Charge Is Quantized — Question 5

Question 5

Counting pump errors changes the delivered charge by whole electron units

Use $e=1.602176634\times10^{-19}$ C, exact. A pump is intended to transfer one electron from left to right in each of $10^8$ cycles at frequency $f=250$ MHz. In this run, 120 cycles transfer no electron and 20 cycles transfer two; all other cycles transfer one. Define signed transferred charge as charge carried to the right, and average conventional current as that charge divided by run time.

1. Find the actual electron count, run duration and signed total transferred charge.

2. Find the magnitude of average current and its relative error from the ideal $ef$. Explain why a very smooth average current is compatible with discrete transfers.

Question 5 - Solution

Counting pump errors changes the delivered charge by whole electron units

(a) Method and calculation

$$N_e=10^8-120+20=99\,999\,900,\quad t=N_{\mathrm{cycles}}/f=.400\,\mathrm s.$$$$Q=-N_ee\approx-1.602175032\times10^{-11}\,\mathrm C.$$

(b) Interpretation and further calculation

$$|I|=N_ee/t\approx4.005437580\times10^{-11}\,\mathrm A,\quad I_{\mathrm{ideal,magnitude}}=ef=4.005441585\times10^{-11}\,\mathrm A.$$$$\frac{|I|-ef}{ef}=\frac{-100}{10^8}=-10^{-6}.$$A time average combines many individually quantized events; it need not appear as separate steps at the measurement time scale.

Physical check

Electron transport rightward corresponds to negative conventional current in the rightward sign convention.