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 = m 0 *[1-(v/c) 2 ] -1/2 Test by curving electron beam with B field, straightening with E field.

Setup -Electron beam from 90 Sr 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/B m= = eRB 2 /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

Calibration Results Magnetic field vs current with linear polynomial overlay. -Calibration Equation: B = 101.7A χ 2 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)dmbetadbeta E E E-302.3E E E E E E E E E E E Clearly mass varies with velocity, but how?

Mass vs. Velocity Relativity predicts mass will vary with velocity as such: m = m o (1- β 2 ) -1/2 We plotted m vs β and m vs (1- β 2 ) -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 m 0 = 9.572* The m vs (1- β 2 ) -1/2 fit gives m 0 = 9.48* The χ 2 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 equation e/m o = e/[m(1- β 2 ) 1/2 ] = U/[RB(D 2 B 2 – U 2 /e 2 ) 1/2 ] Results: B(Tessla)U (kV)e/mo (C/kg)de/mo E E E E E E E E E E E E E E+09 -Conclusion: e/m 0 = (1.68 +/-.04)*10 11 C/kg

Sources of Uncertainty Tiny uncertainty on E, big uncertainty on B: -σ E /E ≈.001 -σ B /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/m 0 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 m o 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 m 0. Choose the one from the plot with the better fit: m 0 = (9.48 +/ )* kg e/m 0 = (1.68 +/-.04)*10 11 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