Question

An electron of charge 'e' moves in a circular orbit of radius r around a nucleus with a frequency \(\nu\). The magnetic moment associated with the orbital motion of the electron is

• A. \(\pi \nu e r^2\)
• B. \(\dfrac{\pi \nu r^2}{e}\)
• C. \(\dfrac{\pi \nu e}{r}\)
• D. \(\dfrac{\pi e r^2}{\nu}\)

Hint:

Current is \(e\nu\) and area is \(\pi r^2\). Multiply.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A charge going round a loop forms a tiny current loop. Its magnetic moment is \(M = I A\).

Step 2: Find the current:
The electron passes a point \(\nu\) times each second, so the current magnitude is \(I = e\nu\).

Step 3: Find the area:
The orbit is a circle, so \(A = \pi r^2\).

Step 4: Multiply:
\[ M = I A = e\nu \times \pi r^2 = \pi \nu e r^2 \]

Step 5: Check the options:
Options 2 and 4 divide by e or \(\nu\), which does not match \(I = e\nu\). Option 3 has r in the denominator and no \(r^2\). Only option 1 fits.

Final Answer:
The orbital magnetic moment is \(\pi \nu e r^2\). \[ \boxed{M = \pi \nu e r^{2}} \]