Let the current 'I' be associated with an electron of charge 'e' moving in a circular orbit of radius ' $r$ ' with speed ' $v$ ' around the positively charged nucleus. The ratio $\frac{r}{v}$ is

The current $I$ for an electron of charge $e$ in a circular orbit equals the charge passing a point per unit time. Since charge $e$ passes a point each revolution period $T$, the current is given by $I = \frac{e}{T}$.

The time period for one orbit is $T = \frac{2\pi r}{v}$, where $r$ is the radius and $v$ is the orbital speed.

Combining these yields $I = \frac{e}{\frac{2\pi r}{v}} = \frac{ev}{2\pi r}$.

Rearranging to solve for the ratio $\frac{r}{v}$ gives $\frac{r}{v} = \frac{e}{2\pi I}$.

This expression corresponds to option C.