Electric Charge — Question 8

Question 8

A charge measurement can establish magnitude without establishing sign

Two independent charge estimates for the same sample give hard allowed intervals $[-.20,+.40]$ nC and $[+.10,+.50]$ nC. The intervals are bounds, not standard deviations. Assume both bounds are valid.

1. Find the joint allowed interval and determine whether the sample sign is established.

2. A calibration correction $b$ is subsequently discovered, with $-.15\le b\le+.05$ nC; the true charge is the intersected estimate plus $b$. Find the full corrected interval and reassess the sign.

Question 8 - Solution

A charge measurement can establish magnitude without establishing sign

(a) Method and calculation

$$I=[-.20,.40]\cap[.10,.50]=[.10,.40]\,\mathrm{nC}.$$$$\min I=.10>0\Rightarrow Q>0\quad\text{before the correction}.$$

(b) Interpretation and further calculation

$$I_{\mathrm{true}}=[.10-.15,.40+.05]=[-.05,.45]\,\mathrm{nC}.$$$$Q<0,\ Q=0,\ Q>0\quad\text{are all compatible with the corrected bounds}.$$A common unknown bias cannot be reduced by intersecting repeated measurements as if it were independent random noise.

Physical check

Hard interval endpoints combine by extremization, without a root-sum-square rule.