Charge Is Conserved — Question 10

Question 10

Hard measurement bounds can reveal a missing charge reservoir

A device is claimed to be a completely isolated, initially neutral system consisting only of components A, B and C. Final charge bounds are $Q_A\in[3.8,4.2]$ nC, $Q_B\in[2.7,3.3]$ nC and $Q_C\in[-1.1,-.9]$ nC. Treat these as simultaneous hard bounds, with all endpoint combinations permitted.

1. Find the allowed final total charge interval and test whether the stated isolation and completeness assumptions are consistent with conservation.

2. If an uncounted reservoir D is added to the initially neutral total system, find its required final charge interval. Explain what this result establishes and what it does not identify.

Question 10 - Solution

Hard measurement bounds can reveal a missing charge reservoir

(a) Method and calculation

$$Q_{\mathrm{ABC,min}}=3.8+2.7-1.1=5.4\,\mathrm{nC}.$$$$Q_{\mathrm{ABC,max}}=4.2+3.3-.9=6.6\,\mathrm{nC},\quad0\notin[5.4,6.6].$$

(b) Interpretation and further calculation

$$Q_D=-Q_{\mathrm{ABC}}\Rightarrow Q_D\in[-6.6,-5.4]\,\mathrm{nC}.$$$$Q_{\mathrm{ABC}}+Q_D=0.$$The observations conflict with the complete isolated three-component model. A missing reservoir of the required negative charge would repair the balance, but the arithmetic does not reveal its location or exclude faulty bounds.

Physical check

A failure of a measured subsystem balance is evidence to examine boundaries and measurements before claiming failure of charge conservation.