage. Write an equation relating the kinetic energy acquired by the electron to the electron charge and voltage. Use V for voltage.

 

11) Substitute your result for the electron speed from (9) into the speed in (10) and symbolically solve the resulting equation for the charge to mass ratio (e/m).  The result should only depend on the voltage (V), magnetic field (B), and path radius (r). Show all of your work in arriving at the correct result:  \frac{e}{m} = \frac{2V}{B^2 r^2}

 

12) There were several sources of uncertainty in measuring values shown in the e/m equation above. If we take only one data point, and enter the values into the equation to calculate e/m then our uncertainty would be substantial. We can estimate that uncertainty relatively quickly because the equation for e/m only contains multiplied and divided values of V, r, and B. That means we can estimate the percent uncertainty in each of those parameters individually and then add those percentages to estimate the total percent uncertainty in the calculated e/m value. Let’s start with the acceleration voltage. The uncertainty in the voltage can be assumed to be 0.1 V because this was the limit of the display on the power supply. The voltage was very roughly 100 V throughout the experiment. Estimate the voltage percent uncertainty.

 

13) We already have an estimate of the possible percent error in the magnetic field from (3). Notice that the magnetic field is squared in the expression for e/m so we need to add this percent error twice. Record the total percent uncertainty contributed by the possible error in the magnetic field value.

 

 

14) Estimate the absolute uncertainty in measuring the radius of the electron path in units of cm. Explain your reasoning. (This is not just a guess, there should be critical thinking about the experimental methods. Keep in mind, the video measures diameter.)

 

 

15) The radius was very roughly 0.05m throughout the experiment. Calculate the percent uncertainty in the radius measurement.

 

16) The radius is also squared in the e/m expression, so double this uncertainty just as you did for the magnetic field and then add up all of the percent uncertainty contributions from V, B, r to get a total percent uncertainty in the e/m value.

 

 

 

Analysis Methods

17) The uncertainty from a single measurement is too high. We want to reduce the uncertainty in our e/m measurement, so we will take another approach. First solve your e/m equation for 2V. (Isolate 2V, so it looks like 2V = ____________).

 

18) If we were to take many data points and plot 2V on the vertical axis and B^2 r^2 on the horizontal axis and fit a linear equation to the data, then the slope of that fit line would be represent what physical quantity?

 

19) What do you expect for a y-intercept of that fit line?

 

 

20) Now we will use data provided in the video to create the plot with 2V on the vertical axis and B^2 r^2 as described above, but first we need to collect and organize the data. Create a spreadsheet to keep track of the acceleration voltage, Helmholtz current, and resulting path radius from trial shown in the video. There should be two radius columns, one for the value you read off the ruler in the video and a second that calculates the actual radius from the conversion factor provided in the video. (Don’t forget to record radius instead of diameter, which is what you see measured in the video).

21) Add a column to your spreadsheet that calculates the magnetic field strength for each trial from the recorded Helmholtz current using the formula that you verified the start of the lab.

22) Add another column that calculates B^2 r^2 for each trial.

23) Add another column that calculates 2V for each trial.

You should have a spread sheet that has the following format with data from the indicated sources:

Current (A) Voltage (V) r (tick marks) Actual r (m) B (T) B2r2 (T2m2) 2V (V)
read on power supply read on power supply read from ruler use conversion factor to calculate from ticks calculate magnetic field from current using equation at the start of lab calculate from the B and r values in previous columns calculate from the voltage
….. ….. ….. ….. ….. ….. …..
….. ….. ….. ….. ….. ….. …..

25) Plot 2V vs. B^2 r^2. Be sure to title your graph and label your axes, including units.

26) Apply a linear fit to the data and record the fit equation and R2 value here:

 

27) Your fit equation may only be showing 1 significant figure in the slope. Change the settings to provide more figures. In Excel you would do this by right clicking on the fit equation, choosing format label, selecting scientific from the drop-down options, and setting the decimals option to 2. What is your experimental result for e/m? Calculate a percent difference from the accepted standard value. Cite any source you used to find the standard charge, mass, or e/m values.

 

 

 

Conclusions

28) Does it appear that this small table-top apparatus can accurately measure the charge to mass ratio of the electron? Explain in terms of a comparison to the standard value and the single-point uncertainty estimate you performed earlier.

 

 

 

Further Questions

29) We understand how to measure the charge to mass ratio, but to know the actual values of electron mass or charge we would need to measure one of the values. Do some research and describe two different ways to independently measure the charge on the electron. Describe each experimental setup, the data collected, and the associated analysis methods required for the measurement and cite any and all sources.

 

 

 

 

 

 

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General Physics Remote Lab Manual Copyright © by Lawrence Davis is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

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