Charge Is Quantized — Question 1

Question 1

An exact elementary charge converts a net charge into an electron imbalance

Use $e=1.602176634\times10^{-19}$ C, exact. A small object has charge $Q=-4.806529902\times10^{-12}$ C, specified as an exact model value. Only electrons have been added to an initially neutral object; the positive constituents stay fixed.

1. Find the number of excess electrons and the charge change needed to return the object to neutrality.

2. If instead the displayed magnitude had only three significant figures, explain why the quotient by $e$ would be an estimate rather than an exact electron count. Estimate the number of electron counts within a rounding half-width of $.005\times10^{-12}$ C.

Question 1 - Solution

An exact elementary charge converts a net charge into an electron imbalance

(a) Method and calculation

$$Q=-Ne\Rightarrow N=\frac{4.806529902\times10^{-12}}{1.602176634\times10^{-19}}=3.0000000\times10^7.$$$$\Delta Q=+Ne=+4.806529902\times10^{-12}\,\mathrm C.$$

(b) Interpretation and further calculation

$$\delta N=\delta Q/e=\frac{.005\times10^{-12}}{1.602176634\times10^{-19}}\approx3.12075\times10^4.$$$$N_{\mathrm{allowed\ count}}\approx2\delta N\approx6.24\times10^4.$$Many adjacent integer electron counts fit a rounded macroscopic charge even though every individual object has a definite integer imbalance in this model.

Physical check

The exactness of $e$ does not remove uncertainty or rounding in a measured charge.