Crossed Fields: Discovery of the Electron — Question 7

Question 7

Correlated field calibration errors can cancel in a charge-to-mass experiment

An experiment infers $\eta=e/m=E^2/(2UB^2)$ from independently specified E, U and B. Small fractional errors are epsilonE, epsilonU and epsilonB.

1. Derive the first-order fractional error in eta and a worst-case bound for independent hard error limits of .5 percent, 1.0 percent and .5 percent respectively.

2. Show what happens if E and B share exactly the same multiplicative calibration factor while U is exact.

Question 7 - Solution

Correlated field calibration errors can cancel in a charge-to-mass experiment

(a) Method and calculation

$$\delta\ln\eta=2\delta\ln E-\delta\ln U-2\delta\ln B.$$$$\delta\eta/\eta\simeq2\epsilon_E-\epsilon_U-2\epsilon_B,\quad|\delta\eta|/\eta\lesssim.0300.$$

(b) Interpretation and further calculation

$$E\prime=kE,\quad B\prime=kB,\quad U\prime=U.$$$$\eta\prime=\frac{k^2E^2}{2Uk^2B^2}=\eta.$$

Physical check

A worst-case sum and a correlated-error cancellation answer different experimental questions.