Question

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

• A. \(\frac{\mathrm{eI}}{2\pi}\)
• B. \(\frac{2\pi}{\mathrm{eI}}\)
• C. \(\frac{\mathrm{e}}{2\pi \mathrm{I}}\)
• D. \(\frac{2e\mathrm{I}}{\pi}\)

Hint:

Write the current as charge divided by the time period of the orbit, and express the period using r and v.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
An electron moving in a circular orbit constitutes a current equal to charge per unit time.
The time taken for one revolution is the period \(T\).

Step 2: Write the Period
For a circular orbit of radius \(r\) and speed \(v\):
\(T = \dfrac{2\pi r}{v}\)

Step 3: Calculation
\(I = \dfrac{e}{T} = \dfrac{e v}{2\pi r}\)
Rearranging for \(\dfrac{r}{v}\):
\(\dfrac{r}{v} = \dfrac{e}{2\pi I}\)

Step 4: Compare the options
The result \(\dfrac{e}{2\pi I}\) matches the third option.

Final Answer:
The ratio of radius to speed is the charge divided by \(2\pi\) times the current. \[ \boxed{\text{(C) }\dfrac{e}{2\pi I}} \]