Relativistic Electron Mass Experiment

John Klumpp

Testing Velocity Dependence of the Electron Mass

• Special relativity: A body’s mass increases with velocity according to the equation:

m = m0*[1-(v/c)2]-1/2

• Test by curving electron beam with B field, straightening with E field.

Setup

-Electron beam from 90Sr source is curved by a B field

-A series of narrow apertures only allow electrons curving with R=15cm to pass by

-Beam then travels through a capacitor. Electrons whose path is not straightened by the E field collide with capacitor plates.

-Scintillator counts electrons which make it through

-Vary E field to find out which one best balances B

What Does This Tell Us?

• We know E, B, and R• We know the forces from the B field and E

field match• Knowing this, we find:

v = E/Bm= = eRB2/E

• By determining which E field matches a series of B fields, we can find out how mass varies with velocity

Calibration• To prevent arcing in the capacitor, we put the

whole apparatus in a vacuum chamber• This means we can’t measure B directly while

doing the experiment• Solution: Use Hall probe to find out in advance

what currents lead to what fields:

I (Amps) Average B (Gauss) Uncertainty

0.744 80.04 0.16

0.943 99.98 0.22

1.140 120.36 0.20

1.334 139.93 0.24

1.533 159.73 1.54

1.731 180.36 0.38

1.822 189.69 0.41

Calibration Results

Magnetic field vs current with linear polynomial overlay.

-Calibration Equation: B = 101.7A + 4.3

-χ2 is 1.79 with 5 degrees of freedom.

The Experiment

• Remove Hall probe, insert experimental apparatus• Take measurements at 80, 100, 120, 140, 160, 180, and

190 Gauss.• For each B field, Take 13 measurements for 100

seconds each• Measurements are in 50V intervals about theoretical

value• Plot each set of measurements, fit a Gaussian to

determine peak position• This peak is the E field which should perfectly balance

the B field

Finding the Peak

• E field vs. counts for B = 80.6T with Gaussian overlay• Peak is 2.78kVm-1, in agreement with theory

• Using this method, we found the electric fields which corresponded to each B field.

• From there we calculated mass and velocity

I (Amps) B(Tessla) Peak (KV) m (kg) dm beta dbeta

0.75 0.0081 2.784 1.14E-30 6.531E-33 0.567 0.017

0.95 0.0101 3.99 1.24E-30 2.3E-32 0.650 0.019

1.14 0.0120 5.17 1.36E-30 8.129E-33 0.708 0.015

1.35 0.0141 6.579 1.48E-30 6.292E-33 0.765 0.013

1.53 0.0160 7.897 1.59E-30 1.105E-32 0.809 0.013

1.73 0.0180 9.117 1.73E-30 3.988E-33 0.832 0.011

1.83 0.0190 9.785 1.8E-30 5.895E-33 0.844 0.011

-Clearly mass varies with velocity, but how?

Mass vs. Velocity• Relativity predicts mass will vary with velocity as such:

m = mo(1- β2)-1/2

• We plotted m vs β and m vs (1- β2)-1/2 to find out if our results match this shape, and to find the rest mass.

-As expected, these fit lines match the data quite well. The m vs β fit gives m0 = 9.572*10-31. The m vs (1- β2)-1/2 fit gives m0 = 9.48*10-31. The χ2 on the two graphs are 32.6 and 16.5 respectively. Each has 5 degrees of freedom.

Charge to Mass Ratio• The charge-to-mass ratio of the electron can be

determined by the equatione/mo = e/[m(1- β2)1/2] = U/[RB(D2B2 – U2/e2)1/2]

• Results:

B(Tessla) U (kV) e/mo (C/kg) de/mo

0.00806 2784 1.71E+11 1.24E+10

0.01008 3990 1.7E+11 1.11E+10

0.01199 5170 1.67E+11 9.99E+09

0.01413 6579 1.68E+11 9.74E+09

0.01604 7897 1.71E+11 1.03E+10

0.01800 9117 1.66E+11 9.58E+09

0.01903 9785 1.66E+11 9.64E+09

-Conclusion: e/m0 = (1.68 +/- .04)*1011 C/kg

Sources of Uncertainty• Tiny uncertainty on E, big uncertainty on B:

-σE/E ≈ .001-σB/B ≈ .025

• B varies within the magnet• Some, but not much, statistical uncertainty from calibration equation• Does the calibration remain stable throughout the experiment?• e/m0 results consistently below accepted value – suggests

systematic uncertainty• On plots to determine E, most points had uncertainty of 5% or

higher (√N statistics) -Could be reduced by taking longer measurements-Uncertainty on location of peak only 0.5%, so not worth it.

• Different results on mo for different plots. Reason?-The uncertainties on the x-values of these fits are different-MatLab does not consider uncertainty on x values when making the plot

Conclusion

• Mass varies with velocity in accordance with special relativity

• We have two values for m0. Choose the one from the plot with the better fit: m0 = (9.48 +/- 0.506)*10-31kg

• e/m0 = (1.68 +/- .04)*1011 C/kg• Most significant source of uncertainty: B• Uncertainty can be minimized by

-Taking extreme care in calibrating the magnet-Ensuring calibration conditions hold throughout experiment