Charge Is Quantized — Question 4

Question 4

Observed charge steps give an upper candidate for the basic unit

A device shows exact charge levels $0,6u,10u,16u$, where $u>0$ is a known calibration unit. Suppose each level is an integer multiple of an unknown positive elementary step $q_0$. No claim is made that the device has visited every allowed level.

1. Find the largest possible $q_0$ consistent with all four levels and show explicitly the corresponding integer labels.

2. Explain why the data do not prove this largest candidate is the true smallest step. Determine how a later level $7u$ changes the largest candidate.

Question 4 - Solution

Observed charge steps give an upper candidate for the basic unit

(a) Method and calculation

$$q_0^{\max}=\gcd(6,10,16)u=2u.$$$$(0,6u,10u,16u)=(0,3,5,8)(2u).$$

(b) Interpretation and further calculation

$$q_0=2u/m,\qquad m=1,2,3,\ldots\quad\text{also fits the original data}.$$$$\gcd(6,10,16,7)u=u.$$A missing smaller step can reflect limited sampling or restricted transfer processes. The added odd level excludes $2u$ but still permits subdivisions of $u$.

Physical check

This exact greatest-common-divisor inference would need an explicit tolerance model for noisy measurements.